%------------------------------------------------------------------------------
% File     : ITP234_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_InsertCorrectness 00806_052000
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0067_VEBT_InsertCorrectness_00806_052000 [Des22]

% Status   : Theorem
% Rating   : 0.22 v8.2.0, 0.10 v8.1.0
% Syntax   : Number of formulae    : 11301 (2982 unt;2390 typ;   0 def)
%            Number of atoms       : 23469 (8187 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 17102 (2544   ~; 455   |;1677   &)
%                                         (2277 <=>;10149  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   17 (   2 avg)
%            Number of types       :  463 ( 462 usr)
%            Number of type conns  : 2284 (1570   >; 714   *;   0   +;   0  <<)
%            Number of predicates  :   74 (  71 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1857 (1857 usr; 358 con; 0-4 aty)
%            Number of variables   : 22874 (22250   !; 624   ?;22874   :)
% SPC      : TF0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 21:34:21.941
%------------------------------------------------------------------------------
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tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J,type,
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tff(ty_n_t__fun_It__Extended____Nat__Oenat_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Extended____Nat__Oenat_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_It__Extended____Nat__Oenat_J_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Real__Oreal_J_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__List__Olist_It__Nat__Onat_J_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__List__Olist_It__Int__Oint_J_Mt__HOL__Obool_J,type,
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tff(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__List__Olist_It__Int__Oint_J_Mt__Nat__Onat_J,type,
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tff(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
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tff(ty_n_t__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Extended____Nat__Oenat_J,type,
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tff(ty_n_t__fun_It__Extended____Nat__Oenat_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Extended____Nat__Oenat_Mt__HOL__Obool_J,type,
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tff(ty_n_t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Extended____Nat__Oenat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Extended____Nat__Oenat_J,type,
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tff(ty_n_t__fun_It__Extended____Nat__Oenat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Extended____Nat__Oenat_Mt__Int__Oint_J,type,
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tff(ty_n_t__Option__Ooption_It__Extended____Nat__Oenat_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__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Int__Oint_J,type,
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tff(ty_n_t__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__List__Olist_It__Extended____Nat__Oenat_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__Set__Oset_It__Extended____Nat__Oenat_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__HOL__Obool_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__Nat__Onat_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__HOL__Obool_Mt__HOL__Obool_J,type,
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tff(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__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__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__Option__Ooption_It__Real__Oreal_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__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__Option__Ooption_It__Int__Oint_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__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__VEBT____Definitions__OVEBT,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__Extended____Nat__Oenat,type,
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tff(ty_n_t__Complex__Ocomplex,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__HOL__Obool,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 (1928)
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tff(sy_c_ATP_058Lamm__aab____,type,
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tff(sy_c_ATP_058Lamm__aad____,type,
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tff(sy_c_ATP_058Lamm__aae____,type,
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tff(sy_c_ATP_058Lamm__aaf____,type,
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tff(sy_c_ATP_058Lamm__aag____,type,
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tff(sy_c_ATP_058Lamm__aai____,type,
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tff(sy_c_ATP_058Lamm__aak____,type,
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tff(sy_c_ATP_058Lamm__aam____,type,
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tff(sy_c_ATP_058Lamm__aan____,type,
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tff(sy_c_ATP_058Lamm__aao____,type,
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tff(sy_c_ATP_058Lamm__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,
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tff(sy_c_ATP_058Lamm__aat____,type,
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tff(sy_c_ATP_058Lamm__aav____,type,
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tff(sy_c_ATP_058Lamm__aaw____,type,
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tff(sy_c_ATP_058Lamm__aax____,type,
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tff(sy_c_ATP_058Lamm__aay____,type,
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tff(sy_c_ATP_058Lamm__aaz____,type,
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tff(sy_c_ATP_058Lamm__aba____,type,
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tff(sy_c_ATP_058Lamm__abb____,type,
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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,
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tff(sy_c_ATP_058Lamm__abf____,type,
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tff(sy_c_ATP_058Lamm__abg____,type,
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tff(sy_c_ATP_058Lamm__abh____,type,
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tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: nat > fun_nat_set_nat ).

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

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: fun_Ex7682446853002821117t_bool ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: fun_Ex7682446853002821117t_bool ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: fun_na819858167508000762st_nat ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: nat > fun_na3408937105440547634t_bool ).

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

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: fun_na8797796813480665199d_enat ).

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

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

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

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

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

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

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_Ex8563670519792538868t_bool > fun_fu739766798173130341t_bool ).

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

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

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

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

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

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_Ex8563670519792538868t_bool > fun_fu739766798173130341t_bool ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: set_complex > fun_set_complex_bool ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: set_Extended_enat > fun_se2015444840911768724t_bool ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: set_int > fun_set_int_bool ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_real_bool ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_int_bool ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: set_real > fun_fu4470565429337120259l_bool ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: set_real > fun_fu6440467031773438595l_bool ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_real > fun_fu66453571216240387l_bool ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: set_real > fun_fu8829865716563604795l_bool ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_nat > fun_fu4970875099359191043t_bool ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: set_nat > fun_fu2548079958729413763t_bool ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: set_nat > fun_fu7165551060251750659t_bool ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: set_nat > fun_fu674922162107492667t_bool ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_complex > fun_fu1598310740840436355x_bool ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: set_complex > fun_fu4101317034166783747x_bool ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: set_Extended_enat > fun_fu739766798173130341t_bool ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: set_real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: set_list_nat > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: set_set_nat > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: set_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: set_int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_int_int ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_real_real ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_Ex5852502975177633731d_enat ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_int_int ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_real_real ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_Ex5852502975177633731d_enat ).

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

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

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

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: set_Extended_enat > fun_na4695037785009495611t_bool ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

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

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

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: set_Extended_enat > fun_na4695037785009495611t_bool ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

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

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

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: nat > fun_co1378794544173401837x_bool ).

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

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_real > fun_fu1424522630487402084l_bool ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_real > fun_fu5569630428773330436l_bool ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_nat > fun_fu8422113094057456412t_bool ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: set_nat > fun_fu9131882981303784088t_bool ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: set_complex > fun_fu5450758583235971936x_bool ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_complex > fun_fu6981909852427741054x_bool ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_int > fun_fu5706916984221235812t_bool ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: set_int > fun_fu6323702069054234244t_bool ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: set_Extended_enat > fun_fu4185298608880980944t_bool ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: set_Extended_enat > fun_fu2303348052976528778t_bool ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: set_real > fun_fu5258761731371428616l_bool ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: set_real > fun_fu946007832778367820l_bool ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: set_nat > fun_fu5590568769550944704t_bool ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: set_nat > fun_fu4664640653530332640t_bool ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_complex > fun_fu5688335724886546948x_bool ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: set_complex > fun_fu5639431007873792710x_bool ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: set_int > fun_fu2875372659714724104t_bool ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: set_int > fun_fu1856459741280782796t_bool ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: set_Extended_enat > fun_fu3901247534510871084t_bool ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: set_Extended_enat > fun_fu2784792389871991434t_bool ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: set_real > fun_fu1424522630487402084l_bool ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: set_real > fun_fu5569630428773330436l_bool ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: set_nat > fun_fu8422113094057456412t_bool ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_nat > fun_fu9131882981303784088t_bool ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_complex > fun_fu5450758583235971936x_bool ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_complex > fun_fu6981909852427741054x_bool ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_int > fun_fu5706916984221235812t_bool ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_int > fun_fu6323702069054234244t_bool ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: set_Extended_enat > fun_fu4185298608880980944t_bool ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: set_Extended_enat > fun_fu2303348052976528778t_bool ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_complex > fun_fu5995210064384465734x_bool ).

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

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

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_Extended_enat > fun_fu3738011059562840620t_bool ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_Extended_enat > fun_fu5527893311006036234t_bool ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: int > fun_int_fun_int_bool ).

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

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: num > fun_na4138050610385705875at_nat ).

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

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_Ex8598408993869023341t_bool ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: set_Extended_enat > fun_se6358542376187484365t_bool ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: set_real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: set_list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: set_set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: set_int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: set_nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_real_bool > fun_real_bool ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_list_nat_bool > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_set_nat_bool > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_int_bool > fun_int_bool ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: set_Extended_enat > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: set_list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: set_set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: set_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: extended_enat > fun_se6358542376187484365t_bool ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: list_nat > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: set_nat > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_list_nat_bool > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_set_nat_bool > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_Ex8563670519792538868t_bool > fun_Ex8598408993869023341t_bool ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_list_nat_bool > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_set_nat_bool > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_Ex8563670519792538868t_bool > fun_Ex8598408993869023341t_bool ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_nat_real ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: real > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: complex > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: extended_enat > fun_fu5112335056569181187at_nat ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: extended_enat > fun_fu7336479014630192387t_real ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: real > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: real > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: complex > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: extended_enat > fun_fu5112335056569181187at_nat ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: extended_enat > fun_fu7336479014630192387t_real ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: real > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: nat > fun_fu8464300013002972271at_int ).

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

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

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

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

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: set_int > fun_fu3139726878529933315t_bool ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: set_int > fun_fu4310402290150085684al_int ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_real_fun_int_int > fun_int_fun_real_int ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: set_real > fun_fu5129330665597888180nt_int ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: set_int > fun_fu3078129256884107011t_bool ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: set_int > fun_fu2946920718754786320at_int ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: set_nat > fun_fu6165624312919156332nt_int ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: set_int > fun_fu4844541694308389507t_bool ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: set_int > fun_fu2836861327644265266ex_int ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_co333780937270390912nt_int > fun_in3657390099950231424ex_int ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: set_complex > fun_fu8510821397121033091x_bool ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: set_complex > fun_fu6400036368084001456nt_int ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: set_int > fun_fu7830736900494809023t_bool ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: set_int > fun_fu5983729185816342290at_int ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_Ex6820543082444009806nt_int > fun_in5181886200629118930at_int ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: set_Extended_enat > fun_fu8444152097983751227t_bool ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: set_Extended_enat > fun_fu6293522500203156436nt_int ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: set_nat > fun_fu9073671928554219267t_bool ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: set_nat > fun_fu5088574307552523139t_bool ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: set_nat > fun_fu4637029388598527722ex_nat ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: set_complex > fun_fu1817986637694864396at_nat ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: set_nat > fun_fu2583778531842586371t_bool ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: set_nat > fun_fu8221675813083619692nt_nat ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: set_int > fun_fu7719728765329718531t_bool ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: set_int > fun_fu5002972218919249680at_nat ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: set_nat > fun_fu8451253408283314807t_bool ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: set_nat > fun_fu8372565977779661586at_nat ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: fun_Ex7123704652682714774at_nat > fun_na40478436225343698at_nat ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: set_Extended_enat > fun_fu2429753552268211515t_bool ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: set_Extended_enat > fun_fu2544567023866575296at_nat ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: set_complex > fun_fu4189668387880940419x_bool ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: set_complex > fun_fu3241646296630235056omplex ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: set_real > fun_fu7272995576155203506omplex ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: set_complex > fun_fu8281188428872102787x_bool ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: set_complex > fun_fu5276431085538847372omplex ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: set_nat > fun_fu5286008751397070186omplex ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_real_nat > fun_fu4405974721632067664al_nat ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: set_complex > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_complex_nat > fun_fu2104585616785278288ex_nat ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_int_nat > fun_fu9051645886418017232nt_nat ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_Ex7660298398204468809at_nat > fun_fu5513983380133311534at_nat ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_Ex1451620480498224165t_real > fun_fu4696490002264930058t_real ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_real_int > fun_fu1915891576700340908al_int ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_nat_int > fun_fu6629554324982842668at_int ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_real_nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_complex_nat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_Ex7660298398204468809at_nat > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_complex_real > fun_complex_bool ).

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

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_Ex1451620480498224165t_real > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_real_int > fun_real_bool ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_complex_int > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: complex > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: complex > fun_fu4692242605864086655ex_int ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: complex > fun_fu5040975675184911655d_enat ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: extended_enat > fun_fu8366861321737653343at_nat ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: extended_enat > fun_fu2469135700067826619at_int ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: extended_enat > fun_fu3686806697881729723t_real ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: extended_enat > fun_fu2295278630349710275d_enat ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: real > fun_fu3768230810189712741al_nat ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: real > fun_fu5645595436833112697al_int ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: nat > fun_nat_fun_nat_nat ).

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_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_Ex7660298398204468809at_nat > fun_fu5112335056569181187at_nat ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_int_nat > fun_fu8905809029841901679nt_nat ).

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

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

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

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_nat_real > fun_na7718645245721260691t_real ).

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

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_na8797796813480665199d_enat > fun_na8797796813480665199d_enat ).

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

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

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

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

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

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

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_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_na8797796813480665199d_enat > fun_na8797796813480665199d_enat ).

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: extended_enat > fun_Ex2441982661509597004d_enat ).

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

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_na7542918316964655096d_enat ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_Ex5852502975177633731d_enat ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_real_real ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_int_int ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_Ex8598408993869023341t_bool ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_nat_real > fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_nat_real ).

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

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

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: extended_enat > fun_na1154345429118027792d_enat ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: int > fun_nat_fun_int_int ).

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

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

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

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: real > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: complex > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: extended_enat > fun_fu5112335056569181187at_nat ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: real > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: complex > fun_fu4213752561377164075ex_int ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: extended_enat > fun_fu6685514224203069699at_int ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: real > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: complex > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: extended_enat > fun_fu5112335056569181187at_nat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: real > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: complex > fun_fu4213752561377164075ex_int ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: extended_enat > fun_fu6685514224203069699at_int ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: set_nat > fun_fu6487107696295728876al_int ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: fun_real_fun_nat_int > fun_nat_fun_real_int ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: set_real > fun_fu1164691120062057232at_int ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: set_nat > fun_fu1666499901826068458ex_int ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_co7370231399432144932at_int > fun_na5967208032043182372ex_int ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: set_complex > fun_fu3438847080485518604at_int ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: set_nat > fun_fu6906207628721142290at_int ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_Ex4633621507750988018at_int > fun_na6773767328148392750at_int ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: set_Extended_enat > fun_fu3445227072600493688at_int ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: set_int > fun_fu4310402290150085684al_int ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: set_real > fun_fu5129330665597888180nt_int ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: set_int > fun_fu2836861327644265266ex_int ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: set_complex > fun_fu6400036368084001456nt_int ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: set_int > fun_fu5983729185816342290at_int ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: set_Extended_enat > fun_fu6293522500203156436nt_int ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: set_nat > fun_fu4637029388598527722ex_nat ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: set_complex > fun_fu1817986637694864396at_nat ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: set_nat > fun_fu8221675813083619692nt_nat ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: set_int > fun_fu5002972218919249680at_nat ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: set_nat > fun_fu8372565977779661586at_nat ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: set_Extended_enat > fun_fu2544567023866575296at_nat ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_na8797796813480665199d_enat > fun_na1154345429118027792d_enat ).

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

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_real_nat > fun_fu4405974721632067664al_nat ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_complex_nat > fun_fu2104585616785278288ex_nat ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_int_nat > fun_fu9051645886418017232nt_nat ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_Ex7660298398204468809at_nat > fun_fu5513983380133311534at_nat ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_real_int > fun_fu1915891576700340908al_int ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_complex_int > fun_fu8837874508708327340ex_int ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_Ex3482447378695272101at_int > fun_fu3023900235201584778at_int ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_nat_complex > fun_fu7905158015251895726omplex ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_complex_complex > fun_fu248857536038387758omplex ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_real_nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_complex_nat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_Ex7660298398204468809at_nat > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_real_int > fun_real_bool ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_complex_int > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_Ex3482447378695272101at_int > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: fun_real_complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_complex_complex > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_int_complex > fun_int_bool ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: int > fun_fu973771059875708133nt_nat ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: real > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: extended_enat > fun_na8797796813480665199d_enat ).

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

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: extended_enat > fun_na7542918316964655096d_enat ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: nat > fun_nat_fun_nat_nat ).

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_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: fun_na8797796813480665199d_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_na8797796813480665199d_enat > fun_na1154345429118027792d_enat ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_nat_real ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_nat_int ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: fun_real_real > fun_real_real ).

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

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: fun_complex_real > fun_complex_real ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: fun_int_real > fun_int_real ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_Ex1451620480498224165t_real > fun_Ex1451620480498224165t_real ).

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

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: nat > fun_fu7002938529518989751at_nat ).

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

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: nat > fun_fu40286998686754419omplex ).

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

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_nat_bool > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_nat_bool > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_nat_bool > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_nat_bool > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: set_nat > fun_fu8464300013002972271at_int ).

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

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: set_nat > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: nat > fun_co3258106519541737638omplex ).

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

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_nat_complex > fun_co3258106519541737638omplex ).

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

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

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

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

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

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_nat_fun_nat_int > fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_nat_fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_nat_fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_nat_fun_nat_real > fun_nat_real ).

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

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

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_nat_complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_nat_real > fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: extended_enat > fun_Ex8979681104671372885d_enat ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: real > fun_re4756235295228751087t_real ).

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

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: nat > fun_fu5303113179588540421d_enat ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: nat > fun_fu5303113179588540421d_enat ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: nat > fun_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: nat > fun_re8542384036533109323t_real ).

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

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: nat > fun_fu6810213084838398785t_real ).

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

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: fun_nat_int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: fun_nat_complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: fun_nat_real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: fun_nat_complex > fun_na6632006986671646835omplex ).

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

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: real > fun_fu3477889739401271834t_real ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: real > fun_fu2357351296865450298t_real ).

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

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

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: complex > fun_co3258106519541737638omplex ).

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

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_int_int > fun_int_int ).

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

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_Ex1451620480498224165t_real > fun_fu7336479014630192387t_real ).

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

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_set_nat_real > fun_fu5354524256550068955t_real ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: fun_Ex3482447378695272101at_int > fun_fu6685514224203069699at_int ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: fun_real_int > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: fun_set_nat_int > fun_fu3492966305048685787at_int ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: fun_nat_real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: int > fun_int_fun_int_bool ).

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

tff(sy_c_ATP_058Lamm__ze____,type,
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tff(sy_c_ATP_058Lamm__zf____,type,
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tff(sy_c_ATP_058Lamm__zg____,type,
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tff(sy_c_ATP_058Lamm__zh____,type,
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tff(sy_c_ATP_058Lamm__zi____,type,
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tff(sy_c_ATP_058Lamm__zj____,type,
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tff(sy_c_ATP_058Lamm__zk____,type,
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tff(sy_c_ATP_058Lamm__zl____,type,
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tff(sy_c_ATP_058Lamm__zm____,type,
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tff(sy_c_ATP_058Lamm__zn____,type,
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tff(sy_c_ATP_058Lamm__zo____,type,
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tff(sy_c_ATP_058Lamm__zp____,type,
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tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__zw____,type,
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tff(sy_c_ATP_058Lamm__zx____,type,
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tff(sy_c_ATP_058Lamm__zy____,type,
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tff(sy_c_ATP_058Lamm__zz____,type,
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tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
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tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
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tff(sy_c_Bit__Operations_Oand__int__rel,type,
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tff(sy_c_Bit__Operations_Oconcat__bit,type,
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tff(sy_c_Divides_Odivmod__nat,type,
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    cons_int: ( int * list_int ) > list_int ).

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

tff(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

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

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

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

tff(sy_c_List_Olist_Oset_001t__Extended____Nat__Oenat,type,
    set_Extended_enat2: list_Extended_enat > set_Extended_enat ).

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

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

tff(sy_c_List_Olist_Oset_001t__Num__Onum,type,
    set_num2: list_num > set_num ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
    set_real2: list_real > set_real ).

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

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

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

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

tff(sy_c_List_Olist__update_001t__Extended____Nat__Oenat,type,
    list_u3071683517702156500d_enat: ( list_Extended_enat * nat * extended_enat ) > list_Extended_enat ).

tff(sy_c_List_Olist__update_001t__Int__Oint,type,
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tff(sy_c_List_Olist__update_001t__Nat__Onat,type,
    list_update_nat: ( list_nat * nat * nat ) > list_nat ).

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

tff(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olistrel1_001t__Int__Oint,type,
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tff(sy_c_List_Olistrel1_001t__Nat__Onat,type,
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tff(sy_c_List_Olistrel1_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olistrel1_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olistrel1p_001t__Int__Oint,type,
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tff(sy_c_List_Olistrel1p_001t__Nat__Onat,type,
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tff(sy_c_List_Onth_001t__Extended____Nat__Oenat,type,
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tff(sy_c_List_Onth_001t__Int__Oint,type,
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tff(sy_c_List_Onth_001t__Nat__Onat,type,
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tff(sy_c_List_Onth_001t__Num__Onum,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Real__Oreal,type,
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tff(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,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_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,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,
    produc4743750530478302277T_VEBT: ( list_VEBT_VEBT * list_VEBT_VEBT ) > list_P7413028617227757229T_VEBT ).

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

tff(sy_c_List_Oreplicate_001t__Extended____Nat__Oenat,type,
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tff(sy_c_List_Oreplicate_001t__Int__Oint,type,
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tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
    replicate_nat: ( nat * nat ) > list_nat ).

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

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

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

tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: fun_int_fun_int_bool > fun_list_int_bool ).

tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
    sorted_wrt_nat: fun_nat_fun_nat_bool > fun_list_nat_bool ).

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

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

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

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

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

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

tff(sy_c_Nat_Onat_Ocase__nat_001t__HOL__Obool,type,
    case_nat_bool: ( bool * fun_nat_bool ) > fun_nat_bool ).

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

tff(sy_c_Nat_Onat_Opred,type,
    pred: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
    semiring_1_Nats_int: set_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
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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,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

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

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Extended____Nat__Oenat,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__Real__Oreal,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Extended____Nat__Oenat_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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_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__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat__Bijection_Olist__decode,type,
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tff(sy_c_Nat__Bijection_Olist__decode__rel,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,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > fun_na1476630412504255114at_nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
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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,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_Num_Oinc,type,
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tff(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: fun_num_nat ).

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

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

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: fun_num_nat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: num > real ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Extended____Nat__Oenat,type,
    none_Extended_enat: option_Extended_enat ).

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tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
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tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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    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_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

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

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

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

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

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_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__Real__Oreal,type,
    diffs_real: fun_nat_real > fun_nat_real ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

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

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

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

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_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_Transfer_Obi__total_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bi_tot896582865486249351at_int: fun_Pr5932607294251420486t_bool > $o ).

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

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

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

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

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun_VE3763956218935105199t_bool ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr817965127378462947t_bool ).

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

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

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

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_bool ).

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

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_bool ).

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

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

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr817965127378462947t_bool ).

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

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

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

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: fun_Pr248683899324676269t_bool > fun_Pr7757163693017731481t_bool ).

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

tff(sy_c_Wellfounded_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

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

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

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

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

tff(sy_c_aa_001t__Complex__Ocomplex_001t__HOL__Obool,type,
    aa_complex_bool: ( fun_complex_bool * complex ) > bool ).

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

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

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

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J,type,
    aa_com1553036987509291606x_bool: ( fun_co1378794544173401837x_bool * complex ) > fun_complex_bool ).

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

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

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

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
    aa_com2211550117870696024t_bool: ( fun_co2381122952212249071t_bool * complex ) > fun_int_bool ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_com3205223359959579713nt_int: ( fun_co333780937270390912nt_int * complex ) > fun_int_int ).

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

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
    aa_com5335841084339320244t_bool: ( fun_co2841405903858895691t_bool * complex ) > fun_nat_bool ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_com2205869648058525157at_int: ( fun_co7370231399432144932at_int * complex ) > fun_nat_int ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_com6383720667567721865at_nat: ( fun_co636942507509095880at_nat * complex ) > fun_nat_nat ).

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

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
    aa_com4766338864777664963omplex: ( fun_co286204877037288194omplex * complex ) > fun_real_complex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
    aa_com8943241093633526360l_bool: ( fun_co2286642472385812463l_bool * complex ) > fun_real_bool ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__Complex__Ocomplex,type,
    aa_Ext8748583412541189208omplex: ( fun_Ex3741802288643809319omplex * extended_enat ) > complex ).

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

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__HOL__Obool,type,
    aa_Ext2772979874139974381t_bool: ( fun_Ex8563670519792538868t_bool * extended_enat ) > bool ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__Int__Oint,type,
    aa_Extended_enat_int: ( fun_Ex3482447378695272101at_int * extended_enat ) > int ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
    aa_Extended_enat_nat: ( fun_Ex7660298398204468809at_nat * extended_enat ) > nat ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
    aa_Ext2541572767299617110t_real: ( fun_Ex1451620480498224165t_real * extended_enat ) > real ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Extended____Nat__Oenat_Mt__HOL__Obool_J,type,
    aa_Ext291666158496884126t_bool: ( fun_Ex8598408993869023341t_bool * extended_enat ) > fun_Ex8563670519792538868t_bool ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
    aa_Ext3185103904533377306t_bool: ( fun_Ex7222163901356174497t_bool * extended_enat ) > fun_int_bool ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_Ext6203864024966492415nt_int: ( fun_Ex6820543082444009806nt_int * extended_enat ) > fun_int_int ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Nat__Onat_Mt__Extended____Nat__Oenat_J,type,
    aa_Ext1337733060012393797d_enat: ( fun_Ex2441982661509597004d_enat * extended_enat ) > fun_na8797796813480665199d_enat ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
    aa_Ext6309394871002001526t_bool: ( fun_Ex7682446853002821117t_bool * extended_enat ) > fun_nat_bool ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_Ext5204510313065437859at_int: ( fun_Ex4633621507750988018at_int * extended_enat ) > fun_nat_int ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_Ext158989295719858759at_nat: ( fun_Ex7123704652682714774at_nat * extended_enat ) > fun_nat_nat ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Extended____Nat__Oenat_J_J,type,
    aa_Ext8984936454148280526d_enat: ( fun_Ex8979681104671372885d_enat * extended_enat ) > fun_na7542918316964655096d_enat ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
    aa_Ext4143593917056713754l_bool: ( fun_Ex4578939836232520097l_bool * extended_enat ) > fun_real_bool ).

tff(sy_c_aa_001t__HOL__Obool_001t__HOL__Obool,type,
    aa_bool_bool: ( fun_bool_bool * bool ) > bool ).

tff(sy_c_aa_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J,type,
    aa_boo8970175072276269406l_bool: ( fun_bo1841576699660343469l_bool * bool ) > fun_bool_bool ).

tff(sy_c_aa_001t__Int__Oint_001t__Complex__Ocomplex,type,
    aa_int_complex: ( fun_int_complex * int ) > complex ).

tff(sy_c_aa_001t__Int__Oint_001t__Extended____Nat__Oenat,type,
    aa_int_Extended_enat: ( fun_in5914218760682845459d_enat * int ) > extended_enat ).

tff(sy_c_aa_001t__Int__Oint_001t__HOL__Obool,type,
    aa_int_bool: ( fun_int_bool * int ) > bool ).

tff(sy_c_aa_001t__Int__Oint_001t__Int__Oint,type,
    aa_int_int: ( fun_int_int * int ) > int ).

tff(sy_c_aa_001t__Int__Oint_001t__Nat__Onat,type,
    aa_int_nat: ( fun_int_nat * int ) > nat ).

tff(sy_c_aa_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_int7462565939943104719nt_int: ( fun_in4211310926729683942nt_int * int ) > product_prod_int_int ).

tff(sy_c_aa_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_int1417691210696471063at_nat: ( fun_in4514472496968388910at_nat * int ) > product_prod_nat_nat ).

tff(sy_c_aa_001t__Int__Oint_001t__Real__Oreal,type,
    aa_int_real: ( fun_int_real * int ) > real ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J,type,
    aa_int5354780257279585880x_bool: ( fun_in5750378413879446255x_bool * int ) > fun_complex_bool ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Complex__Ocomplex_Mt__Int__Oint_J,type,
    aa_int6126987627036542017ex_int: ( fun_in3657390099950231424ex_int * int ) > fun_complex_int ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Extended____Nat__Oenat_Mt__HOL__Obool_J,type,
    aa_int7536572431366371422t_bool: ( fun_in2147091231886866205t_bool * int ) > fun_Ex8563670519792538868t_bool ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Extended____Nat__Oenat_Mt__Int__Oint_J,type,
    aa_int4341511938324142267at_int: ( fun_in5181886200629118930at_int * int ) > fun_Ex3482447378695272101at_int ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
    aa_int_fun_int_bool: ( fun_int_fun_int_bool * int ) > fun_int_bool ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_int_fun_int_int: ( fun_int_fun_int_int * int ) > fun_int_int ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_int2565445551602696444nt_int: ( fun_in1696471567231710611nt_int * int ) > fun_in4211310926729683942nt_int ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    aa_int5842867502692063756at_int: ( fun_in1521351555827773387at_int * int ) > fun_int_fun_nat_int ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
    aa_int6823550873765277145at_int: ( fun_in281396392327853336at_int * int ) > fun_in6793278964266839919at_int ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
    aa_int_fun_nat_bool: ( fun_int_fun_nat_bool * int ) > fun_nat_bool ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_int_fun_nat_int: ( fun_int_fun_nat_int * int ) > fun_nat_int ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_int_fun_nat_nat: ( fun_int_fun_nat_nat * int ) > fun_nat_nat ).

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tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__fun_It__Real__Oreal_Mt__Real__Oreal_J_Mt__fun_It__Real__Oreal_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__fun_It__Real__Oreal_Mt__Real__Oreal_J_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__Real__Oreal_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__Real__Oreal_Mt__Int__Oint_J,type,
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tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
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tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__Real__Oreal_Mt__Int__Oint_J_J,type,
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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_Mt__HOL__Obool_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun6824837968375615225at_nat: ( fun_fu3370445115705697296at_nat * fun_real_fun_nat_nat ) > fun_fu8831944113610316029at_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_Mt__HOL__Obool_J_J_Mt__fun_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    aa_fun8436216498351771861al_nat: ( fun_fu6463170612142064108al_nat * fun_real_fun_nat_nat ) > fun_fu6241273384353077465al_nat ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__HOL__Obool_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__HOL__Obool_J,type,
    aa_fun3565389549526402364t_bool: ( fun_fu7495562154832025027t_bool * fun_set_nat_bool ) > fun_set_nat_bool ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J,type,
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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_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Real__Oreal_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Real__Oreal_J,type,
    aa_fun3569254276315353500t_real: ( fun_fu5354524256550068955t_real * fun_set_nat_real ) > fun_set_nat_real ).

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

tff(sy_c_fEx_001t__Nat__Onat,type,
    fEx_nat: fun_fu7168253649841883962l_bool ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun_bool_bool ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

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

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

tff(sy_c_fequal_001t__Complex__Ocomplex,type,
    fequal_complex: fun_co1378794544173401837x_bool ).

tff(sy_c_fequal_001t__Extended____Nat__Oenat,type,
    fequal_Extended_enat: fun_Ex8598408993869023341t_bool ).

tff(sy_c_fequal_001t__HOL__Obool,type,
    fequal_bool: fun_bo1841576699660343469l_bool ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_bool ).

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

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_bool ).

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_re1255401033344146673l_bool ).

tff(sy_c_fequal_001t__Set__Oset_It__Nat__Onat_J,type,
    fequal_set_nat: fun_se707272970763642173t_bool ).

tff(sy_c_fimplies,type,
    fimplies: fun_bo1841576699660343469l_bool ).

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

tff(sy_c_member_001t__Extended____Nat__Oenat,type,
    member_Extended_enat: extended_enat > fun_se2015444840911768724t_bool ).

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

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

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

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

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

tff(sy_c_member_001t__Num__Onum,type,
    member_num: ( num * set_num ) > $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__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member6698963635872716290st_int: ( produc1186641810826059865st_int * set_Pr765067013931698361st_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member7340969449405702474st_nat: ( produc1828647624359046049st_nat * set_Pr3451248702717554689st_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member6693912407220327184at_nat: ( produc6392793444374437607at_nat * set_Pr1542805901266377927at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    member4439316823752958928T_VEBT: ( produc9211091688327510695T_VEBT * set_Pr1916528119006554503T_VEBT ) > $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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: ( produc859450856879609959at_nat * set_Pr8693737435421807431at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    member373505688050248522BT_nat: ( produc9072475918466114483BT_nat * set_Pr7556676689462069481BT_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    member568628332442017744T_VEBT: ( produc8243902056947475879T_VEBT * set_Pr6192946355708809607T_VEBT ) > $o ).

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

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

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

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: fun_VE7346385682144900493T_bool ).

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

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_i____,type,
    i: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (8855)
tff(fact_0_True,axiom,
    i = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na) ).

% True
tff(fact_1_insprop,axiom,
    nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na),vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,mi),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)) = vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,mi),na)) ).

% insprop
tff(fact_2_bit__split__inv,axiom,
    ! [X: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),D)),D) = X ).

% bit_split_inv
tff(fact_3_tc,axiom,
    pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)))) ).

% tc
tff(fact_4__C161_C,axiom,
    ~ ? [X_1: nat] : pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na))),X_1)) ).

% "161"
tff(fact_5_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).

% list_update_id
tff(fact_6_list__update__id,axiom,
    ! [Xs: list_int,I: nat] : list_update_int(Xs,I,aa_nat_int(nth_int(Xs),I)) = Xs ).

% list_update_id
tff(fact_7_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] : list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,I)) = Xs ).

% list_update_id
tff(fact_8_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_9_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,X: int] :
      ( ( I != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_10_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_11__C162_C,axiom,
    ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na))),X_1)) ).

% "162"
tff(fact_12_list__update__overwrite,axiom,
    ! [Xs: list_int,I: nat,X: int,Y: int] : list_update_int(list_update_int(Xs,I,X),I,Y) = list_update_int(Xs,I,Y) ).

% list_update_overwrite
tff(fact_13_list__update__overwrite,axiom,
    ! [Xs: list_nat,I: nat,X: nat,Y: nat] : list_update_nat(list_update_nat(Xs,I,X),I,Y) = list_update_nat(Xs,I,Y) ).

% list_update_overwrite
tff(fact_14_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I,Y) = list_u1324408373059187874T_VEBT(Xs,I,Y) ).

% list_update_overwrite
tff(fact_15_nsprop,axiom,
    ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na))))
   => ( summary = vEBT_vebt_insert(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)) ) ) ).

% nsprop
tff(fact_16__C11_C,axiom,
    ! [X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na),vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,mi),na))))))
     => pp(aa_nat_bool(vEBT_invar_vebt(X2),na)) ) ).

% "11"
tff(fact_17_False,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),xa)) ).

% False
tff(fact_18_list__update__swap,axiom,
    ! [I: nat,I2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X3: vEBT_VEBT] :
      ( ( I != I2 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I2,X3) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X3),I,X) ) ) ).

% list_update_swap
tff(fact_19_list__update__swap,axiom,
    ! [I: nat,I2: nat,Xs: list_int,X: int,X3: int] :
      ( ( I != I2 )
     => ( list_update_int(list_update_int(Xs,I,X),I2,X3) = list_update_int(list_update_int(Xs,I2,X3),I,X) ) ) ).

% list_update_swap
tff(fact_20_list__update__swap,axiom,
    ! [I: nat,I2: nat,Xs: list_nat,X: nat,X3: nat] :
      ( ( I != I2 )
     => ( list_update_nat(list_update_nat(Xs,I,X),I2,X3) = list_update_nat(list_update_nat(Xs,I2,X3),I,X) ) ) ).

% list_update_swap
tff(fact_21_in__children__def,axiom,
    ! [N: nat,TreeList: list_VEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(N,TreeList),X))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N))) ) ).

% in_children_def
tff(fact_22__C5_Ohyps_C_I7_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),ma)) ).

% "5.hyps"(7)
tff(fact_23__C12_C,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na))),vEBT_vebt_insert(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),summary)),m)) ).

% "12"
tff(fact_24_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_12)) ) ).

% not_min_Null_member
tff(fact_25_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ~ pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ).

% min_Null_member
tff(fact_26__C1_C,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(summary),m)) ).

% "1"
tff(fact_27_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
      <=> pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% both_member_options_equiv_member
tff(fact_28_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
       => pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% valid_member_both_member_options
tff(fact_29__C5_C,axiom,
    ( ( mi = ma )
   => ! [X2: vEBT_VEBT] :
        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(treeList)))
       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_1)) ) ) ).

% "5"
tff(fact_30_abcdef,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)) ).

% abcdef
tff(fact_31_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
      <=> pp(aa_set_nat_bool(member_nat(X),vEBT_set_vebt(T))) ) ) ).

% member_correct
tff(fact_32__C0_C,axiom,
    ! [X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(treeList)))
     => pp(aa_nat_bool(vEBT_invar_vebt(X2),na)) ) ).

% "0"
tff(fact_33__C163_C,axiom,
    ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),i)) ).

% "163"
tff(fact_34_mimaxrel,axiom,
    ( ( xa != mi )
    & ( xa != ma ) ) ).

% mimaxrel
tff(fact_35__C8_C,axiom,
    aa_nat_nat(suc,na) = m ).

% "8"
tff(fact_36_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_37_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
    <=> pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).

% valid_eq
tff(fact_38_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),D))
     => pp(aa_nat_bool(vEBT_VEBT_valid(T),D)) ) ).

% valid_eq1
tff(fact_39_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
     => pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).

% valid_eq2
tff(fact_40_order__refl,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),X)) ).

% order_refl
tff(fact_41_order__refl,axiom,
    ! [X: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),X)) ).

% order_refl
tff(fact_42_order__refl,axiom,
    ! [X: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(X),X)) ).

% order_refl
tff(fact_43_order__refl,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X)) ).

% order_refl
tff(fact_44_order__refl,axiom,
    ! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),X)) ).

% order_refl
tff(fact_45_dual__order_Orefl,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),A)) ).

% dual_order.refl
tff(fact_46_dual__order_Orefl,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),A)) ).

% dual_order.refl
tff(fact_47_dual__order_Orefl,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A),A)) ).

% dual_order.refl
tff(fact_48_dual__order_Orefl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).

% dual_order.refl
tff(fact_49_dual__order_Orefl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).

% dual_order.refl
tff(fact_50_inthall,axiom,
    ! [Xs: list_Extended_enat,P: fun_Ex8563670519792538868t_bool,N: nat] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),set_Extended_enat2(Xs)))
         => pp(aa_Ext2772979874139974381t_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3941691890525107288d_enat(Xs)))
       => pp(aa_Ext2772979874139974381t_bool(P,aa_nat_Extended_enat(nth_Extended_enat(Xs),N))) ) ) ).

% inthall
tff(fact_51_inthall,axiom,
    ! [Xs: list_real,P: fun_real_bool,N: nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
         => pp(aa_real_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
       => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),N))) ) ) ).

% inthall
tff(fact_52_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,N: nat] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),set_set_nat2(Xs)))
         => pp(aa_set_nat_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
       => pp(aa_set_nat_bool(P,aa_nat_set_nat(nth_set_nat(Xs),N))) ) ) ).

% inthall
tff(fact_53_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,N: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).

% inthall
tff(fact_54_inthall,axiom,
    ! [Xs: list_int,P: fun_int_bool,N: nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => pp(aa_int_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% inthall
tff(fact_55_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,N: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% inthall
tff(fact_56_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% deg_not_0
tff(fact_57__C3_C,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "3"
tff(fact_58_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_59_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_imp_le_nat
tff(fact_60_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N2: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)
     => ~ ! [N2: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) ) ).

% even_odd_cases
tff(fact_61_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_tree_deg_neq_0
tff(fact_62_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_0_not
tff(fact_63_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_64_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y2) )
    <=> ( X22 = Y2 ) ) ).

% nat.inject
tff(fact_65_mem__Collect__eq,axiom,
    ! [A: extended_enat,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),collec4429806609662206161d_enat(P)))
    <=> pp(aa_Ext2772979874139974381t_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_66_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(A),collect_real(P)))
    <=> pp(aa_real_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_67_mem__Collect__eq,axiom,
    ! [A: list_nat,P: fun_list_nat_bool] :
      ( member_list_nat(A,collect_list_nat(P))
    <=> pp(aa_list_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_68_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),collect_set_nat(P)))
    <=> pp(aa_set_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_69_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(A),collect_nat(P)))
    <=> pp(aa_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_70_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(A),collect_int(P)))
    <=> pp(aa_int_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_71_Collect__mem__eq,axiom,
    ! [A2: set_Extended_enat] : collec4429806609662206161d_enat(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_72_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_73_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_74_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_75_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_76_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_77_Collect__cong,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(P,X4))
        <=> pp(aa_real_bool(Q,X4)) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_78_Collect__cong,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X4: list_nat] :
          ( pp(aa_list_nat_bool(P,X4))
        <=> pp(aa_list_nat_bool(Q,X4)) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_79_Collect__cong,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_nat_bool(P,X4))
        <=> pp(aa_set_nat_bool(Q,X4)) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_80_Collect__cong,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(P,X4))
        <=> pp(aa_nat_bool(Q,X4)) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_81_Collect__cong,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(P,X4))
        <=> pp(aa_int_bool(Q,X4)) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_82_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_eq
tff(fact_83_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) ) ).

% Suc_mono
tff(fact_84_lessI,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,N))) ).

% lessI
tff(fact_85_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_nat_zero_code
tff(fact_86_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% neq0_conv
tff(fact_87_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_88_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% Suc_le_mono
tff(fact_89_le0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% le0
tff(fact_90_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A)) ).

% bot_nat_0.extremum
tff(fact_91_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc_right
tff(fact_92_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).

% Nat.add_0_right
tff(fact_93_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_94_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% nat_add_left_cancel_less
tff(fact_95_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% nat_add_left_cancel_le
tff(fact_96_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,list_u1324408373059187874T_VEBT(Xs,I,X)) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) ).

% length_list_update
tff(fact_97_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X: int] : aa_list_int_nat(size_size_list_int,list_update_int(Xs,I,X)) = aa_list_int_nat(size_size_list_int,Xs) ).

% length_list_update
tff(fact_98_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : aa_list_nat_nat(size_size_list_nat,list_update_nat(Xs,I,X)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_list_update
tff(fact_99_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_100_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N))) ).

% zero_less_Suc
tff(fact_101_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% add_gr_0
tff(fact_102_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),I))
     => ( list_u1324408373059187874T_VEBT(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_103_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),I))
     => ( list_update_int(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_104_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),I))
     => ( list_update_nat(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_105_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I) = X ) ) ).

% nth_list_update_eq
tff(fact_106_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_107_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_108_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_109_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_int_nat(size_size_list_int,Xs)))
       => ( set_int2(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_110_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_111_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ~ ! [Q2: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)) ) ).

% less_natE
tff(fact_112_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        | ? [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,I3))) ) ) ) ).

% Ex_less_Suc2
tff(fact_113_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M2: nat] : N = aa_nat_nat(suc,M2) ) ).

% gr0_conv_Suc
tff(fact_114_All__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I3))) ) ) ) ).

% All_less_Suc2
tff(fact_115_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)))) ).

% less_add_Suc1
tff(fact_116_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I)))) ).

% less_add_Suc2
tff(fact_117_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).

% gr0_implies_Suc
tff(fact_118_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ? [K2: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_iff_Suc_add
tff(fact_119_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ? [K3: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_imp_Suc_add
tff(fact_120_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_VEBT_VEBT] : aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs2) = N ).

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

% Ex_list_of_length
tff(fact_122_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] : aa_list_nat_nat(size_size_list_nat,Xs2) = N ).

% Ex_list_of_length
tff(fact_123_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) != aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_124_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) != aa_list_int_nat(size_size_list_int,Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_125_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) != aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_126_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( ( M = zero_zero_nat )
        | ? [J2: nat] :
            ( ( M = aa_nat_nat(suc,J2) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_127_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3) = J ) ) ) ).

% less_imp_add_positive
tff(fact_128_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,X) != aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_129_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( aa_num_nat(size_size_num,X) != aa_num_nat(size_size_num,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_130_size__neq__size__imp__neq,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X) != aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_131_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( aa_list_int_nat(size_size_list_int,X) != aa_list_int_nat(size_size_list_int,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_132_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,X) != aa_list_nat_nat(size_size_list_nat,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_133_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).

% not0_implies_Suc
tff(fact_134_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_135_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) ).

% add_Suc_shift
tff(fact_136_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_137_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_138_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_139_zero__induct,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ).

% zero_induct
tff(fact_140_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_141_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X4),zero_zero_nat))
     => ( ! [Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,zero_zero_nat),aa_nat_nat(suc,Y3)))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(suc,X4)),aa_nat_nat(suc,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ) ).

% diff_induct
tff(fact_142_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_143_nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_induct
tff(fact_144_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_145_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_146_add__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc
tff(fact_147_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_148_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).

% plus_nat.add_0
tff(fact_149_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_150_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_151_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_152_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_153_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_154_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_155_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I4: nat] :
            ( ( J = aa_nat_nat(suc,I4) )
           => pp(aa_nat_bool(P,I4)) )
       => ( ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),J))
             => ( pp(aa_nat_bool(P,aa_nat_nat(suc,I4)))
               => pp(aa_nat_bool(P,I4)) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_156_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),aa_nat_nat(suc,I4)))
       => ( ! [I4: nat,J3: nat,K3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),J3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),K3))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),J3))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J3),K3))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),K3)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_157_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_158_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_SucD
tff(fact_159_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_160_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
    <=> ? [M4: nat] :
          ( ( M = aa_nat_nat(suc,M4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M4)) ) ) ).

% Suc_less_eq2
tff(fact_161_All__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,N))
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
           => pp(aa_nat_bool(P,I3)) ) ) ) ).

% All_less_Suc
tff(fact_162_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M))) ) ).

% not_less_eq
tff(fact_163_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_164_Ex__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,N))
        | ? [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
            & pp(aa_nat_bool(P,I3)) ) ) ) ).

% Ex_less_Suc
tff(fact_165_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% less_SucI
tff(fact_166_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_167_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( ( aa_nat_nat(suc,M) != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N)) ) ) ).

% Suc_lessI
tff(fact_168_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K))
     => ~ ! [J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J3))
           => ( K != aa_nat_nat(suc,J3) ) ) ) ).

% Suc_lessE
tff(fact_169_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_lessD
tff(fact_170_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K))
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J3))
             => ( K != aa_nat_nat(suc,J3) ) ) ) ) ).

% Nat.lessE
tff(fact_171_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% less_add_eq_less
tff(fact_172_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_less_add2
tff(fact_173_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_less_add1
tff(fact_174_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

% add_less_mono1
tff(fact_175_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I)) ).

% not_add_less2
tff(fact_176_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I)) ).

% not_add_less1
tff(fact_177_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_less_mono
tff(fact_178_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% add_lessD1
tff(fact_179_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),X4))
       => ( ! [X4: nat,Y3: nat,Z: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y3),Z))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Z)) ) )
         => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,N2),aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_180_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(P,M))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_181_full__nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M5)),N2))
             => pp(aa_nat_bool(P,M5)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% full_nat_induct
tff(fact_182_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% not_less_eq_eq
tff(fact_183_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_184_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        | ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_185_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M6))
     => ? [M3: nat] : M6 = aa_nat_nat(suc,M3) ) ).

% Suc_le_D
tff(fact_186_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N))) ) ).

% le_SucI
tff(fact_187_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_SucE
tff(fact_188_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_leD
tff(fact_189_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ? [K2: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2) ) ).

% nat_le_iff_add
tff(fact_190_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_le_add2
tff(fact_191_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_le_add1
tff(fact_192_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

% add_le_mono1
tff(fact_193_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_le_mono
tff(fact_194_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
     => ? [N2: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) ) ).

% le_Suc_ex
tff(fact_195_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% add_leD2
tff(fact_196_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% add_leD1
tff(fact_197_le__add2,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% le_add2
tff(fact_198_le__add1,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))) ).

% le_add1
tff(fact_199_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).

% add_leE
tff(fact_200_infinite__descent0,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( ~ pp(aa_nat_bool(P,N2))
             => ? [M5: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
                  & ~ pp(aa_nat_bool(P,M5)) ) ) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% infinite_descent0
tff(fact_201_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_202_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_zeroE
tff(fact_203_not__less0,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less0
tff(fact_204_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr0
tff(fact_205_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr0I
tff(fact_206_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ).

% bot_nat_0.extremum_strict
tff(fact_207_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_208_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_209_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_210_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% less_eq_nat.simps(1)
tff(fact_211_ex__least__nat__less,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
            & ! [I5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I5),K3))
               => ~ pp(aa_nat_bool(P,I5)) )
            & pp(aa_nat_bool(P,aa_nat_nat(suc,K3))) ) ) ) ).

% ex_least_nat_less
tff(fact_212_length__pos__if__in__set,axiom,
    ! [X: extended_enat,Xs: list_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),set_Extended_enat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3941691890525107288d_enat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_213_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_real(Xs))) ) ).

% length_pos_if_in_set
tff(fact_214_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_215_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))) ) ).

% length_pos_if_in_set
tff(fact_216_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_int_nat(size_size_list_int,Xs))) ) ).

% length_pos_if_in_set
tff(fact_217_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs))) ) ).

% length_pos_if_in_set
tff(fact_218_length__induct,axiom,
    ! [P: fun_li6006399982103069984T_bool,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs2)))
             => pp(aa_lis2822252094023462681T_bool(P,Ys2)) )
         => pp(aa_lis2822252094023462681T_bool(P,Xs2)) )
     => pp(aa_lis2822252094023462681T_bool(P,Xs)) ) ).

% length_induct
tff(fact_219_length__induct,axiom,
    ! [P: fun_list_int_bool,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_int_nat(size_size_list_int,Ys2)),aa_list_int_nat(size_size_list_int,Xs2)))
             => pp(aa_list_int_bool(P,Ys2)) )
         => pp(aa_list_int_bool(P,Xs2)) )
     => pp(aa_list_int_bool(P,Xs)) ) ).

% length_induct
tff(fact_220_length__induct,axiom,
    ! [P: fun_list_nat_bool,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_nat_nat(size_size_list_nat,Ys2)),aa_list_nat_nat(size_size_list_nat,Xs2)))
             => pp(aa_list_nat_bool(P,Ys2)) )
         => pp(aa_list_nat_bool(P,Xs2)) )
     => pp(aa_list_nat_bool(P,Xs)) ) ).

% length_induct
tff(fact_221_subset__code_I1_J,axiom,
    ! [Xs: list_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(set_Extended_enat2(Xs)),B))
    <=> ! [X5: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),set_Extended_enat2(Xs)))
         => pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),B)) ) ) ).

% subset_code(1)
tff(fact_222_subset__code_I1_J,axiom,
    ! [Xs: list_real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),B))
    <=> ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),set_real2(Xs)))
         => pp(aa_set_real_bool(member_real(X5),B)) ) ) ).

% subset_code(1)
tff(fact_223_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),B))
    <=> ! [X5: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X5),set_set_nat2(Xs)))
         => pp(aa_set_set_nat_bool(member_set_nat(X5),B)) ) ) ).

% subset_code(1)
tff(fact_224_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B)
    <=> ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(Xs)))
         => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),B)) ) ) ).

% subset_code(1)
tff(fact_225_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),B))
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
         => pp(aa_set_nat_bool(member_nat(X5),B)) ) ) ).

% subset_code(1)
tff(fact_226_subset__code_I1_J,axiom,
    ! [Xs: list_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),B))
    <=> ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(Xs)))
         => pp(aa_set_int_bool(member_int(X5),B)) ) ) ).

% subset_code(1)
tff(fact_227_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_228_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_na8797796813480665199d_enat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,N2)),aa_nat_Extended_enat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,N)),aa_nat_Extended_enat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_229_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_230_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_231_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_232_lift__Suc__mono__less,axiom,
    ! [F: fun_na8797796813480665199d_enat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,N2)),aa_nat_Extended_enat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,N)),aa_nat_Extended_enat(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_233_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_234_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_235_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_real,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,aa_nat_nat(suc,N2))),aa_nat_real(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),aa_nat_real(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_236_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(F,aa_nat_nat(suc,N2))),aa_nat_set_nat(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(F,N3)),aa_nat_set_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_237_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N2))),aa_nat_set_int(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N3)),aa_nat_set_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_238_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_239_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_240_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_real,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_241_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(F,N2)),aa_nat_set_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(F,N)),aa_nat_set_nat(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_242_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N2)),aa_nat_set_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N)),aa_nat_set_int(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_243_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_244_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_245_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% le_imp_less_Suc
tff(fact_246_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% less_eq_Suc_le
tff(fact_247_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_Suc_eq_le
tff(fact_248_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_249_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_lessD
tff(fact_250_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,J))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
               => ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
                 => pp(aa_nat_bool(P,N2)) ) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% inc_induct
tff(fact_251_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,I))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) )
         => pp(aa_nat_bool(P,J)) ) ) ) ).

% dec_induct
tff(fact_252_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_eq
tff(fact_253_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N)) ) ).

% Suc_leI
tff(fact_254_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M3)),aa_nat_nat(F,N2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,M)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)))) ) ).

% mono_nat_linear_lb
tff(fact_255_ex__least__nat__le,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N))
            & ! [I5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),K3))
               => ~ pp(aa_nat_bool(P,I5)) )
            & pp(aa_nat_bool(P,K3)) ) ) ) ).

% ex_least_nat_le
tff(fact_256_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys) )
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys,I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_257_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs)))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_258_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_259_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na2351039247863542899T_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_13: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I3),X_13)) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I3),nth_VEBT_VEBT(Xs3,I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_260_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_13: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P,I3),X_13)) )
    <=> ? [Xs3: list_int] :
          ( ( aa_list_int_nat(size_size_list_int,Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_int_bool(aa_nat_fun_int_bool(P,I3),aa_nat_int(nth_int(Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_261_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),X_13)) )
    <=> ? [Xs3: list_nat] :
          ( ( aa_list_nat_nat(size_size_list_nat,Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),aa_nat_nat(nth_nat(Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_262_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys) )
     => ( ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
           => ( nth_VEBT_VEBT(Xs,I4) = nth_VEBT_VEBT(Ys,I4) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_263_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_list_int_nat(size_size_list_int,Xs)))
           => ( aa_nat_int(nth_int(Xs),I4) = aa_nat_int(nth_int(Ys),I4) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_264_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_list_nat_nat(size_size_list_nat,Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I4) = aa_nat_nat(nth_nat(Ys),I4) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_265_set__update__subsetI,axiom,
    ! [Xs: list_Extended_enat,A2: set_Extended_enat,X: extended_enat,I: nat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(set_Extended_enat2(Xs)),A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
       => pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(set_Extended_enat2(list_u3071683517702156500d_enat(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_266_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,X: real,I: nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),A2))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_267_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,X: set_nat,I: nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),A2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
       => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_268_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A2)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X)),A2) ) ) ).

% set_update_subsetI
tff(fact_269_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),A2))
     => ( pp(aa_set_nat_bool(member_nat(X),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_270_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,X: int,I: nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),A2))
     => ( pp(aa_set_int_bool(member_int(X),A2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_271_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X5)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_272_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_bool] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(Xs)))
         => pp(aa_int_bool(P,X5)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_273_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_bool] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X5)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_274_all__nth__imp__all__set,axiom,
    ! [Xs: list_Extended_enat,P: fun_Ex8563670519792538868t_bool,X: extended_enat] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s3941691890525107288d_enat(Xs)))
         => pp(aa_Ext2772979874139974381t_bool(P,aa_nat_Extended_enat(nth_Extended_enat(Xs),I4))) )
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),set_Extended_enat2(Xs)))
       => pp(aa_Ext2772979874139974381t_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_275_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_bool,X: real] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_real(Xs)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),I4))) )
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => pp(aa_real_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_276_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,X: set_nat] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s3254054031482475050et_nat(Xs)))
         => pp(aa_set_nat_bool(P,aa_nat_set_nat(nth_set_nat(Xs),I4))) )
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
       => pp(aa_set_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_277_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I4))) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_278_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_bool,X: int] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_list_int_nat(size_size_list_int,Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I4))) )
     => ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
       => pp(aa_int_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_279_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,X: nat] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_list_nat_nat(size_size_list_nat,Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I4))) )
     => ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
       => pp(aa_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_280_in__set__conv__nth,axiom,
    ! [X: extended_enat,Xs: list_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),set_Extended_enat2(Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3941691890525107288d_enat(Xs)))
          & ( aa_nat_Extended_enat(nth_Extended_enat(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_281_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_real(Xs)))
          & ( aa_nat_real(nth_real(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_282_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs)))
          & ( aa_nat_set_nat(nth_set_nat(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_283_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
          & ( nth_VEBT_VEBT(Xs,I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_284_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs)))
          & ( aa_nat_int(nth_int(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_285_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
          & ( aa_nat_nat(nth_nat(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_286_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
           => pp(aa_VEBT_VEBT_bool(P,X4)) )
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_287_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: fun_int_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
           => pp(aa_int_bool(P,X4)) )
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_288_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
           => pp(aa_nat_bool(P,X4)) )
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_289_nth__mem,axiom,
    ! [N: nat,Xs: list_Extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3941691890525107288d_enat(Xs)))
     => pp(aa_set3347977610044113549t_bool(member_Extended_enat(aa_nat_Extended_enat(nth_Extended_enat(Xs),N)),set_Extended_enat2(Xs))) ) ).

% nth_mem
tff(fact_290_nth__mem,axiom,
    ! [N: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => pp(aa_set_real_bool(member_real(aa_nat_real(nth_real(Xs),N)),set_real2(Xs))) ) ).

% nth_mem
tff(fact_291_nth__mem,axiom,
    ! [N: nat,Xs: list_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => pp(aa_set_set_nat_bool(member_set_nat(aa_nat_set_nat(nth_set_nat(Xs),N)),set_set_nat2(Xs))) ) ).

% nth_mem
tff(fact_292_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,nth_VEBT_VEBT(Xs,N)),set_VEBT_VEBT2(Xs))) ) ).

% nth_mem
tff(fact_293_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
     => pp(aa_set_int_bool(member_int(aa_nat_int(nth_int(Xs),N)),set_int2(Xs))) ) ).

% nth_mem
tff(fact_294_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => pp(aa_set_nat_bool(member_nat(aa_nat_nat(nth_nat(Xs),N)),set_nat2(Xs))) ) ).

% nth_mem
tff(fact_295_order__antisym__conv,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_296_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),X))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_297_order__antisym__conv,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_298_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_299_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_300_linorder__le__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% linorder_le_cases
tff(fact_301_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_le_cases
tff(fact_302_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_le_cases
tff(fact_303_ord__le__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( ( aa_real_real(F,B2) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_304_ord__le__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( ( aa_real_nat(F,B2) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_305_ord__le__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( ( aa_real_int(F,B2) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_306_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: fun_nat_real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( ( aa_nat_real(F,B2) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_307_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: fun_nat_nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( ( aa_nat_nat(F,B2) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_308_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: fun_nat_int,C: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( ( aa_nat_int(F,B2) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_309_ord__le__eq__subst,axiom,
    ! [A: int,B2: int,F: fun_int_real,C: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( ( aa_int_real(F,B2) = C )
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_310_ord__le__eq__subst,axiom,
    ! [A: int,B2: int,F: fun_int_nat,C: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( ( aa_int_nat(F,B2) = C )
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_311_ord__le__eq__subst,axiom,
    ! [A: int,B2: int,F: fun_int_int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( ( aa_int_int(F,B2) = C )
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_312_ord__le__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_set_nat,C: set_nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( ( aa_real_set_nat(F,B2) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(F,X4)),aa_real_set_nat(F,Y3))) )
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_313_ord__eq__le__subst,axiom,
    ! [A: real,F: fun_real_real,B2: real,C: real] :
      ( ( A = aa_real_real(F,B2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_314_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_real_nat,B2: real,C: real] :
      ( ( A = aa_real_nat(F,B2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_real_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_315_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_real_int,B2: real,C: real] :
      ( ( A = aa_real_int(F,B2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_real_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_316_ord__eq__le__subst,axiom,
    ! [A: real,F: fun_nat_real,B2: nat,C: nat] :
      ( ( A = aa_nat_real(F,B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_317_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_nat_nat,B2: nat,C: nat] :
      ( ( A = aa_nat_nat(F,B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_318_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_nat_int,B2: nat,C: nat] :
      ( ( A = aa_nat_int(F,B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_319_ord__eq__le__subst,axiom,
    ! [A: real,F: fun_int_real,B2: int,C: int] :
      ( ( A = aa_int_real(F,B2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_int_real(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_320_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_int_nat,B2: int,C: int] :
      ( ( A = aa_int_nat(F,B2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_int_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_321_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_int_int,B2: int,C: int] :
      ( ( A = aa_int_int(F,B2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_322_ord__eq__le__subst,axiom,
    ! [A: set_nat,F: fun_real_set_nat,B2: real,C: real] :
      ( ( A = aa_real_set_nat(F,B2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(F,X4)),aa_real_set_nat(F,Y3))) )
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),aa_real_set_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_323_linorder__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% linorder_linear
tff(fact_324_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_linear
tff(fact_325_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_linear
tff(fact_326_order__eq__refl,axiom,
    ! [X: real,Y: real] :
      ( ( X = Y )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% order_eq_refl
tff(fact_327_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y)) ) ).

% order_eq_refl
tff(fact_328_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% order_eq_refl
tff(fact_329_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_eq_refl
tff(fact_330_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_eq_refl
tff(fact_331_order__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B2)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_332_order__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,B2)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_333_order__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,B2)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_334_order__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_335_order__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_336_order__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_int,C: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_337_order__subst2,axiom,
    ! [A: int,B2: int,F: fun_int_real,C: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,B2)),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_338_order__subst2,axiom,
    ! [A: int,B2: int,F: fun_int_nat,C: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,B2)),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_339_order__subst2,axiom,
    ! [A: int,B2: int,F: fun_int_int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,B2)),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_340_order__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_set_nat,C: set_nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(F,B2)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(F,X4)),aa_real_set_nat(F,Y3))) )
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_341_order__subst1,axiom,
    ! [A: real,F: fun_real_real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(F,C))) ) ) ) ).

% order_subst1
tff(fact_342_order__subst1,axiom,
    ! [A: real,F: fun_nat_real,B2: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_subst1
tff(fact_343_order__subst1,axiom,
    ! [A: real,F: fun_int_real,B2: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_int_real(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_int_real(F,C))) ) ) ) ).

% order_subst1
tff(fact_344_order__subst1,axiom,
    ! [A: nat,F: fun_real_nat,B2: real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_real_nat(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_real_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_345_order__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_346_order__subst1,axiom,
    ! [A: nat,F: fun_int_nat,B2: int,C: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_int_nat(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_int_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_347_order__subst1,axiom,
    ! [A: int,F: fun_real_int,B2: real,C: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_real_int(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_real_int(F,C))) ) ) ) ).

% order_subst1
tff(fact_348_order__subst1,axiom,
    ! [A: int,F: fun_nat_int,B2: nat,C: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(F,C))) ) ) ) ).

% order_subst1
tff(fact_349_order__subst1,axiom,
    ! [A: int,F: fun_int_int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(F,C))) ) ) ) ).

% order_subst1
tff(fact_350_order__subst1,axiom,
    ! [A: real,F: fun_set_nat_real,B2: set_nat,C: set_nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_set_nat_real(F,B2)))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),C))
       => ( ! [X4: set_nat,Y3: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(F,X4)),aa_set_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_set_nat_real(F,C))) ) ) ) ).

% order_subst1
tff(fact_351_Orderings_Oorder__eq__iff,axiom,
    ! [A: real,B2: real] :
      ( ( A = B2 )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_352_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A = B2 )
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_353_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A = B2 )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
        & pp(aa_set_int_bool(ord_less_eq_set_int(B2),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_354_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_355_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_356_antisym,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_357_antisym,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A))
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_358_antisym,axiom,
    ! [A: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A))
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_359_antisym,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_360_antisym,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_361_dual__order_Otrans,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A)) ) ) ).

% dual_order.trans
tff(fact_362_dual__order_Otrans,axiom,
    ! [B2: set_nat,A: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),B2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),A)) ) ) ).

% dual_order.trans
tff(fact_363_dual__order_Otrans,axiom,
    ! [B2: set_int,A: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C),B2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(C),A)) ) ) ).

% dual_order.trans
tff(fact_364_dual__order_Otrans,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% dual_order.trans
tff(fact_365_dual__order_Otrans,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% dual_order.trans
tff(fact_366_dual__order_Oantisym,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_367_dual__order_Oantisym,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_368_dual__order_Oantisym,axiom,
    ! [B2: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_369_dual__order_Oantisym,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_370_dual__order_Oantisym,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_371_dual__order_Oeq__iff,axiom,
    ! [A: real,B2: real] :
      ( ( A = B2 )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ) ).

% dual_order.eq_iff
tff(fact_372_dual__order_Oeq__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A = B2 )
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2)) ) ) ).

% dual_order.eq_iff
tff(fact_373_dual__order_Oeq__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A = B2 )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A),B2)) ) ) ).

% dual_order.eq_iff
tff(fact_374_dual__order_Oeq__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2)) ) ) ).

% dual_order.eq_iff
tff(fact_375_dual__order_Oeq__iff,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ) ).

% dual_order.eq_iff
tff(fact_376_linorder__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,A: real,B2: real] :
      ( ! [A3: real,B3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A3),B3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) )
     => ( ! [A3: real,B3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B3),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B2)) ) ) ).

% linorder_wlog
tff(fact_377_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A3),B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
     => ( ! [A3: nat,B3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B2)) ) ) ).

% linorder_wlog
tff(fact_378_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A: int,B2: int] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A3),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
     => ( ! [A3: int,B3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(P,B3),A3))
           => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B2)) ) ) ).

% linorder_wlog
tff(fact_379_order__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Z2)) ) ) ).

% order_trans
tff(fact_380_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),Z2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Z2)) ) ) ).

% order_trans
tff(fact_381_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(X),Z2)) ) ) ).

% order_trans
tff(fact_382_order__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z2)) ) ) ).

% order_trans
tff(fact_383_order__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z2)) ) ) ).

% order_trans
tff(fact_384_order_Otrans,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C)) ) ) ).

% order.trans
tff(fact_385_order_Otrans,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),C))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),C)) ) ) ).

% order.trans
tff(fact_386_order_Otrans,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C)) ) ) ).

% order.trans
tff(fact_387_order_Otrans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% order.trans
tff(fact_388_order_Otrans,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% order.trans
tff(fact_389_order__antisym,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_390_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_391_order__antisym,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_392_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_393_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_394_ord__le__eq__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( ( B2 = C )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_395_ord__le__eq__trans,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
     => ( ( B2 = C )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_396_ord__le__eq__trans,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
     => ( ( B2 = C )
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_397_ord__le__eq__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( ( B2 = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_398_ord__le__eq__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( ( B2 = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_399_ord__eq__le__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A = B2 )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_400_ord__eq__le__trans,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ( A = B2 )
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),C))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_401_ord__eq__le__trans,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( A = B2 )
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_402_ord__eq__le__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A = B2 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_403_ord__eq__le__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A = B2 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_404_order__class_Oorder__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X = Y )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_405_order__class_Oorder__eq__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_406_order__class_Oorder__eq__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
        & pp(aa_set_int_bool(ord_less_eq_set_int(Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_407_order__class_Oorder__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_408_order__class_Oorder__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_409_le__cases3,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2)) )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Z2)) )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Z2))
           => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),Y)) )
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),Y))
             => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) )
           => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2))
               => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),X)) )
             => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),X))
                 => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_410_le__cases3,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z2)) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z2)) )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z2))
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),Y)) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),Y))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) )
           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z2))
               => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),X)) )
             => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),X))
                 => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_411_le__cases3,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z2)) )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z2)) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z2))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),Y)) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),Y))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) )
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z2))
               => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),X)) )
             => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),X))
                 => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_412_nle__le,axiom,
    ! [A: real,B2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
        & ( B2 != A ) ) ) ).

% nle_le
tff(fact_413_nle__le,axiom,
    ! [A: nat,B2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
        & ( B2 != A ) ) ) ).

% nle_le
tff(fact_414_nle__le,axiom,
    ! [A: int,B2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
        & ( B2 != A ) ) ) ).

% nle_le
tff(fact_415_set__update__memI,axiom,
    ! [N: nat,Xs: list_Extended_enat,X: extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3941691890525107288d_enat(Xs)))
     => pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),set_Extended_enat2(list_u3071683517702156500d_enat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_416_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => pp(aa_set_real_bool(member_real(X),set_real2(list_update_real(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_417_set__update__memI,axiom,
    ! [N: nat,Xs: list_set_nat,X: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(list_update_set_nat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_418_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_419_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
     => pp(aa_set_int_bool(member_int(X),set_int2(list_update_int(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_420_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => pp(aa_set_nat_bool(member_nat(X),set_nat2(list_update_nat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_421_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_422_order__less__imp__not__less,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
     => ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_423_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_424_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_425_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_426_order__less__imp__not__eq2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_427_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_428_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_429_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_430_order__less__imp__not__eq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_431_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_432_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_433_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      | ( X = Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_less_linear
tff(fact_434_linorder__less__linear,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
      | ( X = Y )
      | pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),X)) ) ).

% linorder_less_linear
tff(fact_435_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      | ( X = Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_less_linear
tff(fact_436_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      | ( X = Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_less_linear
tff(fact_437_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_438_order__less__imp__triv,axiom,
    ! [X: extended_enat,Y: extended_enat,P: bool] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_439_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_440_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_441_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_not_sym
tff(fact_442_order__less__not__sym,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
     => ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),X)) ) ).

% order_less_not_sym
tff(fact_443_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_not_sym
tff(fact_444_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_not_sym
tff(fact_445_order__less__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_446_order__less__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_na8797796813480665199d_enat,C: extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_447_order__less__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_448_order__less__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_int,C: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_449_order__less__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: fun_Ex7660298398204468809at_nat,C: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,B2)),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_450_order__less__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: fun_Ex5852502975177633731d_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,B2)),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_451_order__less__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: fun_Ex1451620480498224165t_real,C: real] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,B2)),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_452_order__less__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: fun_Ex3482447378695272101at_int,C: int] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Extended_enat_int(F,B2)),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Extended_enat_int(F,X4)),aa_Extended_enat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Extended_enat_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_453_order__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,B2)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_454_order__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_re6065469302365115155d_enat,C: extended_enat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,B2)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_455_order__less__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_456_order__less__subst1,axiom,
    ! [A: nat,F: fun_Ex7660298398204468809at_nat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_Extended_enat_nat(F,B2)))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_Extended_enat_nat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_457_order__less__subst1,axiom,
    ! [A: nat,F: fun_real_nat,B2: real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_real_nat(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_real_nat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_458_order__less__subst1,axiom,
    ! [A: nat,F: fun_int_nat,B2: int,C: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_int_nat(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X4)),aa_int_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_int_nat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_459_order__less__subst1,axiom,
    ! [A: extended_enat,F: fun_na8797796813480665199d_enat,B2: nat,C: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_nat_Extended_enat(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_nat_Extended_enat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_460_order__less__subst1,axiom,
    ! [A: extended_enat,F: fun_Ex5852502975177633731d_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_Ext7969882492699152700d_enat(F,B2)))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_Ext7969882492699152700d_enat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_461_order__less__subst1,axiom,
    ! [A: extended_enat,F: fun_re6065469302365115155d_enat,B2: real,C: real] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_462_order__less__subst1,axiom,
    ! [A: extended_enat,F: fun_in5914218760682845459d_enat,B2: int,C: int] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_int_Extended_enat(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,X4)),aa_int_Extended_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_int_Extended_enat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_463_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B2: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_464_order__less__subst1,axiom,
    ! [A: real,F: fun_Ex1451620480498224165t_real,B2: extended_enat,C: extended_enat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_Ext2541572767299617110t_real(F,B2)))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_Ext2541572767299617110t_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_465_order__less__irrefl,axiom,
    ! [X: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X)) ).

% order_less_irrefl
tff(fact_466_order__less__irrefl,axiom,
    ! [X: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),X)) ).

% order_less_irrefl
tff(fact_467_order__less__irrefl,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X)) ).

% order_less_irrefl
tff(fact_468_order__less__irrefl,axiom,
    ! [X: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X)) ).

% order_less_irrefl
tff(fact_469_ord__less__eq__subst,axiom,
    ! [A: nat,B2: nat,F: fun_nat_nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( ( aa_nat_nat(F,B2) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_470_ord__less__eq__subst,axiom,
    ! [A: nat,B2: nat,F: fun_na8797796813480665199d_enat,C: extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( ( aa_nat_Extended_enat(F,B2) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_471_ord__less__eq__subst,axiom,
    ! [A: nat,B2: nat,F: fun_nat_real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( ( aa_nat_real(F,B2) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_472_ord__less__eq__subst,axiom,
    ! [A: nat,B2: nat,F: fun_nat_int,C: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( ( aa_nat_int(F,B2) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_473_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B2: extended_enat,F: fun_Ex7660298398204468809at_nat,C: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( ( aa_Extended_enat_nat(F,B2) = C )
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_474_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B2: extended_enat,F: fun_Ex5852502975177633731d_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( ( aa_Ext7969882492699152700d_enat(F,B2) = C )
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_475_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B2: extended_enat,F: fun_Ex1451620480498224165t_real,C: real] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( ( aa_Ext2541572767299617110t_real(F,B2) = C )
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_476_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B2: extended_enat,F: fun_Ex3482447378695272101at_int,C: int] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( ( aa_Extended_enat_int(F,B2) = C )
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Extended_enat_int(F,X4)),aa_Extended_enat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Extended_enat_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_477_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( ( aa_real_nat(F,B2) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_478_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_re6065469302365115155d_enat,C: extended_enat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( ( aa_rea1194391308913012860d_enat(F,B2) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_479_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_nat_nat,B2: nat,C: nat] :
      ( ( A = aa_nat_nat(F,B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_480_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: fun_na8797796813480665199d_enat,B2: nat,C: nat] :
      ( ( A = aa_nat_Extended_enat(F,B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_nat_Extended_enat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_481_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_nat_real,B2: nat,C: nat] :
      ( ( A = aa_nat_real(F,B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_482_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_nat_int,B2: nat,C: nat] :
      ( ( A = aa_nat_int(F,B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_483_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_Ex7660298398204468809at_nat,B2: extended_enat,C: extended_enat] :
      ( ( A = aa_Extended_enat_nat(F,B2) )
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_Extended_enat_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_484_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: fun_Ex5852502975177633731d_enat,B2: extended_enat,C: extended_enat] :
      ( ( A = aa_Ext7969882492699152700d_enat(F,B2) )
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_Ext7969882492699152700d_enat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_485_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_Ex1451620480498224165t_real,B2: extended_enat,C: extended_enat] :
      ( ( A = aa_Ext2541572767299617110t_real(F,B2) )
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_Ext2541572767299617110t_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_486_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_Ex3482447378695272101at_int,B2: extended_enat,C: extended_enat] :
      ( ( A = aa_Extended_enat_int(F,B2) )
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Extended_enat_int(F,X4)),aa_Extended_enat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_Extended_enat_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_487_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B2: real,C: real] :
      ( ( A = aa_real_nat(F,B2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_real_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_488_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: fun_re6065469302365115155d_enat,B2: real,C: real] :
      ( ( A = aa_rea1194391308913012860d_enat(F,B2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_489_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2)) ) ) ).

% order_less_trans
tff(fact_490_order__less__trans,axiom,
    ! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),Z2))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Z2)) ) ) ).

% order_less_trans
tff(fact_491_order__less__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2)) ) ) ).

% order_less_trans
tff(fact_492_order__less__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2)) ) ) ).

% order_less_trans
tff(fact_493_order__less__asym_H,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A)) ) ).

% order_less_asym'
tff(fact_494_order__less__asym_H,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A)) ) ).

% order_less_asym'
tff(fact_495_order__less__asym_H,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) ) ).

% order_less_asym'
tff(fact_496_order__less__asym_H,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A)) ) ).

% order_less_asym'
tff(fact_497_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_498_linorder__neq__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != Y )
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
        | pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_499_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_500_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_501_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_asym
tff(fact_502_order__less__asym,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
     => ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),X)) ) ).

% order_less_asym
tff(fact_503_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_asym
tff(fact_504_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_asym
tff(fact_505_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE
tff(fact_506_linorder__neqE,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != Y )
     => ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),X)) ) ) ).

% linorder_neqE
tff(fact_507_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neqE
tff(fact_508_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE
tff(fact_509_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_510_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A))
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_511_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_512_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_513_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_514_order_Ostrict__implies__not__eq,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_515_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_516_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_517_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_518_dual__order_Ostrict__trans,axiom,
    ! [B2: extended_enat,A: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,C),B2))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_519_dual__order_Ostrict__trans,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_520_dual__order_Ostrict__trans,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_521_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_522_not__less__iff__gr__or__eq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_523_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_524_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_525_order_Ostrict__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans
tff(fact_526_order_Ostrict__trans,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),C)) ) ) ).

% order.strict_trans
tff(fact_527_order_Ostrict__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans
tff(fact_528_order_Ostrict__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans
tff(fact_529_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),A3))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B2)) ) ) ) ).

% linorder_less_wlog
tff(fact_530_linorder__less__wlog,axiom,
    ! [P: fun_Ex8598408993869023341t_bool,A: extended_enat,B2: extended_enat] :
      ( ! [A3: extended_enat,B3: extended_enat] :
          ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A3),B3))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(P,A3),B3)) )
     => ( ! [A3: extended_enat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(P,A3),A3))
       => ( ! [A3: extended_enat,B3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(P,B3),A3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(P,A3),B3)) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(P,A),B2)) ) ) ) ).

% linorder_less_wlog
tff(fact_531_linorder__less__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,A: real,B2: real] :
      ( ! [A3: real,B3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A3),B3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) )
     => ( ! [A3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),A3))
       => ( ! [A3: real,B3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B3),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B2)) ) ) ) ).

% linorder_less_wlog
tff(fact_532_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A: int,B2: int] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
     => ( ! [A3: int] : pp(aa_int_bool(aa_int_fun_int_bool(P,A3),A3))
       => ( ! [A3: int,B3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(P,B3),A3))
             => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B2)) ) ) ) ).

% linorder_less_wlog
tff(fact_533_exists__least__iff,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ? [N4: nat] :
          ( pp(aa_nat_bool(P,N4))
          & ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N4))
             => ~ pp(aa_nat_bool(P,M2)) ) ) ) ).

% exists_least_iff
tff(fact_534_exists__least__iff,axiom,
    ! [P: fun_Ex8563670519792538868t_bool] :
      ( ? [X_13: extended_enat] : pp(aa_Ext2772979874139974381t_bool(P,X_13))
    <=> ? [N4: extended_enat] :
          ( pp(aa_Ext2772979874139974381t_bool(P,N4))
          & ! [M2: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,M2),N4))
             => ~ pp(aa_Ext2772979874139974381t_bool(P,M2)) ) ) ) ).

% exists_least_iff
tff(fact_535_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% dual_order.irrefl
tff(fact_536_dual__order_Oirrefl,axiom,
    ! [A: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),A)) ).

% dual_order.irrefl
tff(fact_537_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% dual_order.irrefl
tff(fact_538_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% dual_order.irrefl
tff(fact_539_dual__order_Oasym,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2)) ) ).

% dual_order.asym
tff(fact_540_dual__order_Oasym,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A))
     => ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2)) ) ).

% dual_order.asym
tff(fact_541_dual__order_Oasym,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ).

% dual_order.asym
tff(fact_542_dual__order_Oasym,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) ) ).

% dual_order.asym
tff(fact_543_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( ( X != Y )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_cases
tff(fact_544_linorder__cases,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
     => ( ( X != Y )
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),X)) ) ) ).

% linorder_cases
tff(fact_545_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ( X != Y )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_cases
tff(fact_546_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( ( X != Y )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_cases
tff(fact_547_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_548_antisym__conv3,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),X))
     => ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_549_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_550_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_551_less__induct,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),X4))
             => pp(aa_nat_bool(P,Y4)) )
         => pp(aa_nat_bool(P,X4)) )
     => pp(aa_nat_bool(P,A)) ) ).

% less_induct
tff(fact_552_less__induct,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,A: extended_enat] :
      ( ! [X4: extended_enat] :
          ( ! [Y4: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y4),X4))
             => pp(aa_Ext2772979874139974381t_bool(P,Y4)) )
         => pp(aa_Ext2772979874139974381t_bool(P,X4)) )
     => pp(aa_Ext2772979874139974381t_bool(P,A)) ) ).

% less_induct
tff(fact_553_ord__less__eq__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( ( B2 = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_554_ord__less__eq__trans,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( ( B2 = C )
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_555_ord__less__eq__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( ( B2 = C )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_556_ord__less__eq__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( ( B2 = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_557_ord__eq__less__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A = B2 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_558_ord__eq__less__trans,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( A = B2 )
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_559_ord__eq__less__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A = B2 )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_560_ord__eq__less__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A = B2 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_561_order_Oasym,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A)) ) ).

% order.asym
tff(fact_562_order_Oasym,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A)) ) ).

% order.asym
tff(fact_563_order_Oasym,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) ) ).

% order.asym
tff(fact_564_order_Oasym,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A)) ) ).

% order.asym
tff(fact_565_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_566_less__imp__neq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_567_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_568_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_569_dense,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [Z: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),Y)) ) ) ).

% dense
tff(fact_570_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X_12)) ).

% gt_ex
tff(fact_571_gt__ex,axiom,
    ! [X: real] :
    ? [X_12: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X_12)) ).

% gt_ex
tff(fact_572_gt__ex,axiom,
    ! [X: int] :
    ? [X_12: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X_12)) ).

% gt_ex
tff(fact_573_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X)) ).

% lt_ex
tff(fact_574_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X)) ).

% lt_ex
tff(fact_575_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
     => ( ( ( I = J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = X ) )
        & ( ( I != J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = nth_VEBT_VEBT(Xs,J) ) ) ) ) ).

% nth_list_update
tff(fact_576_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
     => ( ( ( I = J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = X ) )
        & ( ( I != J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = aa_nat_int(nth_int(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_577_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( ( ( I = J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = X ) )
        & ( ( I != J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_578_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,X) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = X ) ) ) ).

% list_update_same_conv
tff(fact_579_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
     => ( ( list_update_int(Xs,I,X) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_580_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( ( list_update_nat(Xs,I,X) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_581_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_582_infinite__descent,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa_nat_bool(P,N2))
         => ? [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
              & ~ pp(aa_nat_bool(P,M5)) ) )
     => pp(aa_nat_bool(P,N)) ) ).

% infinite_descent
tff(fact_583_nat__less__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
             => pp(aa_nat_bool(P,M5)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% nat_less_induct
tff(fact_584_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_irrefl_nat
tff(fact_585_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
     => ( S != T ) ) ).

% less_not_refl3
tff(fact_586_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_587_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_not_refl
tff(fact_588_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% nat_neq_iff
tff(fact_589_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B2: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B2)) )
       => ? [X4: nat] :
            ( pp(aa_nat_bool(P,X4))
            & ! [Y4: nat] :
                ( pp(aa_nat_bool(P,Y4))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),X4)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_590_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% nat_le_linear
tff(fact_591_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_592_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% eq_imp_le
tff(fact_593_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ) ).

% le_trans
tff(fact_594_le__refl,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).

% le_refl
tff(fact_595_order__le__imp__less__or__eq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X),Y))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_596_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_597_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_598_order__le__imp__less__or__eq,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_599_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_600_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_601_linorder__le__less__linear,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X),Y))
      | pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_602_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_603_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_604_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_605_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_na8797796813480665199d_enat,C: extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_606_order__less__le__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: fun_Ex5852502975177633731d_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(F,B2)),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_607_order__less__le__subst2,axiom,
    ! [A: real,B2: real,F: fun_re6065469302365115155d_enat,C: extended_enat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,B2)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_608_order__less__le__subst2,axiom,
    ! [A: int,B2: int,F: fun_in5914218760682845459d_enat,C: extended_enat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,B2)),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,X4)),aa_int_Extended_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_609_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_610_order__less__le__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: fun_Ex1451620480498224165t_real,C: real] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,B2)),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_611_order__less__le__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B2)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_612_order__less__le__subst2,axiom,
    ! [A: int,B2: int,F: fun_int_real,C: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,B2)),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_613_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_614_order__less__le__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: fun_Ex7660298398204468809at_nat,C: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,B2)),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_615_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: fun_re6065469302365115155d_enat,B2: real,C: real] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_616_order__less__le__subst1,axiom,
    ! [A: real,F: fun_real_real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_617_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_real_nat,B2: real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_real_nat(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_real_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_618_order__less__le__subst1,axiom,
    ! [A: int,F: fun_real_int,B2: real,C: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_real_int(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_real_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_619_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: fun_na8797796813480665199d_enat,B2: nat,C: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_nat_Extended_enat(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_nat_Extended_enat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_620_order__less__le__subst1,axiom,
    ! [A: real,F: fun_nat_real,B2: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_621_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_622_order__less__le__subst1,axiom,
    ! [A: int,F: fun_nat_int,B2: nat,C: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_623_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: fun_in5914218760682845459d_enat,B2: int,C: int] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_int_Extended_enat(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,X4)),aa_int_Extended_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_int_Extended_enat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_624_order__less__le__subst1,axiom,
    ! [A: real,F: fun_int_real,B2: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_625_order__le__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_re6065469302365115155d_enat,C: extended_enat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,B2)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_626_order__le__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B2)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_627_order__le__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,B2)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_628_order__le__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,B2)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_629_order__le__less__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_na8797796813480665199d_enat,C: extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_630_order__le__less__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_631_order__le__less__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_632_order__le__less__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_int,C: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,B2)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_633_order__le__less__subst2,axiom,
    ! [A: int,B2: int,F: fun_in5914218760682845459d_enat,C: extended_enat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,B2)),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,X4)),aa_int_Extended_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_634_order__le__less__subst2,axiom,
    ! [A: int,B2: int,F: fun_int_real,C: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,B2)),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_635_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: fun_na8797796813480665199d_enat,B2: nat,C: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),aa_nat_Extended_enat(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_nat_Extended_enat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_636_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: fun_Ex5852502975177633731d_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),aa_Ext7969882492699152700d_enat(F,B2)))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_Ext7969882492699152700d_enat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_637_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: fun_re6065469302365115155d_enat,B2: real,C: real] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),aa_rea1194391308913012860d_enat(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_638_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: fun_in5914218760682845459d_enat,B2: int,C: int] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),aa_int_Extended_enat(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,X4)),aa_int_Extended_enat(F,Y3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),aa_int_Extended_enat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_639_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B2: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_640_order__le__less__subst1,axiom,
    ! [A: real,F: fun_Ex1451620480498224165t_real,B2: extended_enat,C: extended_enat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_Ext2541572767299617110t_real(F,B2)))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_Ext2541572767299617110t_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_641_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_642_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B2: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_int_real(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_643_order__le__less__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_644_order__le__less__subst1,axiom,
    ! [A: nat,F: fun_Ex7660298398204468809at_nat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_Extended_enat_nat(F,B2)))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_Extended_enat_nat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_645_order__less__le__trans,axiom,
    ! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Y),Z2))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Z2)) ) ) ).

% order_less_le_trans
tff(fact_646_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2)) ) ) ).

% order_less_le_trans
tff(fact_647_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),Y))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),Z2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),Z2)) ) ) ).

% order_less_le_trans
tff(fact_648_order__less__le__trans,axiom,
    ! [X: set_int,Y: set_int,Z2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z2))
       => pp(aa_set_int_bool(ord_less_set_int(X),Z2)) ) ) ).

% order_less_le_trans
tff(fact_649_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2)) ) ) ).

% order_less_le_trans
tff(fact_650_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2)) ) ) ).

% order_less_le_trans
tff(fact_651_order__le__less__trans,axiom,
    ! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X),Y))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),Z2))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Z2)) ) ) ).

% order_le_less_trans
tff(fact_652_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2)) ) ) ).

% order_le_less_trans
tff(fact_653_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,Y),Z2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),Z2)) ) ) ).

% order_le_less_trans
tff(fact_654_order__le__less__trans,axiom,
    ! [X: set_int,Y: set_int,Z2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_set_int(Y),Z2))
       => pp(aa_set_int_bool(ord_less_set_int(X),Z2)) ) ) ).

% order_le_less_trans
tff(fact_655_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2)) ) ) ).

% order_le_less_trans
tff(fact_656_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2)) ) ) ).

% order_le_less_trans
tff(fact_657_order__neq__le__trans,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( A != B2 )
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2)) ) ) ).

% order_neq_le_trans
tff(fact_658_order__neq__le__trans,axiom,
    ! [A: real,B2: real] :
      ( ( A != B2 )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ) ).

% order_neq_le_trans
tff(fact_659_order__neq__le__trans,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A != B2 )
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),B2)) ) ) ).

% order_neq_le_trans
tff(fact_660_order__neq__le__trans,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A != B2 )
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
       => pp(aa_set_int_bool(ord_less_set_int(A),B2)) ) ) ).

% order_neq_le_trans
tff(fact_661_order__neq__le__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != B2 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2)) ) ) ).

% order_neq_le_trans
tff(fact_662_order__neq__le__trans,axiom,
    ! [A: int,B2: int] :
      ( ( A != B2 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) ) ) ).

% order_neq_le_trans
tff(fact_663_order__le__neq__trans,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
     => ( ( A != B2 )
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2)) ) ) ).

% order_le_neq_trans
tff(fact_664_order__le__neq__trans,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( ( A != B2 )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ) ).

% order_le_neq_trans
tff(fact_665_order__le__neq__trans,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
     => ( ( A != B2 )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),B2)) ) ) ).

% order_le_neq_trans
tff(fact_666_order__le__neq__trans,axiom,
    ! [A: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
     => ( ( A != B2 )
       => pp(aa_set_int_bool(ord_less_set_int(A),B2)) ) ) ).

% order_le_neq_trans
tff(fact_667_order__le__neq__trans,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( ( A != B2 )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2)) ) ) ).

% order_le_neq_trans
tff(fact_668_order__le__neq__trans,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( ( A != B2 )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) ) ) ).

% order_le_neq_trans
tff(fact_669_order__less__imp__le,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X),Y)) ) ).

% order_less_imp_le
tff(fact_670_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% order_less_imp_le
tff(fact_671_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),Y))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y)) ) ).

% order_less_imp_le
tff(fact_672_order__less__imp__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% order_less_imp_le
tff(fact_673_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_less_imp_le
tff(fact_674_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_less_imp_le
tff(fact_675_linorder__not__less,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Y),X)) ) ).

% linorder_not_less
tff(fact_676_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% linorder_not_less
tff(fact_677_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_not_less
tff(fact_678_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_not_less
tff(fact_679_linorder__not__le,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X),Y))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),X)) ) ).

% linorder_not_le
tff(fact_680_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_not_le
tff(fact_681_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_not_le
tff(fact_682_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_not_le
tff(fact_683_order__less__le,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_684_order__less__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_685_order__less__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),Y))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_686_order__less__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_687_order__less__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_688_order__less__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_689_order__le__less,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X),Y))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_690_order__le__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_691_order__le__less,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_692_order__le__less,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_693_order__le__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_694_order__le__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_695_dual__order_Ostrict__implies__order,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),A)) ) ).

% dual_order.strict_implies_order
tff(fact_696_dual__order_Ostrict__implies__order,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ).

% dual_order.strict_implies_order
tff(fact_697_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B2),A))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A)) ) ).

% dual_order.strict_implies_order
tff(fact_698_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B2),A))
     => pp(aa_set_int_bool(ord_less_eq_set_int(B2),A)) ) ).

% dual_order.strict_implies_order
tff(fact_699_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A)) ) ).

% dual_order.strict_implies_order
tff(fact_700_dual__order_Ostrict__implies__order,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A)) ) ).

% dual_order.strict_implies_order
tff(fact_701_order_Ostrict__implies__order,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2)) ) ).

% order.strict_implies_order
tff(fact_702_order_Ostrict__implies__order,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ).

% order.strict_implies_order
tff(fact_703_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),B2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2)) ) ).

% order.strict_implies_order
tff(fact_704_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A),B2)) ) ).

% order.strict_implies_order
tff(fact_705_order_Ostrict__implies__order,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2)) ) ).

% order.strict_implies_order
tff(fact_706_order_Ostrict__implies__order,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ).

% order.strict_implies_order
tff(fact_707_dual__order_Ostrict__iff__not,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),A))
        & ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_708_dual__order_Ostrict__iff__not,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_709_dual__order_Ostrict__iff__not,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B2),A))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A))
        & ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_710_dual__order_Ostrict__iff__not,axiom,
    ! [B2: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B2),A))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_711_dual__order_Ostrict__iff__not,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_712_dual__order_Ostrict__iff__not,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_713_dual__order_Ostrict__trans2,axiom,
    ! [B2: extended_enat,A: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,C),B2))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_714_dual__order_Ostrict__trans2,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_715_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_nat,A: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B2),A))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),B2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_716_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_int,A: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B2),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C),B2))
       => pp(aa_set_int_bool(ord_less_set_int(C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_717_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_718_dual__order_Ostrict__trans2,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_719_dual__order_Ostrict__trans1,axiom,
    ! [B2: extended_enat,A: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),A))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,C),B2))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_720_dual__order_Ostrict__trans1,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_721_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_nat,A: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),B2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_722_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_int,A: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A))
     => ( pp(aa_set_int_bool(ord_less_set_int(C),B2))
       => pp(aa_set_int_bool(ord_less_set_int(C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_723_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_724_dual__order_Ostrict__trans1,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_725_dual__order_Ostrict__iff__order,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),A))
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_726_dual__order_Ostrict__iff__order,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_727_dual__order_Ostrict__iff__order,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B2),A))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A))
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_728_dual__order_Ostrict__iff__order,axiom,
    ! [B2: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B2),A))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A))
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_729_dual__order_Ostrict__iff__order,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_730_dual__order_Ostrict__iff__order,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_731_dual__order_Oorder__iff__strict,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),A))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A))
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_732_dual__order_Oorder__iff__strict,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_733_dual__order_Oorder__iff__strict,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B2),A))
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_734_dual__order_Oorder__iff__strict,axiom,
    ! [B2: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(B2),A))
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_735_dual__order_Oorder__iff__strict,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_736_dual__order_Oorder__iff__strict,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_737_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ! [W: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),W))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),Y))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z2)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2)) ) ) ).

% dense_le_bounded
tff(fact_738_dense__ge__bounded,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
     => ( ! [W: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),W))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),X))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),W)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2)) ) ) ).

% dense_ge_bounded
tff(fact_739_order_Ostrict__iff__not,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
        & ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),A)) ) ) ).

% order.strict_iff_not
tff(fact_740_order_Ostrict__iff__not,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ) ).

% order.strict_iff_not
tff(fact_741_order_Ostrict__iff__not,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),B2))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
        & ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A)) ) ) ).

% order.strict_iff_not
tff(fact_742_order_Ostrict__iff__not,axiom,
    ! [A: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(B2),A)) ) ) ).

% order.strict_iff_not
tff(fact_743_order_Ostrict__iff__not,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A)) ) ) ).

% order.strict_iff_not
tff(fact_744_order_Ostrict__iff__not,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A)) ) ) ).

% order.strict_iff_not
tff(fact_745_order_Ostrict__trans2,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),C))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_746_order_Ostrict__trans2,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans2
tff(fact_747_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),C))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_748_order_Ostrict__trans2,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C))
       => pp(aa_set_int_bool(ord_less_set_int(A),C)) ) ) ).

% order.strict_trans2
tff(fact_749_order_Ostrict__trans2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_750_order_Ostrict__trans2,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans2
tff(fact_751_order_Ostrict__trans1,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_752_order_Ostrict__trans1,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans1
tff(fact_753_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B2),C))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_754_order_Ostrict__trans1,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
     => ( pp(aa_set_int_bool(ord_less_set_int(B2),C))
       => pp(aa_set_int_bool(ord_less_set_int(A),C)) ) ) ).

% order.strict_trans1
tff(fact_755_order_Ostrict__trans1,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_756_order_Ostrict__trans1,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans1
tff(fact_757_order_Ostrict__iff__order,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_758_order_Ostrict__iff__order,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_759_order_Ostrict__iff__order,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),B2))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_760_order_Ostrict__iff__order,axiom,
    ! [A: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_761_order_Ostrict__iff__order,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_762_order_Ostrict__iff__order,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_763_order_Oorder__iff__strict,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_764_order_Oorder__iff__strict,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_765_order_Oorder__iff__strict,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),B2))
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_766_order_Oorder__iff__strict,axiom,
    ! [A: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(A),B2))
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_767_order_Oorder__iff__strict,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_768_order_Oorder__iff__strict,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_769_not__le__imp__less,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Y),X))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y)) ) ).

% not_le_imp_less
tff(fact_770_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% not_le_imp_less
tff(fact_771_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% not_le_imp_less
tff(fact_772_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% not_le_imp_less
tff(fact_773_less__le__not__le,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X),Y))
        & ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_774_less__le__not__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ).

% less_le_not_le
tff(fact_775_less__le__not__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),Y))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
        & ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_776_less__le__not__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(Y),X)) ) ) ).

% less_le_not_le
tff(fact_777_less__le__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_778_less__le__not__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% less_le_not_le
tff(fact_779_dense__le,axiom,
    ! [Y: real,Z2: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Z2)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2)) ) ).

% dense_le
tff(fact_780_dense__ge,axiom,
    ! [Z2: real,Y: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2)) ) ).

% dense_ge
tff(fact_781_antisym__conv2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X),Y))
     => ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_782_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_783_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
     => ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_784_antisym__conv2,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( ~ pp(aa_set_int_bool(ord_less_set_int(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_785_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_786_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_787_antisym__conv1,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_788_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_789_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),Y))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_790_antisym__conv1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ~ pp(aa_set_int_bool(ord_less_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_791_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_792_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_793_nless__le,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
    <=> ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_794_nless__le,axiom,
    ! [A: real,B2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_795_nless__le,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),B2))
    <=> ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_796_nless__le,axiom,
    ! [A: set_int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(ord_less_set_int(A),B2))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_797_nless__le,axiom,
    ! [A: nat,B2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_798_nless__le,axiom,
    ! [A: int,B2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_799_leI,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Y),X)) ) ).

% leI
tff(fact_800_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% leI
tff(fact_801_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% leI
tff(fact_802_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% leI
tff(fact_803_leD,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Y),X))
     => ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Y)) ) ).

% leD
tff(fact_804_leD,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% leD
tff(fact_805_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),X))
     => ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),Y)) ) ).

% leD
tff(fact_806_leD,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ~ pp(aa_set_int_bool(ord_less_set_int(X),Y)) ) ).

% leD
tff(fact_807_leD,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% leD
tff(fact_808_leD,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% leD
tff(fact_809_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I4: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),J3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,I4)),aa_nat_nat(F,J3))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_810_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ( M != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% le_neq_implies_less
tff(fact_811_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_or_eq_imp_le
tff(fact_812_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_813_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_buildup(N)),N)) ) ).

% buildup_gives_valid
tff(fact_814_add__less__same__cancel1,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A)),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel1
tff(fact_815_add__less__same__cancel1,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(B2),A)),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel1
tff(fact_816_add__less__same__cancel1,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel1
tff(fact_817_add__less__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel2
tff(fact_818_add__less__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),B2)),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel2
tff(fact_819_add__less__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel2
tff(fact_820_less__add__same__cancel1,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2)) ) ).

% less_add_same_cancel1
tff(fact_821_less__add__same__cancel1,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(plus_plus_real(A),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) ) ).

% less_add_same_cancel1
tff(fact_822_less__add__same__cancel1,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ).

% less_add_same_cancel1
tff(fact_823_less__add__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2)) ) ).

% less_add_same_cancel2
tff(fact_824_less__add__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(plus_plus_real(B2),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) ) ).

% less_add_same_cancel2
tff(fact_825_less__add__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ).

% less_add_same_cancel2
tff(fact_826_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_827_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_828_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_829_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_830_add__le__same__cancel1,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(B2),A)),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel1
tff(fact_831_add__le__same__cancel1,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A)),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel1
tff(fact_832_add__le__same__cancel1,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel1
tff(fact_833_add__le__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),B2)),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel2
tff(fact_834_add__le__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel2
tff(fact_835_add__le__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel2
tff(fact_836_le__add__same__cancel1,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(plus_plus_real(A),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) ) ).

% le_add_same_cancel1
tff(fact_837_le__add__same__cancel1,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2)) ) ).

% le_add_same_cancel1
tff(fact_838_le__add__same__cancel1,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) ) ).

% le_add_same_cancel1
tff(fact_839_le__add__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(plus_plus_real(B2),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) ) ).

% le_add_same_cancel2
tff(fact_840_le__add__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2)) ) ).

% le_add_same_cancel2
tff(fact_841_le__add__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) ) ).

% le_add_same_cancel2
tff(fact_842_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_843_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_844_add__left__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B2 = C ) ) ).

% add_left_cancel
tff(fact_845_add__left__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B2 = C ) ) ).

% add_left_cancel
tff(fact_846_add__left__cancel,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B2) = aa_real_real(plus_plus_real(A),C) )
    <=> ( B2 = C ) ) ).

% add_left_cancel
tff(fact_847_add__right__cancel,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B2 = C ) ) ).

% add_right_cancel
tff(fact_848_add__right__cancel,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B2 = C ) ) ).

% add_right_cancel
tff(fact_849_add__right__cancel,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B2),A) = aa_real_real(plus_plus_real(C),A) )
    <=> ( B2 = C ) ) ).

% add_right_cancel
tff(fact_850_le__zero__eq,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,N),zero_z5237406670263579293d_enat))
    <=> ( N = zero_z5237406670263579293d_enat ) ) ).

% le_zero_eq
tff(fact_851_le__zero__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_852_not__gr__zero,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_853_not__gr__zero,axiom,
    ! [N: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),N))
    <=> ( N = zero_z5237406670263579293d_enat ) ) ).

% not_gr_zero
tff(fact_854_add__le__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ).

% add_le_cancel_left
tff(fact_855_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2)) ) ).

% add_le_cancel_left
tff(fact_856_add__le__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ).

% add_le_cancel_left
tff(fact_857_add__le__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B2),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ).

% add_le_cancel_right
tff(fact_858_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2)) ) ).

% add_le_cancel_right
tff(fact_859_add__le__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ).

% add_le_cancel_right
tff(fact_860_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_861_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_862_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_863_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_864_add_Oright__neutral,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),zero_z5237406670263579293d_enat) = A ).

% add.right_neutral
tff(fact_865_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(plus_plus_real(A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_866_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_867_add__cancel__left__left,axiom,
    ! [B2: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) = A )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_868_add__cancel__left__left,axiom,
    ! [B2: real,A: real] :
      ( ( aa_real_real(plus_plus_real(B2),A) = A )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_869_add__cancel__left__left,axiom,
    ! [B2: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) = A )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_870_add__cancel__left__left,axiom,
    ! [B2: complex,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(B2),A) = A )
    <=> ( B2 = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_871_add__cancel__left__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = A )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_872_add__cancel__left__right,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(plus_plus_real(A),B2) = A )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_873_add__cancel__left__right,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = A )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_874_add__cancel__left__right,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B2) = A )
    <=> ( B2 = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_875_add__cancel__right__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_876_add__cancel__right__left,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(plus_plus_real(B2),A) )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_877_add__cancel__right__left,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_878_add__cancel__right__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(B2),A) )
    <=> ( B2 = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_879_add__cancel__right__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_880_add__cancel__right__right,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(plus_plus_real(A),B2) )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_881_add__cancel__right__right,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_882_add__cancel__right__right,axiom,
    ! [A: complex,B2: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(A),B2) )
    <=> ( B2 = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_883_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_884_add__eq__0__iff__both__eq__0,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),Y) = zero_z5237406670263579293d_enat )
    <=> ( ( X = zero_z5237406670263579293d_enat )
        & ( Y = zero_z5237406670263579293d_enat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_885_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_886_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( zero_z5237406670263579293d_enat = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),Y) )
    <=> ( ( X = zero_z5237406670263579293d_enat )
        & ( Y = zero_z5237406670263579293d_enat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_887_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_888_add__0,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% add_0
tff(fact_889_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_890_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_891_add__0,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(zero_z5237406670263579293d_enat),A) = A ).

% add_0
tff(fact_892_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2)) ) ).

% add_less_cancel_left
tff(fact_893_add__less__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ).

% add_less_cancel_left
tff(fact_894_add__less__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) ) ).

% add_less_cancel_left
tff(fact_895_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2)) ) ).

% add_less_cancel_right
tff(fact_896_add__less__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B2),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ).

% add_less_cancel_right
tff(fact_897_add__less__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) ) ).

% add_less_cancel_right
tff(fact_898_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_899_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_900_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_901_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
    <=> ( X = zero_zero_real ) ) ).

% zero_reorient
tff(fact_902_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_903_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
    <=> ( X = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_904_zero__reorient,axiom,
    ! [X: extended_enat] :
      ( ( zero_z5237406670263579293d_enat = X )
    <=> ( X = zero_z5237406670263579293d_enat ) ) ).

% zero_reorient
tff(fact_905_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B2: 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),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_906_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B2: 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),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_907_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B2)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B2),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_908_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)),C) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B2),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_909_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_910_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_911_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_real_real(plus_plus_real(I),K) = aa_real_real(plus_plus_real(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_912_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(I),K) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_913_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B2: 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),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)) ) ) ).

% group_cancel.add1
tff(fact_914_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B2: 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),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% group_cancel.add1
tff(fact_915_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B2: real] :
      ( ( A2 = aa_real_real(plus_plus_real(K),A) )
     => ( aa_real_real(plus_plus_real(A2),B2) = aa_real_real(plus_plus_real(K),aa_real_real(plus_plus_real(A),B2)) ) ) ).

% group_cancel.add1
tff(fact_916_group__cancel_Oadd1,axiom,
    ! [A2: extended_enat,K: extended_enat,A: extended_enat,B2: extended_enat] :
      ( ( A2 = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(K),A) )
     => ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A2),B2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(K),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)) ) ) ).

% group_cancel.add1
tff(fact_917_group__cancel_Oadd2,axiom,
    ! [B: nat,K: nat,B2: nat,A: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B2) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),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),B2)) ) ) ).

% group_cancel.add2
tff(fact_918_group__cancel_Oadd2,axiom,
    ! [B: int,K: int,B2: int,A: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B2) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),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),B2)) ) ) ).

% group_cancel.add2
tff(fact_919_group__cancel_Oadd2,axiom,
    ! [B: real,K: real,B2: real,A: real] :
      ( ( B = aa_real_real(plus_plus_real(K),B2) )
     => ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(K),aa_real_real(plus_plus_real(A),B2)) ) ) ).

% group_cancel.add2
tff(fact_920_group__cancel_Oadd2,axiom,
    ! [B: extended_enat,K: extended_enat,B2: extended_enat,A: extended_enat] :
      ( ( B = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(K),B2) )
     => ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(K),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)) ) ) ).

% group_cancel.add2
tff(fact_921_add_Oassoc,axiom,
    ! [A: nat,B2: 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),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) ).

% add.assoc
tff(fact_922_add_Oassoc,axiom,
    ! [A: int,B2: 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),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ).

% add.assoc
tff(fact_923_add_Oassoc,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B2)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B2),C)) ).

% add.assoc
tff(fact_924_add_Oassoc,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)),C) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B2),C)) ).

% add.assoc
tff(fact_925_add_Oleft__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B2 = C ) ) ).

% add.left_cancel
tff(fact_926_add_Oleft__cancel,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B2) = aa_real_real(plus_plus_real(A),C) )
    <=> ( B2 = C ) ) ).

% add.left_cancel
tff(fact_927_add_Oright__cancel,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B2 = C ) ) ).

% add.right_cancel
tff(fact_928_add_Oright__cancel,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B2),A) = aa_real_real(plus_plus_real(C),A) )
    <=> ( B2 = C ) ) ).

% add.right_cancel
tff(fact_929_add_Ocommute,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) ).

% add.commute
tff(fact_930_add_Ocommute,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) ).

% add.commute
tff(fact_931_add_Ocommute,axiom,
    ! [A: real,B2: real] : aa_real_real(plus_plus_real(A),B2) = aa_real_real(plus_plus_real(B2),A) ).

% add.commute
tff(fact_932_add_Ocommute,axiom,
    ! [A: extended_enat,B2: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B2),A) ).

% add.commute
tff(fact_933_add_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),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,B2),C)) ).

% add.left_commute
tff(fact_934_add_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),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,B2),C)) ).

% add.left_commute
tff(fact_935_add_Oleft__commute,axiom,
    ! [B2: real,A: real,C: real] : aa_real_real(plus_plus_real(B2),aa_real_real(plus_plus_real(A),C)) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B2),C)) ).

% add.left_commute
tff(fact_936_add_Oleft__commute,axiom,
    ! [B2: extended_enat,A: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B2),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B2),C)) ).

% add.left_commute
tff(fact_937_add__left__imp__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
tff(fact_938_add__left__imp__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
tff(fact_939_add__left__imp__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B2) = aa_real_real(plus_plus_real(A),C) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
tff(fact_940_add__right__imp__eq,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
tff(fact_941_add__right__imp__eq,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
tff(fact_942_add__right__imp__eq,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B2),A) = aa_real_real(plus_plus_real(C),A) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
tff(fact_943_zero__le,axiom,
    ! [X: extended_enat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),X)) ).

% zero_le
tff(fact_944_zero__le,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X)) ).

% zero_le
tff(fact_945_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr_zeroI
tff(fact_946_gr__zeroI,axiom,
    ! [N: extended_enat] :
      ( ( N != zero_z5237406670263579293d_enat )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),N)) ) ).

% gr_zeroI
tff(fact_947_not__less__zero,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less_zero
tff(fact_948_not__less__zero,axiom,
    ! [N: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,N),zero_z5237406670263579293d_enat)) ).

% not_less_zero
tff(fact_949_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_950_gr__implies__not__zero,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,M),N))
     => ( N != zero_z5237406670263579293d_enat ) ) ).

% gr_implies_not_zero
tff(fact_951_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_952_zero__less__iff__neq__zero,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),N))
    <=> ( N != zero_z5237406670263579293d_enat ) ) ).

% zero_less_iff_neq_zero
tff(fact_953_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,I),J))
        & ( K = L ) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(I),K)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_954_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_955_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_956_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_957_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( I = J )
        & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,K),L)) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(I),K)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_958_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_959_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_960_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_961_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,I),J))
        & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,K),L)) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(I),K)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_962_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_963_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_964_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_965_add__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,C),D))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B2),D))) ) ) ).

% add_mono
tff(fact_966_add__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B2),D))) ) ) ).

% add_mono
tff(fact_967_add__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D))) ) ) ).

% add_mono
tff(fact_968_add__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D))) ) ) ).

% add_mono
tff(fact_969_add__left__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(C),A)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(C),B2))) ) ).

% add_left_mono
tff(fact_970_add__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B2))) ) ).

% add_left_mono
tff(fact_971_add__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2))) ) ).

% add_left_mono
tff(fact_972_add__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))) ) ).

% add_left_mono
tff(fact_973_less__eqE,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
     => ~ ! [C2: extended_enat] : B2 != aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C2) ) ).

% less_eqE
tff(fact_974_less__eqE,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ~ ! [C2: nat] : B2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% less_eqE
tff(fact_975_add__right__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B2),C))) ) ).

% add_right_mono
tff(fact_976_add__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B2),C))) ) ).

% add_right_mono
tff(fact_977_add__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C))) ) ).

% add_right_mono
tff(fact_978_add__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))) ) ).

% add_right_mono
tff(fact_979_le__iff__add,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
    <=> ? [C3: extended_enat] : B2 = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C3) ) ).

% le_iff_add
tff(fact_980_le__iff__add,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
    <=> ? [C3: nat] : B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% le_iff_add
tff(fact_981_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B2)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ).

% add_le_imp_le_left
tff(fact_982_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2)) ) ).

% add_le_imp_le_left
tff(fact_983_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ).

% add_le_imp_le_left
tff(fact_984_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B2),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ).

% add_le_imp_le_right
tff(fact_985_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2)) ) ).

% add_le_imp_le_right
tff(fact_986_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ).

% add_le_imp_le_right
tff(fact_987_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_988_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_989_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_990_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_991_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(zero_z5237406670263579293d_enat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_992_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_993_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_994_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_995_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_996_add_Ocomm__neutral,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),zero_z5237406670263579293d_enat) = A ).

% add.comm_neutral
tff(fact_997_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_998_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_999_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_1000_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1001_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1002_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1003_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1004_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1005_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1006_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1007_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1008_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1009_add__strict__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D))) ) ) ).

% add_strict_mono
tff(fact_1010_add__strict__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,C),D))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B2),D))) ) ) ).

% add_strict_mono
tff(fact_1011_add__strict__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B2),D))) ) ) ).

% add_strict_mono
tff(fact_1012_add__strict__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D))) ) ) ).

% add_strict_mono
tff(fact_1013_add__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2))) ) ).

% add_strict_left_mono
tff(fact_1014_add__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B2))) ) ).

% add_strict_left_mono
tff(fact_1015_add__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))) ) ).

% add_strict_left_mono
tff(fact_1016_add__strict__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C))) ) ).

% add_strict_right_mono
tff(fact_1017_add__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B2),C))) ) ).

% add_strict_right_mono
tff(fact_1018_add__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))) ) ).

% add_strict_right_mono
tff(fact_1019_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2)) ) ).

% add_less_imp_less_left
tff(fact_1020_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B2)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ).

% add_less_imp_less_left
tff(fact_1021_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) ) ).

% add_less_imp_less_left
tff(fact_1022_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2)) ) ).

% add_less_imp_less_right
tff(fact_1023_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B2),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ).

% add_less_imp_less_right
tff(fact_1024_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) ) ).

% add_less_imp_less_right
tff(fact_1025_add__nonpos__eq__0__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X),zero_z5237406670263579293d_enat))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Y),zero_z5237406670263579293d_enat))
       => ( ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),Y) = zero_z5237406670263579293d_enat )
        <=> ( ( X = zero_z5237406670263579293d_enat )
            & ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1026_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( ( aa_real_real(plus_plus_real(X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1027_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),zero_zero_nat))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1028_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1029_add__nonneg__eq__0__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),X))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),Y))
       => ( ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),Y) = zero_z5237406670263579293d_enat )
        <=> ( ( X = zero_z5237406670263579293d_enat )
            & ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1030_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(plus_plus_real(X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1031_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1032_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1033_add__nonpos__nonpos,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),zero_z5237406670263579293d_enat))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),zero_z5237406670263579293d_enat))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)),zero_z5237406670263579293d_enat)) ) ) ).

% add_nonpos_nonpos
tff(fact_1034_add__nonpos__nonpos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),B2)),zero_zero_real)) ) ) ).

% add_nonpos_nonpos
tff(fact_1035_add__nonpos__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),zero_zero_nat)) ) ) ).

% add_nonpos_nonpos
tff(fact_1036_add__nonpos__nonpos,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),zero_zero_int)) ) ) ).

% add_nonpos_nonpos
tff(fact_1037_add__nonneg__nonneg,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),A))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),B2))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2))) ) ) ).

% add_nonneg_nonneg
tff(fact_1038_add__nonneg__nonneg,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(A),B2))) ) ) ).

% add_nonneg_nonneg
tff(fact_1039_add__nonneg__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2))) ) ) ).

% add_nonneg_nonneg
tff(fact_1040_add__nonneg__nonneg,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))) ) ) ).

% add_nonneg_nonneg
tff(fact_1041_add__increasing2,axiom,
    ! [C: extended_enat,B2: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),C))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),A))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C))) ) ) ).

% add_increasing2
tff(fact_1042_add__increasing2,axiom,
    ! [C: real,B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(plus_plus_real(A),C))) ) ) ).

% add_increasing2
tff(fact_1043_add__increasing2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_increasing2
tff(fact_1044_add__increasing2,axiom,
    ! [C: int,B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_increasing2
tff(fact_1045_add__decreasing2,axiom,
    ! [C: extended_enat,A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,C),zero_z5237406670263579293d_enat))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)),B2)) ) ) ).

% add_decreasing2
tff(fact_1046_add__decreasing2,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),B2)) ) ) ).

% add_decreasing2
tff(fact_1047_add__decreasing2,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B2)) ) ) ).

% add_decreasing2
tff(fact_1048_add__decreasing2,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B2)) ) ) ).

% add_decreasing2
tff(fact_1049_add__increasing,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),A))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),C))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C))) ) ) ).

% add_increasing
tff(fact_1050_add__increasing,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(plus_plus_real(A),C))) ) ) ).

% add_increasing
tff(fact_1051_add__increasing,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_increasing
tff(fact_1052_add__increasing,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_increasing
tff(fact_1053_add__decreasing,axiom,
    ! [A: extended_enat,C: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),zero_z5237406670263579293d_enat))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,C),B2))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)),B2)) ) ) ).

% add_decreasing
tff(fact_1054_add__decreasing,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),B2)) ) ) ).

% add_decreasing
tff(fact_1055_add__decreasing,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B2)) ) ) ).

% add_decreasing
tff(fact_1056_add__decreasing,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B2)) ) ) ).

% add_decreasing
tff(fact_1057_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1058_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1059_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1060_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1061_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1062_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1063_add__le__less__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B2),D))) ) ) ).

% add_le_less_mono
tff(fact_1064_add__le__less__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D))) ) ) ).

% add_le_less_mono
tff(fact_1065_add__le__less__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D))) ) ) ).

% add_le_less_mono
tff(fact_1066_add__less__le__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B2),D))) ) ) ).

% add_less_le_mono
tff(fact_1067_add__less__le__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D))) ) ) ).

% add_less_le_mono
tff(fact_1068_add__less__le__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D))) ) ) ).

% add_less_le_mono
tff(fact_1069_pos__add__strict,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% pos_add_strict
tff(fact_1070_pos__add__strict,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),A))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),C))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C))) ) ) ).

% pos_add_strict
tff(fact_1071_pos__add__strict,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(plus_plus_real(A),C))) ) ) ).

% pos_add_strict
tff(fact_1072_pos__add__strict,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% pos_add_strict
tff(fact_1073_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ~ ! [C2: nat] :
            ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1074_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ~ ! [C2: extended_enat] :
            ( ( B2 = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C2) )
           => ( C2 = zero_z5237406670263579293d_enat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1075_add__pos__pos,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2))) ) ) ).

% add_pos_pos
tff(fact_1076_add__pos__pos,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),A))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),B2))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2))) ) ) ).

% add_pos_pos
tff(fact_1077_add__pos__pos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),B2))) ) ) ).

% add_pos_pos
tff(fact_1078_add__pos__pos,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))) ) ) ).

% add_pos_pos
tff(fact_1079_add__neg__neg,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),zero_zero_nat)) ) ) ).

% add_neg_neg
tff(fact_1080_add__neg__neg,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),zero_z5237406670263579293d_enat))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),zero_z5237406670263579293d_enat))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)),zero_z5237406670263579293d_enat)) ) ) ).

% add_neg_neg
tff(fact_1081_add__neg__neg,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),B2)),zero_zero_real)) ) ) ).

% add_neg_neg
tff(fact_1082_add__neg__neg,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),zero_zero_int)) ) ) ).

% add_neg_neg
tff(fact_1083_add__neg__nonpos,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),zero_z5237406670263579293d_enat))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),zero_z5237406670263579293d_enat))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)),zero_z5237406670263579293d_enat)) ) ) ).

% add_neg_nonpos
tff(fact_1084_add__neg__nonpos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),B2)),zero_zero_real)) ) ) ).

% add_neg_nonpos
tff(fact_1085_add__neg__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),zero_zero_nat)) ) ) ).

% add_neg_nonpos
tff(fact_1086_add__neg__nonpos,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),zero_zero_int)) ) ) ).

% add_neg_nonpos
tff(fact_1087_add__nonneg__pos,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),A))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),B2))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2))) ) ) ).

% add_nonneg_pos
tff(fact_1088_add__nonneg__pos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),B2))) ) ) ).

% add_nonneg_pos
tff(fact_1089_add__nonneg__pos,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2))) ) ) ).

% add_nonneg_pos
tff(fact_1090_add__nonneg__pos,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))) ) ) ).

% add_nonneg_pos
tff(fact_1091_add__nonpos__neg,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),zero_z5237406670263579293d_enat))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),zero_z5237406670263579293d_enat))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)),zero_z5237406670263579293d_enat)) ) ) ).

% add_nonpos_neg
tff(fact_1092_add__nonpos__neg,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),B2)),zero_zero_real)) ) ) ).

% add_nonpos_neg
tff(fact_1093_add__nonpos__neg,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),zero_zero_nat)) ) ) ).

% add_nonpos_neg
tff(fact_1094_add__nonpos__neg,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),zero_zero_int)) ) ) ).

% add_nonpos_neg
tff(fact_1095_add__pos__nonneg,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),A))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),B2))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2))) ) ) ).

% add_pos_nonneg
tff(fact_1096_add__pos__nonneg,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),B2))) ) ) ).

% add_pos_nonneg
tff(fact_1097_add__pos__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2))) ) ) ).

% add_pos_nonneg
tff(fact_1098_add__pos__nonneg,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))) ) ) ).

% add_pos_nonneg
tff(fact_1099_add__strict__increasing,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(plus_plus_real(A),C))) ) ) ).

% add_strict_increasing
tff(fact_1100_add__strict__increasing,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_strict_increasing
tff(fact_1101_add__strict__increasing,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_strict_increasing
tff(fact_1102_add__strict__increasing2,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(plus_plus_real(A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1103_add__strict__increasing2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1104_add__strict__increasing2,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1105_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(plus_plus_real(A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_1106_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_1107_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_leaf
tff(fact_1108_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(plus_plus_real(Y),E))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_epsilon
tff(fact_1109_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(X),Y)),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real)) ) ) ).

% add_less_zeroD
tff(fact_1110_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),zero_zero_int)) ) ) ).

% add_less_zeroD
tff(fact_1111_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_1112_subsetI,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),B)) )
     => pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B)) ) ).

% subsetI
tff(fact_1113_subsetI,axiom,
    ! [A2: set_real,B: set_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_set_real_bool(member_real(X4),B)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),B)) ) ).

% subsetI
tff(fact_1114_subsetI,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(X4),B)) )
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B)) ) ).

% subsetI
tff(fact_1115_subsetI,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_set_nat_bool(member_nat(X4),B)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B)) ) ).

% subsetI
tff(fact_1116_subsetI,axiom,
    ! [A2: set_int,B: set_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_set_int_bool(member_int(X4),B)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B)) ) ).

% subsetI
tff(fact_1117_psubsetI,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
     => ( ( A2 != B )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B)) ) ) ).

% psubsetI
tff(fact_1118_psubsetI,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
     => ( ( A2 != B )
       => pp(aa_set_int_bool(ord_less_set_int(A2),B)) ) ) ).

% psubsetI
tff(fact_1119_subset__antisym,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A2))
       => ( A2 = B ) ) ) ).

% subset_antisym
tff(fact_1120_subset__antisym,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A2))
       => ( A2 = B ) ) ) ).

% subset_antisym
tff(fact_1121_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_min_max
tff(fact_1122_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_set_nat_bool(finite_finite_nat,vEBT_VEBT_set_vebt(T))) ) ).

% set_vebt_finite
tff(fact_1123_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
    <=> ( pp(aa_nat_bool(vEBT_V5719532721284313246member(T),X))
        | pp(aa_nat_bool(vEBT_VEBT_membermima(T),X)) ) ) ).

% both_member_options_def
tff(fact_1124_empty__Collect__eq,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( bot_bot_set_list_nat = collect_list_nat(P) )
    <=> ! [X5: list_nat] : ~ pp(aa_list_nat_bool(P,X5)) ) ).

% empty_Collect_eq
tff(fact_1125_empty__Collect__eq,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( bot_bot_set_set_nat = collect_set_nat(P) )
    <=> ! [X5: set_nat] : ~ pp(aa_set_nat_bool(P,X5)) ) ).

% empty_Collect_eq
tff(fact_1126_empty__Collect__eq,axiom,
    ! [P: fun_Ex8563670519792538868t_bool] :
      ( ( bot_bo7653980558646680370d_enat = collec4429806609662206161d_enat(P) )
    <=> ! [X5: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(P,X5)) ) ).

% empty_Collect_eq
tff(fact_1127_empty__Collect__eq,axiom,
    ! [P: fun_real_bool] :
      ( ( bot_bot_set_real = collect_real(P) )
    <=> ! [X5: real] : ~ pp(aa_real_bool(P,X5)) ) ).

% empty_Collect_eq
tff(fact_1128_empty__Collect__eq,axiom,
    ! [P: fun_nat_bool] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X5: nat] : ~ pp(aa_nat_bool(P,X5)) ) ).

% empty_Collect_eq
tff(fact_1129_empty__Collect__eq,axiom,
    ! [P: fun_int_bool] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X5: int] : ~ pp(aa_int_bool(P,X5)) ) ).

% empty_Collect_eq
tff(fact_1130_Collect__empty__eq,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ! [X5: list_nat] : ~ pp(aa_list_nat_bool(P,X5)) ) ).

% Collect_empty_eq
tff(fact_1131_Collect__empty__eq,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ! [X5: set_nat] : ~ pp(aa_set_nat_bool(P,X5)) ) ).

% Collect_empty_eq
tff(fact_1132_Collect__empty__eq,axiom,
    ! [P: fun_Ex8563670519792538868t_bool] :
      ( ( collec4429806609662206161d_enat(P) = bot_bo7653980558646680370d_enat )
    <=> ! [X5: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(P,X5)) ) ).

% Collect_empty_eq
tff(fact_1133_Collect__empty__eq,axiom,
    ! [P: fun_real_bool] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ! [X5: real] : ~ pp(aa_real_bool(P,X5)) ) ).

% Collect_empty_eq
tff(fact_1134_Collect__empty__eq,axiom,
    ! [P: fun_nat_bool] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X5: nat] : ~ pp(aa_nat_bool(P,X5)) ) ).

% Collect_empty_eq
tff(fact_1135_Collect__empty__eq,axiom,
    ! [P: fun_int_bool] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X5: int] : ~ pp(aa_int_bool(P,X5)) ) ).

% Collect_empty_eq
tff(fact_1136_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ! [X5: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(X5),A2))
    <=> ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_1137_all__not__in__conv,axiom,
    ! [A2: set_Extended_enat] :
      ( ! [X5: extended_enat] : ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
    <=> ( A2 = bot_bo7653980558646680370d_enat ) ) ).

% all_not_in_conv
tff(fact_1138_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ! [X5: real] : ~ pp(aa_set_real_bool(member_real(X5),A2))
    <=> ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_1139_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ! [X5: nat] : ~ pp(aa_set_nat_bool(member_nat(X5),A2))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_1140_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ! [X5: int] : ~ pp(aa_set_int_bool(member_int(X5),A2))
    <=> ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_1141_empty__iff,axiom,
    ! [C: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(C),bot_bot_set_set_nat)) ).

% empty_iff
tff(fact_1142_empty__iff,axiom,
    ! [C: extended_enat] : ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),bot_bo7653980558646680370d_enat)) ).

% empty_iff
tff(fact_1143_empty__iff,axiom,
    ! [C: real] : ~ pp(aa_set_real_bool(member_real(C),bot_bot_set_real)) ).

% empty_iff
tff(fact_1144_empty__iff,axiom,
    ! [C: nat] : ~ pp(aa_set_nat_bool(member_nat(C),bot_bot_set_nat)) ).

% empty_iff
tff(fact_1145_empty__iff,axiom,
    ! [C: int] : ~ pp(aa_set_int_bool(member_int(C),bot_bot_set_int)) ).

% empty_iff
tff(fact_1146_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
       => ( pp(aa_nat_bool(vEBT_V5719532721284313246member(Tree),X))
          | pp(aa_nat_bool(vEBT_VEBT_membermima(Tree),X)) ) ) ) ).

% member_valid_both_member_options
tff(fact_1147_empty__subsetI,axiom,
    ! [A2: set_Extended_enat] : pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(bot_bo7653980558646680370d_enat),A2)) ).

% empty_subsetI
tff(fact_1148_empty__subsetI,axiom,
    ! [A2: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(bot_bot_set_real),A2)) ).

% empty_subsetI
tff(fact_1149_empty__subsetI,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,bot_bot_set_nat),A2)) ).

% empty_subsetI
tff(fact_1150_empty__subsetI,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A2)) ).

% empty_subsetI
tff(fact_1151_subset__empty,axiom,
    ! [A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),bot_bo7653980558646680370d_enat))
    <=> ( A2 = bot_bo7653980558646680370d_enat ) ) ).

% subset_empty
tff(fact_1152_subset__empty,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),bot_bot_set_real))
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_1153_subset__empty,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),bot_bot_set_nat))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1154_subset__empty,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),bot_bot_set_int))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_1155_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_1156_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : pp(aa_set_nat_bool(finite_finite_nat,set_nat2(Xs))) ).

% List.finite_set
tff(fact_1157_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : pp(aa_set_complex_bool(finite3207457112153483333omplex,set_complex2(Xs))) ).

% List.finite_set
tff(fact_1158_List_Ofinite__set,axiom,
    ! [Xs: list_int] : pp(aa_set_int_bool(finite_finite_int,set_int2(Xs))) ).

% List.finite_set
tff(fact_1159_List_Ofinite__set,axiom,
    ! [Xs: list_Extended_enat] : pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,set_Extended_enat2(Xs))) ).

% List.finite_set
tff(fact_1160_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),S2))
            & ~ ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),S2))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1161_ex__min__if__finite,axiom,
    ! [S2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
            & ~ ? [Xa: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),S2))
                  & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1162_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),S2))
            & ~ ? [Xa: real] :
                  ( pp(aa_set_real_bool(member_real(Xa),S2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1163_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),S2))
            & ~ ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),S2))
                  & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1164_not__psubset__empty,axiom,
    ! [A2: set_Extended_enat] : ~ pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(A2),bot_bo7653980558646680370d_enat)) ).

% not_psubset_empty
tff(fact_1165_not__psubset__empty,axiom,
    ! [A2: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A2),bot_bot_set_real)) ).

% not_psubset_empty
tff(fact_1166_not__psubset__empty,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),bot_bot_set_nat)) ).

% not_psubset_empty
tff(fact_1167_not__psubset__empty,axiom,
    ! [A2: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A2),bot_bot_set_int)) ).

% not_psubset_empty
tff(fact_1168_psubsetD,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,C: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(A2),B))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),A2))
       => pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),B)) ) ) ).

% psubsetD
tff(fact_1169_psubsetD,axiom,
    ! [A2: set_real,B: set_real,C: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A2),B))
     => ( pp(aa_set_real_bool(member_real(C),A2))
       => pp(aa_set_real_bool(member_real(C),B)) ) ) ).

% psubsetD
tff(fact_1170_psubsetD,axiom,
    ! [A2: set_set_nat,B: set_set_nat,C: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),B))
     => ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
       => pp(aa_set_set_nat_bool(member_set_nat(C),B)) ) ) ).

% psubsetD
tff(fact_1171_psubsetD,axiom,
    ! [A2: set_nat,B: set_nat,C: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B))
     => ( pp(aa_set_nat_bool(member_nat(C),A2))
       => pp(aa_set_nat_bool(member_nat(C),B)) ) ) ).

% psubsetD
tff(fact_1172_psubsetD,axiom,
    ! [A2: set_int,B: set_int,C: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B))
     => ( pp(aa_set_int_bool(member_int(C),A2))
       => pp(aa_set_int_bool(member_int(C),B)) ) ) ).

% psubsetD
tff(fact_1173_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),X6))
           => ? [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),X6))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Xa)) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,X6)) ) ) ).

% infinite_growing
tff(fact_1174_infinite__growing,axiom,
    ! [X6: set_Extended_enat] :
      ( ( X6 != bot_bo7653980558646680370d_enat )
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),X6))
           => ? [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),X6))
                & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Xa)) ) )
       => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,X6)) ) ) ).

% infinite_growing
tff(fact_1175_infinite__growing,axiom,
    ! [X6: set_real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),X6))
           => ? [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),X6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Xa)) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,X6)) ) ) ).

% infinite_growing
tff(fact_1176_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),X6))
           => ? [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),X6))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Xa)) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,X6)) ) ) ).

% infinite_growing
tff(fact_1177_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ? [X5: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(X5),A2))
    <=> ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_1178_ex__in__conv,axiom,
    ! [A2: set_Extended_enat] :
      ( ? [X5: extended_enat] : pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
    <=> ( A2 != bot_bo7653980558646680370d_enat ) ) ).

% ex_in_conv
tff(fact_1179_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ? [X5: real] : pp(aa_set_real_bool(member_real(X5),A2))
    <=> ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_1180_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ? [X5: nat] : pp(aa_set_nat_bool(member_nat(X5),A2))
    <=> ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_1181_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ? [X5: int] : pp(aa_set_int_bool(member_int(X5),A2))
    <=> ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_1182_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y3: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(Y3),A2))
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_1183_equals0I,axiom,
    ! [A2: set_Extended_enat] :
      ( ! [Y3: extended_enat] : ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y3),A2))
     => ( A2 = bot_bo7653980558646680370d_enat ) ) ).

% equals0I
tff(fact_1184_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y3: real] : ~ pp(aa_set_real_bool(member_real(Y3),A2))
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_1185_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y3: nat] : ~ pp(aa_set_nat_bool(member_nat(Y3),A2))
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_1186_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y3: int] : ~ pp(aa_set_int_bool(member_int(Y3),A2))
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_1187_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ pp(aa_set_set_nat_bool(member_set_nat(A),A2)) ) ).

% equals0D
tff(fact_1188_equals0D,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( ( A2 = bot_bo7653980558646680370d_enat )
     => ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2)) ) ).

% equals0D
tff(fact_1189_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ pp(aa_set_real_bool(member_real(A),A2)) ) ).

% equals0D
tff(fact_1190_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ pp(aa_set_nat_bool(member_nat(A),A2)) ) ).

% equals0D
tff(fact_1191_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ pp(aa_set_int_bool(member_int(A),A2)) ) ).

% equals0D
tff(fact_1192_emptyE,axiom,
    ! [A: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(A),bot_bot_set_set_nat)) ).

% emptyE
tff(fact_1193_emptyE,axiom,
    ! [A: extended_enat] : ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),bot_bo7653980558646680370d_enat)) ).

% emptyE
tff(fact_1194_emptyE,axiom,
    ! [A: real] : ~ pp(aa_set_real_bool(member_real(A),bot_bot_set_real)) ).

% emptyE
tff(fact_1195_emptyE,axiom,
    ! [A: nat] : ~ pp(aa_set_nat_bool(member_nat(A),bot_bot_set_nat)) ).

% emptyE
tff(fact_1196_emptyE,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(member_int(A),bot_bot_set_int)) ).

% emptyE
tff(fact_1197_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A),bot_bo7653980558646680370d_enat))
     => ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_uniqueI
tff(fact_1198_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),bot_bot_set_real))
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_1199_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),bot_bot_set_nat))
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1200_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),bot_bot_set_int))
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_1201_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),bot_bot_nat))
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1202_bot_Oextremum__unique,axiom,
    ! [A: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A),bot_bo7653980558646680370d_enat))
    <=> ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_unique
tff(fact_1203_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),bot_bot_set_real))
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_1204_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),bot_bot_set_nat))
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_1205_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),bot_bot_set_int))
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_1206_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),bot_bot_nat))
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_1207_bot_Oextremum,axiom,
    ! [A: set_Extended_enat] : pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(bot_bo7653980558646680370d_enat),A)) ).

% bot.extremum
tff(fact_1208_bot_Oextremum,axiom,
    ! [A: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(bot_bot_set_real),A)) ).

% bot.extremum
tff(fact_1209_bot_Oextremum,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,bot_bot_set_nat),A)) ).

% bot.extremum
tff(fact_1210_bot_Oextremum,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A)) ).

% bot.extremum
tff(fact_1211_bot_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,bot_bot_nat),A)) ).

% bot.extremum
tff(fact_1212_bot_Onot__eq__extremum,axiom,
    ! [A: set_Extended_enat] :
      ( ( A != bot_bo7653980558646680370d_enat )
    <=> pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(bot_bo7653980558646680370d_enat),A)) ) ).

% bot.not_eq_extremum
tff(fact_1213_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_set_real(bot_bot_set_real),A)) ) ).

% bot.not_eq_extremum
tff(fact_1214_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,bot_bot_set_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_1215_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_set_int(bot_bot_set_int),A)) ) ).

% bot.not_eq_extremum
tff(fact_1216_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,bot_bot_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_1217_bot_Onot__eq__extremum,axiom,
    ! [A: extended_enat] :
      ( ( A != bot_bo4199563552545308370d_enat )
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,bot_bo4199563552545308370d_enat),A)) ) ).

% bot.not_eq_extremum
tff(fact_1218_bot_Oextremum__strict,axiom,
    ! [A: set_Extended_enat] : ~ pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(A),bot_bo7653980558646680370d_enat)) ).

% bot.extremum_strict
tff(fact_1219_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A),bot_bot_set_real)) ).

% bot.extremum_strict
tff(fact_1220_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),bot_bot_set_nat)) ).

% bot.extremum_strict
tff(fact_1221_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A),bot_bot_set_int)) ).

% bot.extremum_strict
tff(fact_1222_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),bot_bot_nat)) ).

% bot.extremum_strict
tff(fact_1223_bot_Oextremum__strict,axiom,
    ! [A: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),bot_bo4199563552545308370d_enat)) ).

% bot.extremum_strict
tff(fact_1224_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A2 ) ).

% finite_list
tff(fact_1225_finite__list,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A2 ) ).

% finite_list
tff(fact_1226_finite__list,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A2 ) ).

% finite_list
tff(fact_1227_finite__list,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ? [Xs2: list_int] : set_int2(Xs2) = A2 ) ).

% finite_list
tff(fact_1228_finite__list,axiom,
    ! [A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ? [Xs2: list_Extended_enat] : set_Extended_enat2(Xs2) = A2 ) ).

% finite_list
tff(fact_1229_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1230_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1231_linordered__field__no__ub,axiom,
    ! [X2: real] :
    ? [X_12: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),X_12)) ).

% linordered_field_no_ub
tff(fact_1232_linordered__field__no__lb,axiom,
    ! [X2: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ).

% linordered_field_no_lb
tff(fact_1233_subset__iff__psubset__eq,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B))
        | ( A2 = B ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1234_subset__iff__psubset__eq,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(A2),B))
        | ( A2 = B ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1235_subset__psubset__trans,axiom,
    ! [A2: set_nat,B: set_nat,C4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B),C4))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),C4)) ) ) ).

% subset_psubset_trans
tff(fact_1236_subset__psubset__trans,axiom,
    ! [A2: set_int,B: set_int,C4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
     => ( pp(aa_set_int_bool(ord_less_set_int(B),C4))
       => pp(aa_set_int_bool(ord_less_set_int(A2),C4)) ) ) ).

% subset_psubset_trans
tff(fact_1237_subset__not__subset__eq,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
        & ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A2)) ) ) ).

% subset_not_subset_eq
tff(fact_1238_subset__not__subset__eq,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(B),A2)) ) ) ).

% subset_not_subset_eq
tff(fact_1239_psubset__subset__trans,axiom,
    ! [A2: set_nat,B: set_nat,C4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),C4))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),C4)) ) ) ).

% psubset_subset_trans
tff(fact_1240_psubset__subset__trans,axiom,
    ! [A2: set_int,B: set_int,C4: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C4))
       => pp(aa_set_int_bool(ord_less_set_int(A2),C4)) ) ) ).

% psubset_subset_trans
tff(fact_1241_psubset__imp__subset,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B)) ) ).

% psubset_imp_subset
tff(fact_1242_psubset__imp__subset,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B)) ) ).

% psubset_imp_subset
tff(fact_1243_Collect__mono__iff,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(P)),collect_real(Q)))
    <=> ! [X5: real] :
          ( pp(aa_real_bool(P,X5))
         => pp(aa_real_bool(Q,X5)) ) ) ).

% Collect_mono_iff
tff(fact_1244_Collect__mono__iff,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q))
    <=> ! [X5: list_nat] :
          ( pp(aa_list_nat_bool(P,X5))
         => pp(aa_list_nat_bool(Q,X5)) ) ) ).

% Collect_mono_iff
tff(fact_1245_Collect__mono__iff,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q)))
    <=> ! [X5: set_nat] :
          ( pp(aa_set_nat_bool(P,X5))
         => pp(aa_set_nat_bool(Q,X5)) ) ) ).

% Collect_mono_iff
tff(fact_1246_Collect__mono__iff,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q)))
    <=> ! [X5: nat] :
          ( pp(aa_nat_bool(P,X5))
         => pp(aa_nat_bool(Q,X5)) ) ) ).

% Collect_mono_iff
tff(fact_1247_Collect__mono__iff,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(P)),collect_int(Q)))
    <=> ! [X5: int] :
          ( pp(aa_int_bool(P,X5))
         => pp(aa_int_bool(Q,X5)) ) ) ).

% Collect_mono_iff
tff(fact_1248_set__eq__subset,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( A2 = B )
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A2)) ) ) ).

% set_eq_subset
tff(fact_1249_set__eq__subset,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( A2 = B )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
        & pp(aa_set_int_bool(ord_less_eq_set_int(B),A2)) ) ) ).

% set_eq_subset
tff(fact_1250_subset__trans,axiom,
    ! [A2: set_nat,B: set_nat,C4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),C4))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C4)) ) ) ).

% subset_trans
tff(fact_1251_subset__trans,axiom,
    ! [A2: set_int,B: set_int,C4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C4))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4)) ) ) ).

% subset_trans
tff(fact_1252_Collect__mono,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(P,X4))
         => pp(aa_real_bool(Q,X4)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(P)),collect_real(Q))) ) ).

% Collect_mono
tff(fact_1253_Collect__mono,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X4: list_nat] :
          ( pp(aa_list_nat_bool(P,X4))
         => pp(aa_list_nat_bool(Q,X4)) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q)) ) ).

% Collect_mono
tff(fact_1254_Collect__mono,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_nat_bool(P,X4))
         => pp(aa_set_nat_bool(Q,X4)) )
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q))) ) ).

% Collect_mono
tff(fact_1255_Collect__mono,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(P,X4))
         => pp(aa_nat_bool(Q,X4)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q))) ) ).

% Collect_mono
tff(fact_1256_Collect__mono,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(P,X4))
         => pp(aa_int_bool(Q,X4)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(P)),collect_int(Q))) ) ).

% Collect_mono
tff(fact_1257_subset__refl,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),A2)) ).

% subset_refl
tff(fact_1258_subset__refl,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A2),A2)) ).

% subset_refl
tff(fact_1259_subset__iff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
    <=> ! [T2: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(T2),A2))
         => pp(aa_set3347977610044113549t_bool(member_Extended_enat(T2),B)) ) ) ).

% subset_iff
tff(fact_1260_subset__iff,axiom,
    ! [A2: set_real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
    <=> ! [T2: real] :
          ( pp(aa_set_real_bool(member_real(T2),A2))
         => pp(aa_set_real_bool(member_real(T2),B)) ) ) ).

% subset_iff
tff(fact_1261_subset__iff,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B))
    <=> ! [T2: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(T2),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(T2),B)) ) ) ).

% subset_iff
tff(fact_1262_subset__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
    <=> ! [T2: nat] :
          ( pp(aa_set_nat_bool(member_nat(T2),A2))
         => pp(aa_set_nat_bool(member_nat(T2),B)) ) ) ).

% subset_iff
tff(fact_1263_subset__iff,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
    <=> ! [T2: int] :
          ( pp(aa_set_int_bool(member_int(T2),A2))
         => pp(aa_set_int_bool(member_int(T2),B)) ) ) ).

% subset_iff
tff(fact_1264_psubset__eq,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
        & ( A2 != B ) ) ) ).

% psubset_eq
tff(fact_1265_psubset__eq,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
        & ( A2 != B ) ) ) ).

% psubset_eq
tff(fact_1266_equalityD2,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( A2 = B )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A2)) ) ).

% equalityD2
tff(fact_1267_equalityD2,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( A2 = B )
     => pp(aa_set_int_bool(ord_less_eq_set_int(B),A2)) ) ).

% equalityD2
tff(fact_1268_equalityD1,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( A2 = B )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B)) ) ).

% equalityD1
tff(fact_1269_equalityD1,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( A2 = B )
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B)) ) ).

% equalityD1
tff(fact_1270_subset__eq,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
    <=> ! [X5: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
         => pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),B)) ) ) ).

% subset_eq
tff(fact_1271_subset__eq,axiom,
    ! [A2: set_real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
    <=> ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),A2))
         => pp(aa_set_real_bool(member_real(X5),B)) ) ) ).

% subset_eq
tff(fact_1272_subset__eq,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B))
    <=> ! [X5: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X5),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(X5),B)) ) ) ).

% subset_eq
tff(fact_1273_subset__eq,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),A2))
         => pp(aa_set_nat_bool(member_nat(X5),B)) ) ) ).

% subset_eq
tff(fact_1274_subset__eq,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
    <=> ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),A2))
         => pp(aa_set_int_bool(member_int(X5),B)) ) ) ).

% subset_eq
tff(fact_1275_equalityE,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( A2 = B )
     => ~ ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
         => ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A2)) ) ) ).

% equalityE
tff(fact_1276_equalityE,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( A2 = B )
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
         => ~ pp(aa_set_int_bool(ord_less_eq_set_int(B),A2)) ) ) ).

% equalityE
tff(fact_1277_psubsetE,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B))
     => ~ ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A2)) ) ) ).

% psubsetE
tff(fact_1278_psubsetE,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B))
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
         => pp(aa_set_int_bool(ord_less_eq_set_int(B),A2)) ) ) ).

% psubsetE
tff(fact_1279_subsetD,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,C: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),A2))
       => pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),B)) ) ) ).

% subsetD
tff(fact_1280_subsetD,axiom,
    ! [A2: set_real,B: set_real,C: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
     => ( pp(aa_set_real_bool(member_real(C),A2))
       => pp(aa_set_real_bool(member_real(C),B)) ) ) ).

% subsetD
tff(fact_1281_subsetD,axiom,
    ! [A2: set_set_nat,B: set_set_nat,C: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B))
     => ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
       => pp(aa_set_set_nat_bool(member_set_nat(C),B)) ) ) ).

% subsetD
tff(fact_1282_subsetD,axiom,
    ! [A2: set_nat,B: set_nat,C: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
     => ( pp(aa_set_nat_bool(member_nat(C),A2))
       => pp(aa_set_nat_bool(member_nat(C),B)) ) ) ).

% subsetD
tff(fact_1283_subsetD,axiom,
    ! [A2: set_int,B: set_int,C: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
     => ( pp(aa_set_int_bool(member_int(C),A2))
       => pp(aa_set_int_bool(member_int(C),B)) ) ) ).

% subsetD
tff(fact_1284_in__mono,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
       => pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),B)) ) ) ).

% in_mono
tff(fact_1285_in__mono,axiom,
    ! [A2: set_real,B: set_real,X: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => pp(aa_set_real_bool(member_real(X),B)) ) ) ).

% in_mono
tff(fact_1286_in__mono,axiom,
    ! [A2: set_set_nat,B: set_set_nat,X: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
       => pp(aa_set_set_nat_bool(member_set_nat(X),B)) ) ) ).

% in_mono
tff(fact_1287_in__mono,axiom,
    ! [A2: set_nat,B: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
     => ( pp(aa_set_nat_bool(member_nat(X),A2))
       => pp(aa_set_nat_bool(member_nat(X),B)) ) ) ).

% in_mono
tff(fact_1288_in__mono,axiom,
    ! [A2: set_int,B: set_int,X: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
     => ( pp(aa_set_int_bool(member_int(X),A2))
       => pp(aa_set_int_bool(member_int(X),B)) ) ) ).

% in_mono
tff(fact_1289_finite__has__maximal,axiom,
    ! [A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
            & ! [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),A2))
               => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1290_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1291_finite__has__maximal,axiom,
    ! [A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
            & ! [Xa: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Xa),A2))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1292_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( member_set_int(X4,A2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1293_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1294_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1295_finite__has__minimal,axiom,
    ! [A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
            & ! [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),A2))
               => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1296_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1297_finite__has__minimal,axiom,
    ! [A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
            & ! [Xa: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Xa),A2))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1298_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( member_set_int(X4,A2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1299_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1300_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1301_finite__nat__set__iff__bounded__le,axiom,
    ! [N5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
    <=> ? [M2: nat] :
        ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),M2)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1302_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ! [M2: nat] :
        ? [N4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N4))
          & pp(aa_set_nat_bool(member_nat(N4),S2)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_1303_finite__nat__set__iff__bounded,axiom,
    ! [N5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
    <=> ? [M2: nat] :
        ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),M2)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1304_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ! [M2: nat] :
        ? [N4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N4))
          & pp(aa_set_nat_bool(member_nat(N4),S2)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_1305_bounded__nat__set__is__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),N)) )
     => pp(aa_set_nat_bool(finite_finite_nat,N5)) ) ).

% bounded_nat_set_is_finite
tff(fact_1306_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),M3))
         => ? [N6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N6))
              & pp(aa_set_nat_bool(member_nat(N6),S2)) ) )
     => ~ pp(aa_set_nat_bool(finite_finite_nat,S2)) ) ).

% unbounded_k_infinite
tff(fact_1307_finite__psubset__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [A4: set_nat] :
            ( pp(aa_set_nat_bool(finite_finite_nat,A4))
           => ( ! [B4: set_nat] :
                  ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B4),A4))
                 => pp(aa_set_nat_bool(P,B4)) )
             => pp(aa_set_nat_bool(P,A4)) ) )
       => pp(aa_set_nat_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1308_finite__psubset__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [A4: set_complex] :
            ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
           => ( ! [B4: set_complex] :
                  ( pp(aa_set_complex_bool(ord_less_set_complex(B4),A4))
                 => pp(aa_set_complex_bool(P,B4)) )
             => pp(aa_set_complex_bool(P,A4)) ) )
       => pp(aa_set_complex_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1309_finite__psubset__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [A4: set_int] :
            ( pp(aa_set_int_bool(finite_finite_int,A4))
           => ( ! [B4: set_int] :
                  ( pp(aa_set_int_bool(ord_less_set_int(B4),A4))
                 => pp(aa_set_int_bool(P,B4)) )
             => pp(aa_set_int_bool(P,A4)) ) )
       => pp(aa_set_int_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1310_finite__psubset__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [A4: set_Extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
           => ( ! [B4: set_Extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(B4),A4))
                 => pp(aa_set3347977610044113549t_bool(P,B4)) )
             => pp(aa_set3347977610044113549t_bool(P,A4)) ) )
       => pp(aa_set3347977610044113549t_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1311_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),S2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X2)),aa_complex_nat(F,lattic5364784637807008409ex_nat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1312_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ~ ? [X2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),S2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,X2)),aa_Extended_enat_nat(F,lattic3845382081240766429at_nat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1313_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),S2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X2)),aa_real_nat(F,lattic5055836439445974935al_nat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1314_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),S2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,aa_set_nat_nat(lattic7446932960582359483at_nat(F),S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1315_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),S2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(F,lattic8446286672483414039nt_nat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1316_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),S2))
              & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_com3683005661447327866d_enat(F,X2)),aa_com3683005661447327866d_enat(F,lattic7796887085614042845d_enat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1317_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ~ ? [X2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),S2))
              & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X2)),aa_Ext7969882492699152700d_enat(F,lattic1996716550891908761d_enat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1318_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_re6065469302365115155d_enat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),S2))
              & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X2)),aa_rea1194391308913012860d_enat(F,lattic9066027731366277983d_enat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1319_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: fun_na8797796813480665199d_enat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),S2))
              & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X2)),aa_nat_Extended_enat(F,aa_set_nat_nat(lattic8926238025367240251d_enat(F),S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1320_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: fun_in5914218760682845459d_enat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),S2))
              & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,X2)),aa_int_Extended_enat(F,lattic6042659972569420511d_enat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1321_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo6469247547558333820t_bool) ).

% bot_set_def
tff(fact_1322_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo8343409700046576086t_bool) ).

% bot_set_def
tff(fact_1323_bot__set__def,axiom,
    bot_bo7653980558646680370d_enat = collec4429806609662206161d_enat(bot_bo8629865092333707656t_bool) ).

% bot_set_def
tff(fact_1324_bot__set__def,axiom,
    bot_bot_set_real = collect_real(bot_bo4181092155253064880l_bool) ).

% bot_set_def
tff(fact_1325_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_bool) ).

% bot_set_def
tff(fact_1326_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_bool) ).

% bot_set_def
tff(fact_1327_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_1328_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M7)
     => ? [N2: nat] :
        ! [X2: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X2,M7)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,X2)),N2)) ) ) ).

% finite_maxlen
tff(fact_1329_finite__maxlen,axiom,
    ! [M7: set_list_int] :
      ( finite3922522038869484883st_int(M7)
     => ? [N2: nat] :
        ! [X2: list_int] :
          ( member_list_int(X2,M7)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_int_nat(size_size_list_int,X2)),N2)) ) ) ).

% finite_maxlen
tff(fact_1330_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( finite8100373058378681591st_nat(M7)
     => ? [N2: nat] :
        ! [X2: list_nat] :
          ( member_list_nat(X2,M7)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_nat_nat(size_size_list_nat,X2)),N2)) ) ) ).

% finite_maxlen
tff(fact_1331_bounded__Max__nat,axiom,
    ! [P: fun_nat_bool,X: nat,M7: nat] :
      ( pp(aa_nat_bool(P,X))
     => ( ! [X4: nat] :
            ( pp(aa_nat_bool(P,X4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M7)) )
       => ~ ! [M3: nat] :
              ( pp(aa_nat_bool(P,M3))
             => ~ ! [X2: nat] :
                    ( pp(aa_nat_bool(P,X2))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),M3)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1332_finite__has__maximal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
            & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),X4))
            & ! [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),A2))
               => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1333_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1334_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ? [X4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),X4))
            & ! [Xa: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Xa),A2))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1335_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X4: set_int] :
            ( member_set_int(X4,A2)
            & pp(aa_set_int_bool(ord_less_eq_set_int(A),X4))
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1336_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X4))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1337_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X4))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1338_finite__has__minimal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
            & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X4),A))
            & ! [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),A2))
               => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1339_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),A))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1340_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ? [X4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X4),A))
            & ! [Xa: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Xa),A2))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1341_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X4: set_int] :
            ( member_set_int(X4,A2)
            & pp(aa_set_int_bool(ord_less_eq_set_int(X4),A))
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1342_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),A))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1343_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),A))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1344_finite_OemptyI,axiom,
    pp(aa_set_complex_bool(finite3207457112153483333omplex,bot_bot_set_complex)) ).

% finite.emptyI
tff(fact_1345_finite_OemptyI,axiom,
    pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,bot_bo7653980558646680370d_enat)) ).

% finite.emptyI
tff(fact_1346_finite_OemptyI,axiom,
    pp(aa_set_real_bool(finite_finite_real,bot_bot_set_real)) ).

% finite.emptyI
tff(fact_1347_finite_OemptyI,axiom,
    pp(aa_set_nat_bool(finite_finite_nat,bot_bot_set_nat)) ).

% finite.emptyI
tff(fact_1348_finite_OemptyI,axiom,
    pp(aa_set_int_bool(finite_finite_int,bot_bot_set_int)) ).

% finite.emptyI
tff(fact_1349_infinite__imp__nonempty,axiom,
    ! [S2: set_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( S2 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_1350_infinite__imp__nonempty,axiom,
    ! [S2: set_Extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( S2 != bot_bo7653980558646680370d_enat ) ) ).

% infinite_imp_nonempty
tff(fact_1351_infinite__imp__nonempty,axiom,
    ! [S2: set_real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,S2))
     => ( S2 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
tff(fact_1352_infinite__imp__nonempty,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( S2 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_1353_infinite__imp__nonempty,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
     => ( S2 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_1354_finite__transitivity__chain,axiom,
    ! [A2: set_set_nat,R: fun_se707272970763642173t_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( ! [X4: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),X4))
       => ( ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),Y3))
             => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,Y3),Z))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),Z)) ) )
         => ( ! [X4: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
               => ? [Y4: set_nat] :
                    ( pp(aa_set_set_nat_bool(member_set_nat(Y4),A2))
                    & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),Y4)) ) )
           => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1355_finite__transitivity__chain,axiom,
    ! [A2: set_complex,R: fun_co1378794544173401837x_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] : ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),X4))
       => ( ! [X4: complex,Y3: complex,Z: complex] :
              ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),Y3))
             => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y3),Z))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),Z)) ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),A2))
               => ? [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),A2))
                    & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),Y4)) ) )
           => ( A2 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1356_finite__transitivity__chain,axiom,
    ! [A2: set_Extended_enat,R: fun_Ex8598408993869023341t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [X4: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(R,X4),X4))
       => ( ! [X4: extended_enat,Y3: extended_enat,Z: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(R,X4),Y3))
             => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(R,Y3),Z))
               => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(R,X4),Z)) ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
               => ? [Y4: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y4),A2))
                    & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(R,X4),Y4)) ) )
           => ( A2 = bot_bo7653980558646680370d_enat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1357_finite__transitivity__chain,axiom,
    ! [A2: set_real,R: fun_re1255401033344146673l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X4: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),X4))
       => ( ! [X4: real,Y3: real,Z: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),Y3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y3),Z))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),Z)) ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),A2))
               => ? [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),A2))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),Y4)) ) )
           => ( A2 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1358_finite__transitivity__chain,axiom,
    ! [A2: set_nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X4: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),X4))
       => ( ! [X4: nat,Y3: nat,Z: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y3),Z))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Z)) ) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),A2))
               => ? [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),A2))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y4)) ) )
           => ( A2 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1359_finite__transitivity__chain,axiom,
    ! [A2: set_int,R: fun_int_fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(R,X4),X4))
       => ( ! [X4: int,Y3: int,Z: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Y3))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(R,Y3),Z))
               => pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Z)) ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),A2))
               => ? [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),A2))
                    & pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Y4)) ) )
           => ( A2 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1360_finite__subset,axiom,
    ! [A2: set_complex,B: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% finite_subset
tff(fact_1361_finite__subset,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2)) ) ) ).

% finite_subset
tff(fact_1362_finite__subset,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% finite_subset
tff(fact_1363_finite__subset,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% finite_subset
tff(fact_1364_infinite__super,axiom,
    ! [S2: set_complex,T3: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,T3)) ) ) ).

% infinite_super
tff(fact_1365_infinite__super,axiom,
    ! [S2: set_Extended_enat,T3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
     => ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
       => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3)) ) ) ).

% infinite_super
tff(fact_1366_infinite__super,axiom,
    ! [S2: set_nat,T3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,T3)) ) ) ).

% infinite_super
tff(fact_1367_infinite__super,axiom,
    ! [S2: set_int,T3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T3))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
       => ~ pp(aa_set_int_bool(finite_finite_int,T3)) ) ) ).

% infinite_super
tff(fact_1368_rev__finite__subset,axiom,
    ! [B: set_complex,A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% rev_finite_subset
tff(fact_1369_rev__finite__subset,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2)) ) ) ).

% rev_finite_subset
tff(fact_1370_rev__finite__subset,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
       => pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% rev_finite_subset
tff(fact_1371_rev__finite__subset,axiom,
    ! [B: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
       => pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% rev_finite_subset
tff(fact_1372_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,lattic8794016678065449205x_real(F,S2))),aa_complex_real(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1373_arg__min__least,axiom,
    ! [S2: set_Extended_enat,Y: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y),S2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,lattic1189837152898106425t_real(F,S2))),aa_Ext2541572767299617110t_real(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1374_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(member_real(Y),S2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,lattic8440615504127631091l_real(F,S2))),aa_real_real(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1375_arg__min__least,axiom,
    ! [S2: set_nat,Y: nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(member_nat(Y),S2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,aa_set_nat_nat(lattic488527866317076247t_real(F),S2))),aa_nat_real(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1376_arg__min__least,axiom,
    ! [S2: set_int,Y: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ( pp(aa_set_int_bool(member_int(Y),S2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,lattic2675449441010098035t_real(F,S2))),aa_int_real(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1377_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,lattic5364784637807008409ex_nat(F,S2))),aa_complex_nat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1378_arg__min__least,axiom,
    ! [S2: set_Extended_enat,Y: extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y),S2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,lattic3845382081240766429at_nat(F,S2))),aa_Extended_enat_nat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1379_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(member_real(Y),S2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,lattic5055836439445974935al_nat(F,S2))),aa_real_nat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1380_arg__min__least,axiom,
    ! [S2: set_nat,Y: nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(member_nat(Y),S2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,aa_set_nat_nat(lattic7446932960582359483at_nat(F),S2))),aa_nat_nat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1381_arg__min__least,axiom,
    ! [S2: set_int,Y: int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ( pp(aa_set_int_bool(member_int(Y),S2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,lattic8446286672483414039nt_nat(F,S2))),aa_int_nat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1382_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),zero_zero_nat))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B2)) ) ) ) ).

% Euclid_induct
tff(fact_1383_nat__descend__induct,axiom,
    ! [N: nat,P: fun_nat_bool,M: nat] :
      ( ! [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K3))
         => pp(aa_nat_bool(P,K3)) )
     => ( ! [K3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N))
           => ( ! [I5: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),I5))
                 => pp(aa_nat_bool(P,I5)) )
             => pp(aa_nat_bool(P,K3)) ) )
       => pp(aa_nat_bool(P,M)) ) ) ).

% nat_descend_induct
tff(fact_1384_subset__emptyI,axiom,
    ! [A2: set_set_nat] :
      ( ! [X4: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),bot_bot_set_set_nat)) ) ).

% subset_emptyI
tff(fact_1385_subset__emptyI,axiom,
    ! [A2: set_Extended_enat] :
      ( ! [X4: extended_enat] : ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
     => pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),bot_bo7653980558646680370d_enat)) ) ).

% subset_emptyI
tff(fact_1386_subset__emptyI,axiom,
    ! [A2: set_real] :
      ( ! [X4: real] : ~ pp(aa_set_real_bool(member_real(X4),A2))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),bot_bot_set_real)) ) ).

% subset_emptyI
tff(fact_1387_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X4: nat] : ~ pp(aa_set_nat_bool(member_nat(X4),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),bot_bot_set_nat)) ) ).

% subset_emptyI
tff(fact_1388_subset__emptyI,axiom,
    ! [A2: set_int] :
      ( ! [X4: int] : ~ pp(aa_set_int_bool(member_int(X4),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),bot_bot_set_int)) ) ).

% subset_emptyI
tff(fact_1389_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N2: nat] : X != aa_nat_nat(suc,N2) ) ).

% list_decode.cases
tff(fact_1390_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va: nat] : X != aa_nat_nat(suc,aa_nat_nat(suc,Va)) ) ) ).

% vebt_buildup.cases
tff(fact_1391_exists__least__lemma,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_nat_bool(P,zero_zero_nat))
     => ( ? [X_1: nat] : pp(aa_nat_bool(P,X_1))
       => ? [N2: nat] :
            ( ~ pp(aa_nat_bool(P,N2))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_1392_add__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) )
    <=> ( A = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_1393_add__0__iff,axiom,
    ! [B2: real,A: real] :
      ( ( B2 = aa_real_real(plus_plus_real(B2),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_1394_add__0__iff,axiom,
    ! [B2: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_1395_add__0__iff,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 = aa_complex_complex(plus_plus_complex(B2),A) )
    <=> ( A = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_1396_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_1397_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_1398_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_1399_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_1400_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_zero_nat)) ).

% less_numeral_extra(3)
tff(fact_1401_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),zero_z5237406670263579293d_enat)) ).

% less_numeral_extra(3)
tff(fact_1402_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_zero_real)) ).

% less_numeral_extra(3)
tff(fact_1403_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_numeral_extra(3)
tff(fact_1404_field__lbound__gt__zero,axiom,
    ! [D1: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
       => ? [E: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D1))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1405_verit__la__disequality,axiom,
    ! [A: real,B2: real] :
      ( ( A = B2 )
      | ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
      | ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ).

% verit_la_disequality
tff(fact_1406_verit__la__disequality,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A)) ) ).

% verit_la_disequality
tff(fact_1407_verit__la__disequality,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A)) ) ).

% verit_la_disequality
tff(fact_1408_verit__comp__simplify1_I2_J,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1409_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1410_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1411_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1412_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1413_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1414_verit__comp__simplify1_I1_J,axiom,
    ! [A: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1415_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1416_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1417_is__num__normalize_I1_J,axiom,
    ! [A: int,B2: 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),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ).

% is_num_normalize(1)
tff(fact_1418_is__num__normalize_I1_J,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B2)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B2),C)) ).

% is_num_normalize(1)
tff(fact_1419_le__numeral__extra_I3_J,axiom,
    pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),zero_z5237406670263579293d_enat)) ).

% le_numeral_extra(3)
tff(fact_1420_le__numeral__extra_I3_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_zero_real)) ).

% le_numeral_extra(3)
tff(fact_1421_le__numeral__extra_I3_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_zero_nat)) ).

% le_numeral_extra(3)
tff(fact_1422_le__numeral__extra_I3_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% le_numeral_extra(3)
tff(fact_1423_verit__comp__simplify1_I3_J,axiom,
    ! [B5: extended_enat,A5: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B5),A5))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1424_verit__comp__simplify1_I3_J,axiom,
    ! [B5: real,A5: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B5),A5))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1425_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A5: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B5),A5))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1426_verit__comp__simplify1_I3_J,axiom,
    ! [B5: int,A5: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),A5))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1427_Collect__empty__eq__bot,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ( P = bot_bo6469247547558333820t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1428_Collect__empty__eq__bot,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ( P = bot_bo8343409700046576086t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1429_Collect__empty__eq__bot,axiom,
    ! [P: fun_Ex8563670519792538868t_bool] :
      ( ( collec4429806609662206161d_enat(P) = bot_bo7653980558646680370d_enat )
    <=> ( P = bot_bo8629865092333707656t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1430_Collect__empty__eq__bot,axiom,
    ! [P: fun_real_bool] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ( P = bot_bo4181092155253064880l_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1431_Collect__empty__eq__bot,axiom,
    ! [P: fun_nat_bool] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ( P = bot_bot_fun_nat_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1432_Collect__empty__eq__bot,axiom,
    ! [P: fun_int_bool] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ( P = bot_bot_fun_int_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1433_bot__empty__eq,axiom,
    ! [X2: set_nat] :
      ( pp(aa_set_nat_bool(bot_bo8343409700046576086t_bool,X2))
    <=> pp(aa_set_set_nat_bool(member_set_nat(X2),bot_bot_set_set_nat)) ) ).

% bot_empty_eq
tff(fact_1434_bot__empty__eq,axiom,
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(bot_bo8629865092333707656t_bool,X2))
    <=> pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),bot_bo7653980558646680370d_enat)) ) ).

% bot_empty_eq
tff(fact_1435_bot__empty__eq,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(bot_bo4181092155253064880l_bool,X2))
    <=> pp(aa_set_real_bool(member_real(X2),bot_bot_set_real)) ) ).

% bot_empty_eq
tff(fact_1436_bot__empty__eq,axiom,
    ! [X2: nat] :
      ( pp(aa_nat_bool(bot_bot_fun_nat_bool,X2))
    <=> pp(aa_set_nat_bool(member_nat(X2),bot_bot_set_nat)) ) ).

% bot_empty_eq
tff(fact_1437_bot__empty__eq,axiom,
    ! [X2: int] :
      ( pp(aa_int_bool(bot_bot_fun_int_bool,X2))
    <=> pp(aa_set_int_bool(member_int(X2),bot_bot_set_int)) ) ).

% bot_empty_eq
tff(fact_1438_triangle__Suc,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,N)),aa_nat_nat(suc,N)) ).

% triangle_Suc
tff(fact_1439_complete__interval,axiom,
    ! [A: extended_enat,B2: extended_enat,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(P,A))
       => ( ~ pp(aa_Ext2772979874139974381t_bool(P,B2))
         => ? [C2: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),C2))
              & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,C2),B2))
              & ! [X2: extended_enat] :
                  ( ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),X2))
                    & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),C2)) )
                 => pp(aa_Ext2772979874139974381t_bool(P,X2)) )
              & ! [D3: extended_enat] :
                  ( ! [X4: extended_enat] :
                      ( ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),X4))
                        & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),D3)) )
                     => pp(aa_Ext2772979874139974381t_bool(P,X4)) )
                 => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,D3),C2)) ) ) ) ) ) ).

% complete_interval
tff(fact_1440_complete__interval,axiom,
    ! [A: real,B2: real,P: fun_real_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(P,A))
       => ( ~ pp(aa_real_bool(P,B2))
         => ? [C2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),B2))
              & ! [X2: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),C2)) )
                 => pp(aa_real_bool(P,X2)) )
              & ! [D3: real] :
                  ( ! [X4: real] :
                      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
                        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),D3)) )
                     => pp(aa_real_bool(P,X4)) )
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D3),C2)) ) ) ) ) ) ).

% complete_interval
tff(fact_1441_complete__interval,axiom,
    ! [A: nat,B2: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(P,A))
       => ( ~ pp(aa_nat_bool(P,B2))
         => ? [C2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),B2))
              & ! [X2: nat] :
                  ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X2))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),C2)) )
                 => pp(aa_nat_bool(P,X2)) )
              & ! [D3: nat] :
                  ( ! [X4: nat] :
                      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X4))
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),D3)) )
                     => pp(aa_nat_bool(P,X4)) )
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D3),C2)) ) ) ) ) ) ).

% complete_interval
tff(fact_1442_complete__interval,axiom,
    ! [A: int,B2: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_int_bool(P,A))
       => ( ~ pp(aa_int_bool(P,B2))
         => ? [C2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),B2))
              & ! [X2: int] :
                  ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X2))
                    & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),C2)) )
                 => pp(aa_int_bool(P,X2)) )
              & ! [D3: int] :
                  ( ! [X4: int] :
                      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X4))
                        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),D3)) )
                     => pp(aa_int_bool(P,X4)) )
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D3),C2)) ) ) ) ) ) ).

% complete_interval
tff(fact_1443_pinf_I6_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z),X2))
     => ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X2),T)) ) ).

% pinf(6)
tff(fact_1444_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),T)) ) ).

% pinf(6)
tff(fact_1445_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),T)) ) ).

% pinf(6)
tff(fact_1446_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),T)) ) ).

% pinf(6)
tff(fact_1447_pinf_I8_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z),X2))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,T),X2)) ) ).

% pinf(8)
tff(fact_1448_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X2)) ) ).

% pinf(8)
tff(fact_1449_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X2)) ) ).

% pinf(8)
tff(fact_1450_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X2)) ) ).

% pinf(8)
tff(fact_1451_minf_I6_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),Z))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X2),T)) ) ).

% minf(6)
tff(fact_1452_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),T)) ) ).

% minf(6)
tff(fact_1453_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),T)) ) ).

% minf(6)
tff(fact_1454_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),T)) ) ).

% minf(6)
tff(fact_1455_minf_I8_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),Z))
     => ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,T),X2)) ) ).

% minf(8)
tff(fact_1456_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X2)) ) ).

% minf(8)
tff(fact_1457_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X2)) ) ).

% minf(8)
tff(fact_1458_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X2)) ) ).

% minf(8)
tff(fact_1459_count__notin,axiom,
    ! [X: extended_enat,Xs: list_Extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),set_Extended_enat2(Xs)))
     => ( aa_Extended_enat_nat(count_101369445342291426d_enat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1460_count__notin,axiom,
    ! [X: real,Xs: list_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
     => ( aa_real_nat(count_list_real(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1461_count__notin,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
     => ( aa_set_nat_nat(count_list_set_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1462_count__notin,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
     => ( aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1463_count__notin,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
     => ( aa_int_nat(count_list_int(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1464_count__notin,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => ( aa_nat_nat(count_list_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1465__C10_C,axiom,
    vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,mi),ma))),deg,list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na),vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,mi),na))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na))),vEBT_vebt_insert(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),summary)) ).

% "10"
tff(fact_1466_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_1467_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,N))))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList2,S3) ) ).

% deg_SUcn_Node
tff(fact_1468_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg))
     => ( ( Mi = Ma )
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
             => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_1)) )
          & ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_1469_max__bot2,axiom,
    ! [X: set_Extended_enat] : ord_ma4205026669011143323d_enat(X,bot_bo7653980558646680370d_enat) = X ).

% max_bot2
tff(fact_1470_max__bot2,axiom,
    ! [X: set_real] : ord_max_set_real(X,bot_bot_set_real) = X ).

% max_bot2
tff(fact_1471_max__bot2,axiom,
    ! [X: set_nat] : ord_max_set_nat(X,bot_bot_set_nat) = X ).

% max_bot2
tff(fact_1472_max__bot2,axiom,
    ! [X: set_int] : ord_max_set_int(X,bot_bot_set_int) = X ).

% max_bot2
tff(fact_1473_max__bot2,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),bot_bot_nat) = X ).

% max_bot2
tff(fact_1474_max__bot,axiom,
    ! [X: set_Extended_enat] : ord_ma4205026669011143323d_enat(bot_bo7653980558646680370d_enat,X) = X ).

% max_bot
tff(fact_1475_max__bot,axiom,
    ! [X: set_real] : ord_max_set_real(bot_bot_set_real,X) = X ).

% max_bot
tff(fact_1476_max__bot,axiom,
    ! [X: set_nat] : ord_max_set_nat(bot_bot_set_nat,X) = X ).

% max_bot
tff(fact_1477_max__bot,axiom,
    ! [X: set_int] : ord_max_set_int(bot_bot_set_int,X) = X ).

% max_bot
tff(fact_1478_max__bot,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),X) = X ).

% max_bot
tff(fact_1479_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)) ).

% max_Suc_Suc
tff(fact_1480_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_1481_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_1482_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) )
    <=> ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_1483_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_1484_max__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).

% max_0L
tff(fact_1485_max__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).

% max_0R
tff(fact_1486_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_1487_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb)),X))
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_1488_max__def,axiom,
    ! [A: real,B2: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
       => ( aa_real_real(ord_max_real(A),B2) = B2 ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
       => ( aa_real_real(ord_max_real(A),B2) = A ) ) ) ).

% max_def
tff(fact_1489_max__def,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
       => ( ord_max_set_nat(A,B2) = B2 ) )
      & ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
       => ( ord_max_set_nat(A,B2) = A ) ) ) ).

% max_def
tff(fact_1490_max__def,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
       => ( ord_max_set_int(A,B2) = B2 ) )
      & ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
       => ( ord_max_set_int(A,B2) = A ) ) ) ).

% max_def
tff(fact_1491_max__def,axiom,
    ! [A: nat,B2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = B2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = A ) ) ) ).

% max_def
tff(fact_1492_max__def,axiom,
    ! [A: int,B2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = B2 ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = A ) ) ) ).

% max_def
tff(fact_1493_max__absorb1,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => ( aa_real_real(ord_max_real(X),Y) = X ) ) ).

% max_absorb1
tff(fact_1494_max__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),X))
     => ( ord_max_set_nat(X,Y) = X ) ) ).

% max_absorb1
tff(fact_1495_max__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ( ord_max_set_int(X,Y) = X ) ) ).

% max_absorb1
tff(fact_1496_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_1497_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = X ) ) ).

% max_absorb1
tff(fact_1498_max__absorb2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( aa_real_real(ord_max_real(X),Y) = Y ) ) ).

% max_absorb2
tff(fact_1499_max__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
     => ( ord_max_set_nat(X,Y) = Y ) ) ).

% max_absorb2
tff(fact_1500_max__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( ord_max_set_int(X,Y) = Y ) ) ).

% max_absorb2
tff(fact_1501_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_1502_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_1503_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z2: real] : aa_real_real(plus_plus_real(aa_real_real(ord_max_real(X),Y)),Z2) = aa_real_real(ord_max_real(aa_real_real(plus_plus_real(X),Z2)),aa_real_real(plus_plus_real(Y),Z2)) ).

% max_add_distrib_left
tff(fact_1504_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z2)) ).

% max_add_distrib_left
tff(fact_1505_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z2) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z2)) ).

% max_add_distrib_left
tff(fact_1506_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z2: real] : aa_real_real(plus_plus_real(X),aa_real_real(ord_max_real(Y),Z2)) = aa_real_real(ord_max_real(aa_real_real(plus_plus_real(X),Y)),aa_real_real(plus_plus_real(X),Z2)) ).

% max_add_distrib_right
tff(fact_1507_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z2)) ).

% max_add_distrib_right
tff(fact_1508_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z2)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z2)) ).

% max_add_distrib_right
tff(fact_1509_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q3)) ).

% nat_add_max_left
tff(fact_1510_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ).

% nat_add_max_right
tff(fact_1511_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux)) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_1512_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X2)) ) ).

% minf(7)
tff(fact_1513_minf_I7_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),Z))
     => ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,T),X2)) ) ).

% minf(7)
tff(fact_1514_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X2)) ) ).

% minf(7)
tff(fact_1515_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X2)) ) ).

% minf(7)
tff(fact_1516_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),T)) ) ).

% minf(5)
tff(fact_1517_minf_I5_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),Z))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),T)) ) ).

% minf(5)
tff(fact_1518_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),T)) ) ).

% minf(5)
tff(fact_1519_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),T)) ) ).

% minf(5)
tff(fact_1520_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1521_minf_I4_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),Z))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1522_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1523_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1524_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1525_minf_I3_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),Z))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1526_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1527_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1528_minf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q4: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P2,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q4,X4)) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
           => ( ( pp(aa_nat_bool(P,X2))
                | pp(aa_nat_bool(Q,X2)) )
            <=> ( pp(aa_nat_bool(P2,X2))
                | pp(aa_nat_bool(Q4,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1529_minf_I2_J,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,P2: fun_Ex8563670519792538868t_bool,Q: fun_Ex8563670519792538868t_bool,Q4: fun_Ex8563670519792538868t_bool] :
      ( ? [Z3: extended_enat] :
        ! [X4: extended_enat] :
          ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Z3))
         => ( pp(aa_Ext2772979874139974381t_bool(P,X4))
          <=> pp(aa_Ext2772979874139974381t_bool(P2,X4)) ) )
     => ( ? [Z3: extended_enat] :
          ! [X4: extended_enat] :
            ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Z3))
           => ( pp(aa_Ext2772979874139974381t_bool(Q,X4))
            <=> pp(aa_Ext2772979874139974381t_bool(Q4,X4)) ) )
       => ? [Z: extended_enat] :
          ! [X2: extended_enat] :
            ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),Z))
           => ( ( pp(aa_Ext2772979874139974381t_bool(P,X2))
                | pp(aa_Ext2772979874139974381t_bool(Q,X2)) )
            <=> ( pp(aa_Ext2772979874139974381t_bool(P2,X2))
                | pp(aa_Ext2772979874139974381t_bool(Q4,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1530_minf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q4: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q4,X4)) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
           => ( ( pp(aa_real_bool(P,X2))
                | pp(aa_real_bool(Q,X2)) )
            <=> ( pp(aa_real_bool(P2,X2))
                | pp(aa_real_bool(Q4,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1531_minf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q4: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P2,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q4,X4)) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
           => ( ( pp(aa_int_bool(P,X2))
                | pp(aa_int_bool(Q,X2)) )
            <=> ( pp(aa_int_bool(P2,X2))
                | pp(aa_int_bool(Q4,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1532_minf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q4: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P2,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q4,X4)) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
           => ( ( pp(aa_nat_bool(P,X2))
                & pp(aa_nat_bool(Q,X2)) )
            <=> ( pp(aa_nat_bool(P2,X2))
                & pp(aa_nat_bool(Q4,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1533_minf_I1_J,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,P2: fun_Ex8563670519792538868t_bool,Q: fun_Ex8563670519792538868t_bool,Q4: fun_Ex8563670519792538868t_bool] :
      ( ? [Z3: extended_enat] :
        ! [X4: extended_enat] :
          ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Z3))
         => ( pp(aa_Ext2772979874139974381t_bool(P,X4))
          <=> pp(aa_Ext2772979874139974381t_bool(P2,X4)) ) )
     => ( ? [Z3: extended_enat] :
          ! [X4: extended_enat] :
            ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),Z3))
           => ( pp(aa_Ext2772979874139974381t_bool(Q,X4))
            <=> pp(aa_Ext2772979874139974381t_bool(Q4,X4)) ) )
       => ? [Z: extended_enat] :
          ! [X2: extended_enat] :
            ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),Z))
           => ( ( pp(aa_Ext2772979874139974381t_bool(P,X2))
                & pp(aa_Ext2772979874139974381t_bool(Q,X2)) )
            <=> ( pp(aa_Ext2772979874139974381t_bool(P2,X2))
                & pp(aa_Ext2772979874139974381t_bool(Q4,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1534_minf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q4: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q4,X4)) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
           => ( ( pp(aa_real_bool(P,X2))
                & pp(aa_real_bool(Q,X2)) )
            <=> ( pp(aa_real_bool(P2,X2))
                & pp(aa_real_bool(Q4,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1535_minf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q4: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P2,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q4,X4)) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
           => ( ( pp(aa_int_bool(P,X2))
                & pp(aa_int_bool(Q,X2)) )
            <=> ( pp(aa_int_bool(P2,X2))
                & pp(aa_int_bool(Q4,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1536_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X2)) ) ).

% pinf(7)
tff(fact_1537_pinf_I7_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z),X2))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,T),X2)) ) ).

% pinf(7)
tff(fact_1538_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X2)) ) ).

% pinf(7)
tff(fact_1539_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X2)) ) ).

% pinf(7)
tff(fact_1540_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),T)) ) ).

% pinf(5)
tff(fact_1541_pinf_I5_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z),X2))
     => ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),T)) ) ).

% pinf(5)
tff(fact_1542_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),T)) ) ).

% pinf(5)
tff(fact_1543_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),T)) ) ).

% pinf(5)
tff(fact_1544_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1545_pinf_I4_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1546_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1547_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1548_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1549_pinf_I3_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1550_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1551_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1552_pinf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q4: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P2,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q4,X4)) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
           => ( ( pp(aa_nat_bool(P,X2))
                | pp(aa_nat_bool(Q,X2)) )
            <=> ( pp(aa_nat_bool(P2,X2))
                | pp(aa_nat_bool(Q4,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1553_pinf_I2_J,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,P2: fun_Ex8563670519792538868t_bool,Q: fun_Ex8563670519792538868t_bool,Q4: fun_Ex8563670519792538868t_bool] :
      ( ? [Z3: extended_enat] :
        ! [X4: extended_enat] :
          ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z3),X4))
         => ( pp(aa_Ext2772979874139974381t_bool(P,X4))
          <=> pp(aa_Ext2772979874139974381t_bool(P2,X4)) ) )
     => ( ? [Z3: extended_enat] :
          ! [X4: extended_enat] :
            ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z3),X4))
           => ( pp(aa_Ext2772979874139974381t_bool(Q,X4))
            <=> pp(aa_Ext2772979874139974381t_bool(Q4,X4)) ) )
       => ? [Z: extended_enat] :
          ! [X2: extended_enat] :
            ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z),X2))
           => ( ( pp(aa_Ext2772979874139974381t_bool(P,X2))
                | pp(aa_Ext2772979874139974381t_bool(Q,X2)) )
            <=> ( pp(aa_Ext2772979874139974381t_bool(P2,X2))
                | pp(aa_Ext2772979874139974381t_bool(Q4,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1554_pinf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q4: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q4,X4)) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
           => ( ( pp(aa_real_bool(P,X2))
                | pp(aa_real_bool(Q,X2)) )
            <=> ( pp(aa_real_bool(P2,X2))
                | pp(aa_real_bool(Q4,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1555_pinf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q4: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P2,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q4,X4)) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
           => ( ( pp(aa_int_bool(P,X2))
                | pp(aa_int_bool(Q,X2)) )
            <=> ( pp(aa_int_bool(P2,X2))
                | pp(aa_int_bool(Q4,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1556_pinf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q4: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P2,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q4,X4)) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
           => ( ( pp(aa_nat_bool(P,X2))
                & pp(aa_nat_bool(Q,X2)) )
            <=> ( pp(aa_nat_bool(P2,X2))
                & pp(aa_nat_bool(Q4,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1557_pinf_I1_J,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,P2: fun_Ex8563670519792538868t_bool,Q: fun_Ex8563670519792538868t_bool,Q4: fun_Ex8563670519792538868t_bool] :
      ( ? [Z3: extended_enat] :
        ! [X4: extended_enat] :
          ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z3),X4))
         => ( pp(aa_Ext2772979874139974381t_bool(P,X4))
          <=> pp(aa_Ext2772979874139974381t_bool(P2,X4)) ) )
     => ( ? [Z3: extended_enat] :
          ! [X4: extended_enat] :
            ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z3),X4))
           => ( pp(aa_Ext2772979874139974381t_bool(Q,X4))
            <=> pp(aa_Ext2772979874139974381t_bool(Q4,X4)) ) )
       => ? [Z: extended_enat] :
          ! [X2: extended_enat] :
            ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z),X2))
           => ( ( pp(aa_Ext2772979874139974381t_bool(P,X2))
                & pp(aa_Ext2772979874139974381t_bool(Q,X2)) )
            <=> ( pp(aa_Ext2772979874139974381t_bool(P2,X2))
                & pp(aa_Ext2772979874139974381t_bool(Q4,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1558_pinf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q4: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q4,X4)) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
           => ( ( pp(aa_real_bool(P,X2))
                & pp(aa_real_bool(Q,X2)) )
            <=> ( pp(aa_real_bool(P2,X2))
                & pp(aa_real_bool(Q4,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1559_pinf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q4: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P2,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q4,X4)) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
           => ( ( pp(aa_int_bool(P,X2))
                & pp(aa_int_bool(Q,X2)) )
            <=> ( pp(aa_int_bool(P2,X2))
                & pp(aa_int_bool(Q4,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1560_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B3))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A)) ) ).

% ex_gt_or_lt
tff(fact_1561_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),X)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))) ).

% count_le_length
tff(fact_1562_count__le__length,axiom,
    ! [Xs: list_int,X: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(count_list_int(Xs),X)),aa_list_int_nat(size_size_list_int,Xs))) ).

% count_le_length
tff(fact_1563_count__le__length,axiom,
    ! [Xs: list_nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(count_list_nat(Xs),X)),aa_list_nat_nat(size_size_list_nat,Xs))) ).

% count_le_length
tff(fact_1564_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),X)) ).

% vebt_member.simps(4)
tff(fact_1565_max_Oabsorb3,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = A ) ) ).

% max.absorb3
tff(fact_1566_max_Oabsorb3,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A))
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B2) = A ) ) ).

% max.absorb3
tff(fact_1567_max_Oabsorb3,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
     => ( aa_real_real(ord_max_real(A),B2) = A ) ) ).

% max.absorb3
tff(fact_1568_max_Oabsorb3,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = A ) ) ).

% max.absorb3
tff(fact_1569_max_Oabsorb4,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = B2 ) ) ).

% max.absorb4
tff(fact_1570_max_Oabsorb4,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B2) = B2 ) ) ).

% max.absorb4
tff(fact_1571_max_Oabsorb4,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( aa_real_real(ord_max_real(A),B2) = B2 ) ) ).

% max.absorb4
tff(fact_1572_max_Oabsorb4,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = B2 ) ) ).

% max.absorb4
tff(fact_1573_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z2)) ) ) ).

% max_less_iff_conj
tff(fact_1574_max__less__iff__conj,axiom,
    ! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X),Y)),Z2))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X),Z2))
        & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Y),Z2)) ) ) ).

% max_less_iff_conj
tff(fact_1575_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ord_max_real(X),Y)),Z2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z2)) ) ) ).

% max_less_iff_conj
tff(fact_1576_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z2)) ) ) ).

% max_less_iff_conj
tff(fact_1577_max_Oabsorb1,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => ( aa_real_real(ord_max_real(A),B2) = A ) ) ).

% max.absorb1
tff(fact_1578_max_Oabsorb1,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = A ) ) ).

% max.absorb1
tff(fact_1579_max_Oabsorb1,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = A ) ) ).

% max.absorb1
tff(fact_1580_max_Oabsorb2,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( aa_real_real(ord_max_real(A),B2) = B2 ) ) ).

% max.absorb2
tff(fact_1581_max_Oabsorb2,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = B2 ) ) ).

% max.absorb2
tff(fact_1582_max_Oabsorb2,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = B2 ) ) ).

% max.absorb2
tff(fact_1583_max_Obounded__iff,axiom,
    ! [B2: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ord_max_real(B2),C)),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A)) ) ) ).

% max.bounded_iff
tff(fact_1584_max_Obounded__iff,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C)),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% max.bounded_iff
tff(fact_1585_max_Obounded__iff,axiom,
    ! [B2: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C)),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% max.bounded_iff
tff(fact_1586_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X)) ).

% vebt_member.simps(3)
tff(fact_1587_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S),X) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S) ).

% vebt_insert.simps(3)
tff(fact_1588_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb,Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_1589_max_OcoboundedI2,axiom,
    ! [C: real,B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(ord_max_real(A),B2))) ) ).

% max.coboundedI2
tff(fact_1590_max_OcoboundedI2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2))) ) ).

% max.coboundedI2
tff(fact_1591_max_OcoboundedI2,axiom,
    ! [C: int,B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2))) ) ).

% max.coboundedI2
tff(fact_1592_max_OcoboundedI1,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(ord_max_real(A),B2))) ) ).

% max.coboundedI1
tff(fact_1593_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2))) ) ).

% max.coboundedI1
tff(fact_1594_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2))) ) ).

% max.coboundedI1
tff(fact_1595_max_Oabsorb__iff2,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
    <=> ( aa_real_real(ord_max_real(A),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_1596_max_Oabsorb__iff2,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_1597_max_Oabsorb__iff2,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_1598_max_Oabsorb__iff1,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
    <=> ( aa_real_real(ord_max_real(A),B2) = A ) ) ).

% max.absorb_iff1
tff(fact_1599_max_Oabsorb__iff1,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = A ) ) ).

% max.absorb_iff1
tff(fact_1600_max_Oabsorb__iff1,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = A ) ) ).

% max.absorb_iff1
tff(fact_1601_le__max__iff__disj,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),aa_real_real(ord_max_real(X),Y)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),X))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),Y)) ) ) ).

% le_max_iff_disj
tff(fact_1602_le__max__iff__disj,axiom,
    ! [Z2: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),X))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),Y)) ) ) ).

% le_max_iff_disj
tff(fact_1603_le__max__iff__disj,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),X))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),Y)) ) ) ).

% le_max_iff_disj
tff(fact_1604_max_Ocobounded2,axiom,
    ! [B2: real,A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(ord_max_real(A),B2))) ).

% max.cobounded2
tff(fact_1605_max_Ocobounded2,axiom,
    ! [B2: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2))) ).

% max.cobounded2
tff(fact_1606_max_Ocobounded2,axiom,
    ! [B2: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2))) ).

% max.cobounded2
tff(fact_1607_max_Ocobounded1,axiom,
    ! [A: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(ord_max_real(A),B2))) ).

% max.cobounded1
tff(fact_1608_max_Ocobounded1,axiom,
    ! [A: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2))) ).

% max.cobounded1
tff(fact_1609_max_Ocobounded1,axiom,
    ! [A: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2))) ).

% max.cobounded1
tff(fact_1610_max_Oorder__iff,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
    <=> ( A = aa_real_real(ord_max_real(A),B2) ) ) ).

% max.order_iff
tff(fact_1611_max_Oorder__iff,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) ) ) ).

% max.order_iff
tff(fact_1612_max_Oorder__iff,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) ) ) ).

% max.order_iff
tff(fact_1613_max_OboundedI,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ord_max_real(B2),C)),A)) ) ) ).

% max.boundedI
tff(fact_1614_max_OboundedI,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C)),A)) ) ) ).

% max.boundedI
tff(fact_1615_max_OboundedI,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C)),A)) ) ) ).

% max.boundedI
tff(fact_1616_max_OboundedE,axiom,
    ! [B2: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ord_max_real(B2),C)),A))
     => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A)) ) ) ).

% max.boundedE
tff(fact_1617_max_OboundedE,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C)),A))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% max.boundedE
tff(fact_1618_max_OboundedE,axiom,
    ! [B2: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C)),A))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% max.boundedE
tff(fact_1619_max_OorderI,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(ord_max_real(A),B2) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ).

% max.orderI
tff(fact_1620_max_OorderI,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A)) ) ).

% max.orderI
tff(fact_1621_max_OorderI,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A)) ) ).

% max.orderI
tff(fact_1622_max_OorderE,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => ( A = aa_real_real(ord_max_real(A),B2) ) ) ).

% max.orderE
tff(fact_1623_max_OorderE,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) ) ) ).

% max.orderE
tff(fact_1624_max_OorderE,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) ) ) ).

% max.orderE
tff(fact_1625_max_Omono,axiom,
    ! [C: real,A: real,D: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ord_max_real(C),D)),aa_real_real(ord_max_real(A),B2))) ) ) ).

% max.mono
tff(fact_1626_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2))) ) ) ).

% max.mono
tff(fact_1627_max_Omono,axiom,
    ! [C: int,A: int,D: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2))) ) ) ).

% max.mono
tff(fact_1628_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2))) ) ).

% max.strict_coboundedI2
tff(fact_1629_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B2: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,C),B2))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,C),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B2))) ) ).

% max.strict_coboundedI2
tff(fact_1630_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(ord_max_real(A),B2))) ) ).

% max.strict_coboundedI2
tff(fact_1631_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2))) ) ).

% max.strict_coboundedI2
tff(fact_1632_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2))) ) ).

% max.strict_coboundedI1
tff(fact_1633_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,C),A))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,C),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B2))) ) ).

% max.strict_coboundedI1
tff(fact_1634_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(ord_max_real(A),B2))) ) ).

% max.strict_coboundedI1
tff(fact_1635_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2))) ) ).

% max.strict_coboundedI1
tff(fact_1636_max_Ostrict__order__iff,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) )
        & ( A != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_1637_max_Ostrict__order__iff,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A))
    <=> ( ( A = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B2) )
        & ( A != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_1638_max_Ostrict__order__iff,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
    <=> ( ( A = aa_real_real(ord_max_real(A),B2) )
        & ( A != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_1639_max_Ostrict__order__iff,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) )
        & ( A != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_1640_max_Ostrict__boundedE,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C)),A))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_1641_max_Ostrict__boundedE,axiom,
    ! [B2: extended_enat,C: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B2),C)),A))
     => ~ ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A))
         => ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_1642_max_Ostrict__boundedE,axiom,
    ! [B2: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ord_max_real(B2),C)),A))
     => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_1643_max_Ostrict__boundedE,axiom,
    ! [B2: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C)),A))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_1644_less__max__iff__disj,axiom,
    ! [Z2: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),Y)) ) ) ).

% less_max_iff_disj
tff(fact_1645_less__max__iff__disj,axiom,
    ! [Z2: extended_enat,X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z2),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X),Y)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z2),X))
        | pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z2),Y)) ) ) ).

% less_max_iff_disj
tff(fact_1646_less__max__iff__disj,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),aa_real_real(ord_max_real(X),Y)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),Y)) ) ) ).

% less_max_iff_disj
tff(fact_1647_less__max__iff__disj,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),Y)) ) ) ).

% less_max_iff_disj
tff(fact_1648_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts,S),X) = vEBT_Node(Info,zero_zero_nat,Ts,S) ).

% vebt_insert.simps(2)
tff(fact_1649_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_1650_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_1651_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
     => ( nth_Pr744662078594809490T_VEBT(enumerate_VEBT_VEBT(N,Xs),M) = produc599794634098209291T_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),nth_VEBT_VEBT(Xs,M)) ) ) ).

% nth_enumerate_eq
tff(fact_1652_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_list_int_nat(size_size_list_int,Xs)))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(N,Xs),M) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),aa_nat_int(nth_int(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_1653_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( nth_Pr7617993195940197384at_nat(enumerate_nat(N,Xs),M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),aa_nat_nat(nth_nat(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_1654_find__Some__iff2,axiom,
    ! [X: product_prod_nat_nat,P: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat] :
      ( ( some_P7363390416028606310at_nat(X) = find_P8199882355184865565at_nat(P,Xs) )
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s5460976970255530739at_nat(Xs)))
          & pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,I3)))
          & ( X = nth_Pr7617993195940197384at_nat(Xs,I3) )
          & ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),I3))
             => ~ pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,J2))) ) ) ) ).

% find_Some_iff2
tff(fact_1655_find__Some__iff2,axiom,
    ! [X: num,P: fun_num_bool,Xs: list_num] :
      ( ( some_num(X) = find_num(P,Xs) )
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_num(Xs)))
          & pp(aa_num_bool(P,nth_num(Xs,I3)))
          & ( X = nth_num(Xs,I3) )
          & ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),I3))
             => ~ pp(aa_num_bool(P,nth_num(Xs,J2))) ) ) ) ).

% find_Some_iff2
tff(fact_1656_find__Some__iff2,axiom,
    ! [X: vEBT_VEBT,P: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT] :
      ( ( some_VEBT_VEBT(X) = find_VEBT_VEBT(P,Xs) )
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
          & pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I3)))
          & ( X = nth_VEBT_VEBT(Xs,I3) )
          & ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),I3))
             => ~ pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,J2))) ) ) ) ).

% find_Some_iff2
tff(fact_1657_find__Some__iff2,axiom,
    ! [X: int,P: fun_int_bool,Xs: list_int] :
      ( ( some_int(X) = find_int(P,Xs) )
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs)))
          & pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I3)))
          & ( X = aa_nat_int(nth_int(Xs),I3) )
          & ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),I3))
             => ~ pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),J2))) ) ) ) ).

% find_Some_iff2
tff(fact_1658_find__Some__iff2,axiom,
    ! [X: nat,P: fun_nat_bool,Xs: list_nat] :
      ( ( some_nat(X) = find_nat(P,Xs) )
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
          & pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I3)))
          & ( X = aa_nat_nat(nth_nat(Xs),I3) )
          & ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),I3))
             => ~ pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),J2))) ) ) ) ).

% find_Some_iff2
tff(fact_1659_find__Some__iff,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( find_P8199882355184865565at_nat(P,Xs) = some_P7363390416028606310at_nat(X) )
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s5460976970255530739at_nat(Xs)))
          & pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,I3)))
          & ( X = nth_Pr7617993195940197384at_nat(Xs,I3) )
          & ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),I3))
             => ~ pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,J2))) ) ) ) ).

% find_Some_iff
tff(fact_1660_find__Some__iff,axiom,
    ! [P: fun_num_bool,Xs: list_num,X: num] :
      ( ( find_num(P,Xs) = some_num(X) )
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_num(Xs)))
          & pp(aa_num_bool(P,nth_num(Xs,I3)))
          & ( X = nth_num(Xs,I3) )
          & ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),I3))
             => ~ pp(aa_num_bool(P,nth_num(Xs,J2))) ) ) ) ).

% find_Some_iff
tff(fact_1661_find__Some__iff,axiom,
    ! [P: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( find_VEBT_VEBT(P,Xs) = some_VEBT_VEBT(X) )
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
          & pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I3)))
          & ( X = nth_VEBT_VEBT(Xs,I3) )
          & ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),I3))
             => ~ pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,J2))) ) ) ) ).

% find_Some_iff
tff(fact_1662_find__Some__iff,axiom,
    ! [P: fun_int_bool,Xs: list_int,X: int] :
      ( ( find_int(P,Xs) = some_int(X) )
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs)))
          & pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I3)))
          & ( X = aa_nat_int(nth_int(Xs),I3) )
          & ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),I3))
             => ~ pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),J2))) ) ) ) ).

% find_Some_iff
tff(fact_1663_find__Some__iff,axiom,
    ! [P: fun_nat_bool,Xs: list_nat,X: nat] :
      ( ( find_nat(P,Xs) = some_nat(X) )
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
          & pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I3)))
          & ( X = aa_nat_nat(nth_nat(Xs),I3) )
          & ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),I3))
             => ~ pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),J2))) ) ) ) ).

% find_Some_iff
tff(fact_1664__C7_C,axiom,
    ( ( mi != ma )
   => ! [I5: nat] :
        ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m)))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I5 )
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na))) )
          & ! [Y4: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Y4),na) = I5 )
                & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Y4),na))) )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),Y4))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),ma)) ) ) ) ) ) ).

% "7"
tff(fact_1665_listrel1__iff__update,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( member6693912407220327184at_nat(produc5943733680697469783at_nat(Xs,Ys),listre4828114922151135584at_nat(R2))
    <=> ? [Y5: product_prod_nat_nat,N4: nat] :
          ( member8206827879206165904at_nat(produc6161850002892822231at_nat(nth_Pr7617993195940197384at_nat(Xs,N4),Y5),R2)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),size_s5460976970255530739at_nat(Xs)))
          & ( Ys = list_u6180841689913720943at_nat(Xs,N4,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_1666_listrel1__iff__update,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Xs,Ys),listrel1_VEBT_VEBT(R2))
    <=> ? [Y5: vEBT_VEBT,N4: nat] :
          ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,N4),Y5),R2)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)))
          & ( Ys = list_u1324408373059187874T_VEBT(Xs,N4,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_1667_listrel1__iff__update,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R2))
    <=> ? [Y5: int,N4: nat] :
          ( member5262025264175285858nt_int(product_Pair_int_int(aa_nat_int(nth_int(Xs),N4),Y5),R2)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),aa_list_int_nat(size_size_list_int,Xs)))
          & ( Ys = list_update_int(Xs,N4,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_1668_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R2))
    <=> ? [Y5: nat,N4: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),N4)),Y5),R2)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),aa_list_nat_nat(size_size_list_nat,Xs)))
          & ( Ys = list_update_nat(Xs,N4,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_1669_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat,X22: product_prod_nat_nat] : size_o8335143837870341156at_nat(X,some_P7363390416028606310at_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1670_option_Osize__gen_I2_J,axiom,
    ! [X: fun_num_nat,X22: num] : size_option_num(X,some_num(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1671_intind,axiom,
    ! [I: nat,N: nat,P: fun_nat_bool,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_nat_bool(P,X))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(replicate_nat(N,X)),I))) ) ) ).

% intind
tff(fact_1672_intind,axiom,
    ! [I: nat,N: nat,P: fun_int_bool,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_int_bool(P,X))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(replicate_int(N,X)),I))) ) ) ).

% intind
tff(fact_1673_intind,axiom,
    ! [I: nat,N: nat,P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_VEBT_VEBT_bool(P,X))
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X),I))) ) ) ).

% intind
tff(fact_1674_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary) ).

% vebt_insert.simps(4)
tff(fact_1675_gen__length__def,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] : aa_lis7044684121217598542BT_nat(gen_length_VEBT_VEBT(N),Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)) ).

% gen_length_def
tff(fact_1676_gen__length__def,axiom,
    ! [N: nat,Xs: list_int] : aa_list_int_nat(gen_length_int(N),Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_list_int_nat(size_size_list_int,Xs)) ).

% gen_length_def
tff(fact_1677_gen__length__def,axiom,
    ! [N: nat,Xs: list_nat] : aa_list_nat_nat(gen_length_nat(N),Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)) ).

% gen_length_def
tff(fact_1678__C2_C,axiom,
    aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m) ).

% "2"
tff(fact_1679__C5_Ohyps_C_I8_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),deg))) ).

% "5.hyps"(8)
tff(fact_1680__C5_Oprems_C,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),deg))) ).

% "5.prems"
tff(fact_1681__092_060open_062i_A_060_A2_A_094_Am_092_060close_062,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,i),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m))) ).

% \<open>i < 2 ^ m\<close>
tff(fact_1682__C5_OIH_C_I1_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(treeList)))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(X2),na))
        & ! [Xa: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),na)))
           => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_insert(X2,Xa)),na)) ) ) ) ).

% "5.IH"(1)
tff(fact_1683__C4_C,axiom,
    ! [I5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m)))
     => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I5)),X_13))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),I5)) ) ) ).

% "4"
tff(fact_1684_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))) ) ).

% high_bound_aux
tff(fact_1685_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ) ) ).

% member_bound
tff(fact_1686_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% numeral_le_iff
tff(fact_1687_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(M)),numeral_numeral_real(N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% numeral_le_iff
tff(fact_1688_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% numeral_le_iff
tff(fact_1689_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(M)),numeral_numeral_int(N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% numeral_le_iff
tff(fact_1690_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% numeral_less_iff
tff(fact_1691_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% numeral_less_iff
tff(fact_1692_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,numeral_numeral_int(M)),numeral_numeral_int(N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% numeral_less_iff
tff(fact_1693_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(M)),numeral_numeral_real(N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% numeral_less_iff
tff(fact_1694_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,plus_plus_num(M,N)) ).

% numeral_plus_numeral
tff(fact_1695_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)) = numera1916890842035813515d_enat(plus_plus_num(M,N)) ).

% numeral_plus_numeral
tff(fact_1696_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(M)),numeral_numeral_int(N)) = numeral_numeral_int(plus_plus_num(M,N)) ).

% numeral_plus_numeral
tff(fact_1697_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(plus_plus_real(numeral_numeral_real(M)),numeral_numeral_real(N)) = numeral_numeral_real(plus_plus_num(M,N)) ).

% numeral_plus_numeral
tff(fact_1698_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W2)),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,plus_plus_num(V,W2))),Z2) ).

% add_numeral_left
tff(fact_1699_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(V)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(W2)),Z2)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(plus_plus_num(V,W2))),Z2) ).

% add_numeral_left
tff(fact_1700_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(W2)),Z2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(plus_plus_num(V,W2))),Z2) ).

% add_numeral_left
tff(fact_1701_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: real] : aa_real_real(plus_plus_real(numeral_numeral_real(V)),aa_real_real(plus_plus_real(numeral_numeral_real(W2)),Z2)) = aa_real_real(plus_plus_real(numeral_numeral_real(plus_plus_num(V,W2))),Z2) ).

% add_numeral_left
tff(fact_1702_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_1703_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,X)),X)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_1704_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
         => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,Y)),X)) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_1705_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
         => ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_vebt_insert(T,X)),Y))
           => ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_1706_replicate__eq__replicate,axiom,
    ! [M: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(M,X) = replicate_VEBT_VEBT(N,Y) )
    <=> ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_1707_length__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] : aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,replicate_VEBT_VEBT(N,X)) = N ).

% length_replicate
tff(fact_1708_length__replicate,axiom,
    ! [N: nat,X: int] : aa_list_int_nat(size_size_list_int,replicate_int(N,X)) = N ).

% length_replicate
tff(fact_1709_length__replicate,axiom,
    ! [N: nat,X: nat] : aa_list_nat_nat(size_size_list_nat,replicate_nat(N,X)) = N ).

% length_replicate
tff(fact_1710_length__enumerate,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(enumerate_VEBT_VEBT(N,Xs)) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) ).

% length_enumerate
tff(fact_1711_length__enumerate,axiom,
    ! [N: nat,Xs: list_int] : size_s2970893825323803983at_int(enumerate_int(N,Xs)) = aa_list_int_nat(size_size_list_int,Xs) ).

% length_enumerate
tff(fact_1712_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] : size_s5460976970255530739at_nat(enumerate_nat(N,Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_enumerate
tff(fact_1713_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_1714__C5_OIH_C_I2_J,axiom,
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m)))
     => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_insert(summary,X)),m)) ) ).

% "5.IH"(2)
tff(fact_1715_Suc__numeral,axiom,
    ! [N: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,plus_plus_num(N,one)) ).

% Suc_numeral
tff(fact_1716_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(V) ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(U) ) ) ) ).

% max_number_of(1)
tff(fact_1717_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)))
       => ( aa_real_real(ord_max_real(numeral_numeral_real(U)),numeral_numeral_real(V)) = numeral_numeral_real(V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)))
       => ( aa_real_real(ord_max_real(numeral_numeral_real(U)),numeral_numeral_real(V)) = numeral_numeral_real(U) ) ) ) ).

% max_number_of(1)
tff(fact_1718_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = aa_num_nat(numeral_numeral_nat,V) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = aa_num_nat(numeral_numeral_nat,U) ) ) ) ).

% max_number_of(1)
tff(fact_1719_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(U)),numeral_numeral_int(V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,numeral_numeral_int(U)),numeral_numeral_int(V)) = numeral_numeral_int(V) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(U)),numeral_numeral_int(V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,numeral_numeral_int(U)),numeral_numeral_int(V)) = numeral_numeral_int(U) ) ) ) ).

% max_number_of(1)
tff(fact_1720_max__0__1_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(4)
tff(fact_1721_max__0__1_I4_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X)),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(4)
tff(fact_1722_max__0__1_I4_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,numeral_numeral_int(X)),zero_zero_int) = numeral_numeral_int(X) ).

% max_0_1(4)
tff(fact_1723_max__0__1_I4_J,axiom,
    ! [X: num] : aa_real_real(ord_max_real(numeral_numeral_real(X)),zero_zero_real) = numeral_numeral_real(X) ).

% max_0_1(4)
tff(fact_1724_max__0__1_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(3)
tff(fact_1725_max__0__1_I3_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(3)
tff(fact_1726_max__0__1_I3_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),numeral_numeral_int(X)) = numeral_numeral_int(X) ).

% max_0_1(3)
tff(fact_1727_max__0__1_I3_J,axiom,
    ! [X: num] : aa_real_real(ord_max_real(zero_zero_real),numeral_numeral_real(X)) = numeral_numeral_real(X) ).

% max_0_1(3)
tff(fact_1728__C6_C,axiom,
    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),ma))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),deg))) ) ).

% "6"
tff(fact_1729_myIHs,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(treeList)))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(X),na))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),na)))
         => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_insert(X,Xa2)),na)) ) ) ) ).

% myIHs
tff(fact_1730_in__set__replicate,axiom,
    ! [X: extended_enat,N: nat,Y: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),set_Extended_enat2(replic7216382294607269926d_enat(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1731_in__set__replicate,axiom,
    ! [X: real,N: nat,Y: real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(replicate_real(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1732_in__set__replicate,axiom,
    ! [X: set_nat,N: nat,Y: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(replicate_set_nat(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1733_in__set__replicate,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(replicate_int(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1734_in__set__replicate,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(replicate_nat(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1735_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1736_Bex__set__replicate,axiom,
    ! [N: nat,A: int,P: fun_int_bool] :
      ( ? [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(replicate_int(N,A))))
          & pp(aa_int_bool(P,X5)) )
    <=> ( pp(aa_int_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1737_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ? [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(replicate_nat(N,A))))
          & pp(aa_nat_bool(P,X5)) )
    <=> ( pp(aa_nat_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1738_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ? [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
          & pp(aa_VEBT_VEBT_bool(P,X5)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1739_Ball__set__replicate,axiom,
    ! [N: nat,A: int,P: fun_int_bool] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(replicate_int(N,A))))
         => pp(aa_int_bool(P,X5)) )
    <=> ( pp(aa_int_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1740_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(replicate_nat(N,A))))
         => pp(aa_nat_bool(P,X5)) )
    <=> ( pp(aa_nat_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1741_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
         => pp(aa_VEBT_VEBT_bool(P,X5)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1742_nth__replicate,axiom,
    ! [I: nat,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_nat(nth_nat(replicate_nat(N,X)),I) = X ) ) ).

% nth_replicate
tff(fact_1743_nth__replicate,axiom,
    ! [I: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_int(nth_int(replicate_int(N,X)),I) = X ) ) ).

% nth_replicate
tff(fact_1744_nth__replicate,axiom,
    ! [I: nat,N: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X),I) = X ) ) ).

% nth_replicate
tff(fact_1745_highlowprop,axiom,
    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m)))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),na))) ) ).

% highlowprop
tff(fact_1746_add__2__eq__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc
tff(fact_1747_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc'
tff(fact_1748_numeral__Bit0,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,bit0(N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_Bit0
tff(fact_1749_numeral__Bit0,axiom,
    ! [N: num] : numera1916890842035813515d_enat(bit0(N)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N)),numera1916890842035813515d_enat(N)) ).

% numeral_Bit0
tff(fact_1750_numeral__Bit0,axiom,
    ! [N: num] : numeral_numeral_int(bit0(N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(N)),numeral_numeral_int(N)) ).

% numeral_Bit0
tff(fact_1751_numeral__Bit0,axiom,
    ! [N: num] : numeral_numeral_real(bit0(N)) = aa_real_real(plus_plus_real(numeral_numeral_real(N)),numeral_numeral_real(N)) ).

% numeral_Bit0
tff(fact_1752_listrel1__eq__len,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Xs,Ys),listrel1_VEBT_VEBT(R2))
     => ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys) ) ) ).

% listrel1_eq_len
tff(fact_1753_listrel1__eq__len,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R2))
     => ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys) ) ) ).

% listrel1_eq_len
tff(fact_1754_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R2))
     => ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) ) ) ).

% listrel1_eq_len
tff(fact_1755_numeral__2__eq__2,axiom,
    aa_num_nat(numeral_numeral_nat,bit0(one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_1756_find__None__iff,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,Xs: list_Extended_enat] :
      ( ( find_Extended_enat(P,Xs) = none_Extended_enat )
    <=> ~ ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),set_Extended_enat2(Xs)))
            & pp(aa_Ext2772979874139974381t_bool(P,X5)) ) ) ).

% find_None_iff
tff(fact_1757_find__None__iff,axiom,
    ! [P: fun_real_bool,Xs: list_real] :
      ( ( find_real(P,Xs) = none_real )
    <=> ~ ? [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),set_real2(Xs)))
            & pp(aa_real_bool(P,X5)) ) ) ).

% find_None_iff
tff(fact_1758_find__None__iff,axiom,
    ! [P: fun_set_nat_bool,Xs: list_set_nat] :
      ( ( find_set_nat(P,Xs) = none_set_nat )
    <=> ~ ? [X5: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X5),set_set_nat2(Xs)))
            & pp(aa_set_nat_bool(P,X5)) ) ) ).

% find_None_iff
tff(fact_1759_find__None__iff,axiom,
    ! [P: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT] :
      ( ( find_VEBT_VEBT(P,Xs) = none_VEBT_VEBT )
    <=> ~ ? [X5: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(Xs)))
            & pp(aa_VEBT_VEBT_bool(P,X5)) ) ) ).

% find_None_iff
tff(fact_1760_find__None__iff,axiom,
    ! [P: fun_int_bool,Xs: list_int] :
      ( ( find_int(P,Xs) = none_int )
    <=> ~ ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),set_int2(Xs)))
            & pp(aa_int_bool(P,X5)) ) ) ).

% find_None_iff
tff(fact_1761_find__None__iff,axiom,
    ! [P: fun_nat_bool,Xs: list_nat] :
      ( ( find_nat(P,Xs) = none_nat )
    <=> ~ ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
            & pp(aa_nat_bool(P,X5)) ) ) ).

% find_None_iff
tff(fact_1762_find__None__iff,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat] :
      ( ( find_P8199882355184865565at_nat(P,Xs) = none_P5556105721700978146at_nat )
    <=> ~ ? [X5: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(X5,set_Pr5648618587558075414at_nat(Xs))
            & pp(aa_Pro2232559814647311250t_bool(P,X5)) ) ) ).

% find_None_iff
tff(fact_1763_find__None__iff,axiom,
    ! [P: fun_num_bool,Xs: list_num] :
      ( ( find_num(P,Xs) = none_num )
    <=> ~ ? [X5: num] :
            ( member_num(X5,set_num2(Xs))
            & pp(aa_num_bool(P,X5)) ) ) ).

% find_None_iff
tff(fact_1764_find__None__iff2,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,Xs: list_Extended_enat] :
      ( ( none_Extended_enat = find_Extended_enat(P,Xs) )
    <=> ~ ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),set_Extended_enat2(Xs)))
            & pp(aa_Ext2772979874139974381t_bool(P,X5)) ) ) ).

% find_None_iff2
tff(fact_1765_find__None__iff2,axiom,
    ! [P: fun_real_bool,Xs: list_real] :
      ( ( none_real = find_real(P,Xs) )
    <=> ~ ? [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),set_real2(Xs)))
            & pp(aa_real_bool(P,X5)) ) ) ).

% find_None_iff2
tff(fact_1766_find__None__iff2,axiom,
    ! [P: fun_set_nat_bool,Xs: list_set_nat] :
      ( ( none_set_nat = find_set_nat(P,Xs) )
    <=> ~ ? [X5: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X5),set_set_nat2(Xs)))
            & pp(aa_set_nat_bool(P,X5)) ) ) ).

% find_None_iff2
tff(fact_1767_find__None__iff2,axiom,
    ! [P: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT] :
      ( ( none_VEBT_VEBT = find_VEBT_VEBT(P,Xs) )
    <=> ~ ? [X5: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(Xs)))
            & pp(aa_VEBT_VEBT_bool(P,X5)) ) ) ).

% find_None_iff2
tff(fact_1768_find__None__iff2,axiom,
    ! [P: fun_int_bool,Xs: list_int] :
      ( ( none_int = find_int(P,Xs) )
    <=> ~ ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),set_int2(Xs)))
            & pp(aa_int_bool(P,X5)) ) ) ).

% find_None_iff2
tff(fact_1769_find__None__iff2,axiom,
    ! [P: fun_nat_bool,Xs: list_nat] :
      ( ( none_nat = find_nat(P,Xs) )
    <=> ~ ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
            & pp(aa_nat_bool(P,X5)) ) ) ).

% find_None_iff2
tff(fact_1770_find__None__iff2,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat] :
      ( ( none_P5556105721700978146at_nat = find_P8199882355184865565at_nat(P,Xs) )
    <=> ~ ? [X5: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(X5,set_Pr5648618587558075414at_nat(Xs))
            & pp(aa_Pro2232559814647311250t_bool(P,X5)) ) ) ).

% find_None_iff2
tff(fact_1771_find__None__iff2,axiom,
    ! [P: fun_num_bool,Xs: list_num] :
      ( ( none_num = find_num(P,Xs) )
    <=> ~ ? [X5: num] :
            ( member_num(X5,set_num2(Xs))
            & pp(aa_num_bool(P,X5)) ) ) ).

% find_None_iff2
tff(fact_1772_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_1773_option_Osize__gen_I1_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1774_option_Osize__gen_I1_J,axiom,
    ! [X: fun_num_nat] : size_option_num(X,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1775_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_1776_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))
    <=> ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_1777_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))
     => ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_1778_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_1779_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_1780_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,plus_plus_num(V,one))),N) ).

% Suc_nat_number_of_add
tff(fact_1781_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_complex != numera6690914467698888265omplex(N) ).

% zero_neq_numeral
tff(fact_1782_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).

% zero_neq_numeral
tff(fact_1783_zero__neq__numeral,axiom,
    ! [N: num] : zero_z5237406670263579293d_enat != numera1916890842035813515d_enat(N) ).

% zero_neq_numeral
tff(fact_1784_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_int != numeral_numeral_int(N) ).

% zero_neq_numeral
tff(fact_1785_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_real != numeral_numeral_real(N) ).

% zero_neq_numeral
tff(fact_1786_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc9217583074051984497at_nat] :
      ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B3: nat,Acc: nat] : X != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1787_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_1788_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_1789_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1790_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1791_find__cong,axiom,
    ! [Xs: list_Extended_enat,Ys: list_Extended_enat,P: fun_Ex8563670519792538868t_bool,Q: fun_Ex8563670519792538868t_bool] :
      ( ( Xs = Ys )
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),set_Extended_enat2(Ys)))
           => ( pp(aa_Ext2772979874139974381t_bool(P,X4))
            <=> pp(aa_Ext2772979874139974381t_bool(Q,X4)) ) )
       => ( find_Extended_enat(P,Xs) = find_Extended_enat(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1792_find__cong,axiom,
    ! [Xs: list_real,Ys: list_real,P: fun_real_bool,Q: fun_real_bool] :
      ( ( Xs = Ys )
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),set_real2(Ys)))
           => ( pp(aa_real_bool(P,X4))
            <=> pp(aa_real_bool(Q,X4)) ) )
       => ( find_real(P,Xs) = find_real(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1793_find__cong,axiom,
    ! [Xs: list_set_nat,Ys: list_set_nat,P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ( Xs = Ys )
     => ( ! [X4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),set_set_nat2(Ys)))
           => ( pp(aa_set_nat_bool(P,X4))
            <=> pp(aa_set_nat_bool(Q,X4)) ) )
       => ( find_set_nat(P,Xs) = find_set_nat(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1794_find__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,Q: fun_VEBT_VEBT_bool] :
      ( ( Xs = Ys )
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Ys)))
           => ( pp(aa_VEBT_VEBT_bool(P,X4))
            <=> pp(aa_VEBT_VEBT_bool(Q,X4)) ) )
       => ( find_VEBT_VEBT(P,Xs) = find_VEBT_VEBT(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1795_find__cong,axiom,
    ! [Xs: list_int,Ys: list_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ( Xs = Ys )
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),set_int2(Ys)))
           => ( pp(aa_int_bool(P,X4))
            <=> pp(aa_int_bool(Q,X4)) ) )
       => ( find_int(P,Xs) = find_int(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1796_find__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: fun_nat_bool,Q: fun_nat_bool] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Ys)))
           => ( pp(aa_nat_bool(P,X4))
            <=> pp(aa_nat_bool(Q,X4)) ) )
       => ( find_nat(P,Xs) = find_nat(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1797_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(N)),zero_z5237406670263579293d_enat)) ).

% not_numeral_le_zero
tff(fact_1798_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),zero_zero_real)) ).

% not_numeral_le_zero
tff(fact_1799_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_le_zero
tff(fact_1800_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(N)),zero_zero_int)) ).

% not_numeral_le_zero
tff(fact_1801_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(N))) ).

% zero_le_numeral
tff(fact_1802_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),numeral_numeral_real(N))) ).

% zero_le_numeral
tff(fact_1803_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_le_numeral
tff(fact_1804_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),numeral_numeral_int(N))) ).

% zero_le_numeral
tff(fact_1805_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_less_zero
tff(fact_1806_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(N)),zero_z5237406670263579293d_enat)) ).

% not_numeral_less_zero
tff(fact_1807_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,numeral_numeral_int(N)),zero_zero_int)) ).

% not_numeral_less_zero
tff(fact_1808_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(N)),zero_zero_real)) ).

% not_numeral_less_zero
tff(fact_1809_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_less_numeral
tff(fact_1810_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(N))) ).

% zero_less_numeral
tff(fact_1811_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),numeral_numeral_int(N))) ).

% zero_less_numeral
tff(fact_1812_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),numeral_numeral_real(N))) ).

% zero_less_numeral
tff(fact_1813_replicate__eqI,axiom,
    ! [Xs: list_Extended_enat,N: nat,X: extended_enat] :
      ( ( size_s3941691890525107288d_enat(Xs) = N )
     => ( ! [Y3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y3),set_Extended_enat2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replic7216382294607269926d_enat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_1814_replicate__eqI,axiom,
    ! [Xs: list_real,N: nat,X: real] :
      ( ( size_size_list_real(Xs) = N )
     => ( ! [Y3: real] :
            ( pp(aa_set_real_bool(member_real(Y3),set_real2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_real(N,X) ) ) ) ).

% replicate_eqI
tff(fact_1815_replicate__eqI,axiom,
    ! [Xs: list_set_nat,N: nat,X: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = N )
     => ( ! [Y3: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(Y3),set_set_nat2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_set_nat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_1816_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = N )
     => ( ! [Y3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y3),set_VEBT_VEBT2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_VEBT_VEBT(N,X) ) ) ) ).

% replicate_eqI
tff(fact_1817_replicate__eqI,axiom,
    ! [Xs: list_int,N: nat,X: int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = N )
     => ( ! [Y3: int] :
            ( pp(aa_set_int_bool(member_int(Y3),set_int2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_int(N,X) ) ) ) ).

% replicate_eqI
tff(fact_1818_replicate__eqI,axiom,
    ! [Xs: list_nat,N: nat,X: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = N )
     => ( ! [Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y3),set_nat2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_nat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_1819_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
         => ( X4 = X ) )
     => ( replicate_VEBT_VEBT(aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_1820_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => ( X4 = X ) )
     => ( replicate_int(aa_list_int_nat(size_size_list_int,Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_1821_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => ( X4 = X ) )
     => ( replicate_nat(aa_list_nat_nat(size_size_list_nat,Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_1822_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),X)) ).

% vebt_member.simps(2)
tff(fact_1823_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy))) ).

% VEBT_internal.minNull.simps(4)
tff(fact_1824_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I4: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)))
                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I4)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I4: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I4 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N) = I4 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X4))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_1825_num_Osize_I5_J,axiom,
    ! [X22: num] : aa_num_nat(size_size_num,bit0(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_1826_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_1827_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I4: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)))
                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I4)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I4: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I4 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N) = I4 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X4))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_1828_length__code,axiom,
    size_s6755466524823107622T_VEBT = gen_length_VEBT_VEBT(zero_zero_nat) ).

% length_code
tff(fact_1829_length__code,axiom,
    size_size_list_int = gen_length_int(zero_zero_nat) ).

% length_code
tff(fact_1830_length__code,axiom,
    size_size_list_nat = gen_length_nat(zero_zero_nat) ).

% length_code
tff(fact_1831_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1832_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1833_zero__less__power2,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,bit0(one)))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_1834_zero__less__power2,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one)))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_1835_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1836_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1837_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1838_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_1839_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_zero_int))
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_1840_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_1841_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_1842_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_1843_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_1844_power__mono__iff,axiom,
    ! [A: real,B2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B2),N)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ) ) ) ).

% power_mono_iff
tff(fact_1845_power__mono__iff,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2)) ) ) ) ) ).

% power_mono_iff
tff(fact_1846_power__mono__iff,axiom,
    ! [A: int,B2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B2),N)))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ) ) ) ).

% power_mono_iff
tff(fact_1847_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_1848_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_1849_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_1850_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(power_power_real(A),N) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_1851_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(A),N) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_1852_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( aa_nat_int(power_power_int(A),N) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_1853_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
        & ( ( X = Mi )
          | ( X = Ma )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)))
            & pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ) ).

% member_inv
tff(fact_1854_pow__sum,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2) ).

% pow_sum
tff(fact_1855_high__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% high_def
tff(fact_1856__C9_C,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,bit0(one))) = na ).

% "9"
tff(fact_1857_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_1858_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_1859_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B2),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
tff(fact_1860_divide__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( aa_real_real(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B2),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
tff(fact_1861_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C),A) = aa_complex_complex(divide1717551699836669952omplex(C),B2) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
tff(fact_1862_divide__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( aa_real_real(divide_divide_real(C),A) = aa_real_real(divide_divide_real(C),B2) )
    <=> ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
tff(fact_1863_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1864_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_1865_div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1866_div__by__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_1867_divide__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_1868_divide__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(divide_divide_real(A),B2) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_1869_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_1870_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_1871_div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_1872_div__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_1873_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(zero_z5237406670263579293d_enat),aa_nat_nat(suc,N)) = zero_z5237406670263579293d_enat ).

% power_0_Suc
tff(fact_1874_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,N)) = zero_zero_nat ).

% power_0_Suc
tff(fact_1875_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_real(power_power_real(zero_zero_real),aa_nat_nat(suc,N)) = zero_zero_real ).

% power_0_Suc
tff(fact_1876_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,N)) = zero_zero_complex ).

% power_0_Suc
tff(fact_1877_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,N)) = zero_zero_int ).

% power_0_Suc
tff(fact_1878_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_Extended_enat(power_8040749407984259932d_enat(zero_z5237406670263579293d_enat),aa_num_nat(numeral_numeral_nat,K)) = zero_z5237406670263579293d_enat ).

% power_zero_numeral
tff(fact_1879_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_1880_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(power_power_real(zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_1881_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_1882_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_1883_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_1884_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1885_power__Suc0__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1886_power__Suc0__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1887_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( X = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_1888_power__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_1889_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1890_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg)))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(Info,Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_1891_add__divide__distrib,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B2)),C) = aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B2),C)) ).

% add_divide_distrib
tff(fact_1892_divide__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(A),B2)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) ) ) ) ).

% divide_le_0_iff
tff(fact_1893_divide__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B2),C))) ) ) ).

% divide_right_mono
tff(fact_1894_zero__le__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) ) ) ) ).

% zero_le_divide_iff
tff(fact_1895_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_1896_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_nonpos
tff(fact_1897_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_nonneg
tff(fact_1898_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_1899_divide__right__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B2),C)),aa_real_real(divide_divide_real(A),C))) ) ) ).

% divide_right_mono_neg
tff(fact_1900_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% divide_neg_neg
tff(fact_1901_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_neg_pos
tff(fact_1902_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_pos_neg
tff(fact_1903_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% divide_pos_pos
tff(fact_1904_divide__less__0__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(A),B2)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) ) ) ) ).

% divide_less_0_iff
tff(fact_1905_divide__less__cancel,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B2),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_1906_zero__less__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real)) ) ) ) ).

% zero_less_divide_iff
tff(fact_1907_divide__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B2),C))) ) ) ).

% divide_strict_right_mono
tff(fact_1908_divide__strict__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B2),C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1909_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_pos
tff(fact_1910_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_1911_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_1912_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_neg
tff(fact_1913_divide__le__cancel,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B2),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ) ) ).

% divide_le_cancel
tff(fact_1914_frac__less2,axiom,
    ! [X: real,Y: real,W2: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W2),Z2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(X),Z2)),aa_real_real(divide_divide_real(Y),W2))) ) ) ) ) ).

% frac_less2
tff(fact_1915_frac__less,axiom,
    ! [X: real,Y: real,W2: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W2),Z2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(X),Z2)),aa_real_real(divide_divide_real(Y),W2))) ) ) ) ) ).

% frac_less
tff(fact_1916_frac__le,axiom,
    ! [Y: real,X: real,W2: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W2),Z2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Z2)),aa_real_real(divide_divide_real(Y),W2))) ) ) ) ) ).

% frac_le
tff(fact_1917_field__sum__of__halves,axiom,
    ! [X: real] : aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X),numeral_numeral_real(bit0(one)))),aa_real_real(divide_divide_real(X),numeral_numeral_real(bit0(one)))) = X ).

% field_sum_of_halves
tff(fact_1918_half__gt__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),numeral_numeral_real(bit0(one))))) ) ).

% half_gt_zero
tff(fact_1919_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),numeral_numeral_real(bit0(one)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% half_gt_zero_iff
tff(fact_1920_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),Y)),numeral_numeral_real(bit0(one))))) ) ).

% field_less_half_sum
tff(fact_1921_power__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_1922_power__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(power_power_real(A),N) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_1923_power__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),N) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_1924_power__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),N) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_1925_power__mono,axiom,
    ! [A: real,B2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B2),N))) ) ) ).

% power_mono
tff(fact_1926_power__mono,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N))) ) ) ).

% power_mono
tff(fact_1927_power__mono,axiom,
    ! [A: int,B2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B2),N))) ) ) ).

% power_mono
tff(fact_1928_zero__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),N))) ) ).

% zero_le_power
tff(fact_1929_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% zero_le_power
tff(fact_1930_zero__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_le_power
tff(fact_1931_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% zero_less_power
tff(fact_1932_zero__less__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),N))) ) ).

% zero_less_power
tff(fact_1933_zero__less__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_less_power
tff(fact_1934_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_1935_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B2),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ) ).

% power_less_imp_less_base
tff(fact_1936_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2)) ) ) ).

% power_less_imp_less_base
tff(fact_1937_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B2),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) ) ) ).

% power_less_imp_less_base
tff(fact_1938_power__inject__base,axiom,
    ! [A: real,N: nat,B2: real] :
      ( ( aa_nat_real(power_power_real(A),aa_nat_nat(suc,N)) = aa_nat_real(power_power_real(B2),aa_nat_nat(suc,N)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
         => ( A = B2 ) ) ) ) ).

% power_inject_base
tff(fact_1939_power__inject__base,axiom,
    ! [A: nat,N: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(suc,N)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
         => ( A = B2 ) ) ) ) ).

% power_inject_base
tff(fact_1940_power__inject__base,axiom,
    ! [A: int,N: nat,B2: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_nat_int(power_power_int(B2),aa_nat_nat(suc,N)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
         => ( A = B2 ) ) ) ) ).

% power_inject_base
tff(fact_1941_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,N))),aa_nat_real(power_power_real(B2),aa_nat_nat(suc,N))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ) ).

% power_le_imp_le_base
tff(fact_1942_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2)) ) ) ).

% power_le_imp_le_base
tff(fact_1943_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(B2),aa_nat_nat(suc,N))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ) ).

% power_le_imp_le_base
tff(fact_1944_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_Extended_enat(power_8040749407984259932d_enat(zero_z5237406670263579293d_enat),N) = zero_z5237406670263579293d_enat ) ) ).

% zero_power
tff(fact_1945_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ).

% zero_power
tff(fact_1946_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(power_power_real(zero_zero_real),N) = zero_zero_real ) ) ).

% zero_power
tff(fact_1947_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ).

% zero_power
tff(fact_1948_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ).

% zero_power
tff(fact_1949_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K))) ) ).

% power_gt_expt
tff(fact_1950_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N))) ) ).

% nat_one_le_power
tff(fact_1951_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B2: real] :
      ( ( aa_nat_real(power_power_real(A),N) = aa_nat_real(power_power_real(B2),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1952_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1953_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B2: int] :
      ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B2),N) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1954_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
         => ( ( aa_nat_real(power_power_real(A),N) = aa_nat_real(power_power_real(B2),N) )
          <=> ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1955_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N) )
          <=> ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1956_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
         => ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B2),N) )
          <=> ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1957_zero__power2,axiom,
    aa_nat_Extended_enat(power_8040749407984259932d_enat(zero_z5237406670263579293d_enat),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_z5237406670263579293d_enat ).

% zero_power2
tff(fact_1958_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ).

% zero_power2
tff(fact_1959_zero__power2,axiom,
    aa_nat_real(power_power_real(zero_zero_real),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_real ).

% zero_power2
tff(fact_1960_zero__power2,axiom,
    aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_complex ).

% zero_power2
tff(fact_1961_zero__power2,axiom,
    aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_int ).

% zero_power2
tff(fact_1962_less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% less_exp
tff(fact_1963_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M))) ) ).

% self_le_ge2_pow
tff(fact_1964_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% power2_nat_le_eq_le
tff(fact_1965_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% power2_nat_le_imp_le
tff(fact_1966_zero__le__power2,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% zero_le_power2
tff(fact_1967_zero__le__power2,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% zero_le_power2
tff(fact_1968_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1969_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1970_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1971_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_1972_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_1973_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_1974_power__strict__mono,axiom,
    ! [A: real,B2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B2),N))) ) ) ) ).

% power_strict_mono
tff(fact_1975_power__strict__mono,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N))) ) ) ) ).

% power_strict_mono
tff(fact_1976_power__strict__mono,axiom,
    ! [A: int,B2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B2),N))) ) ) ) ).

% power_strict_mono
tff(fact_1977_power2__less__0,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_zero_real)) ).

% power2_less_0
tff(fact_1978_power2__less__0,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_zero_int)) ).

% power2_less_0
tff(fact_1979_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_1980_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_1981_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_1982_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% sum_power2_ge_zero
tff(fact_1983_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% sum_power2_ge_zero
tff(fact_1984_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_1985_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_1986_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_real)) ).

% not_sum_power2_lt_zero
tff(fact_1987_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_int)) ).

% not_sum_power2_lt_zero
tff(fact_1988_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1989_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1990_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_1991_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,bit0(one))) = M ).

% add_self_div_2
tff(fact_1992_div2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% div2_Suc_Suc
tff(fact_1993_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),X))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_1994_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) ) ).

% div_less
tff(fact_1995_div__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,zero_zero_nat)) = M ).

% div_by_Suc_0
tff(fact_1996_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N)) ) ) ).

% set_n_deg_not_0
tff(fact_1997_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% div_2_gt_zero
tff(fact_1998_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_1999_div__exp__eq,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_2000_div__exp__eq,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_2001_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_2002_bits__div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_2003_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_2004_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_2005_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_2006_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_2007_div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% div_by_1
tff(fact_2008_div__by__1,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),one_one_real) = A ).

% div_by_1
tff(fact_2009_divide__eq__1__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = one_one_complex )
    <=> ( ( B2 != zero_zero_complex )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_2010_divide__eq__1__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(divide_divide_real(A),B2) = one_one_real )
    <=> ( ( B2 != zero_zero_real )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_2011_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_2012_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_2013_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),A) = one_one_int ) ) ).

% div_self
tff(fact_2014_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% div_self
tff(fact_2015_one__eq__divide__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A),B2) )
    <=> ( ( B2 != zero_zero_complex )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_2016_one__eq__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(A),B2) )
    <=> ( ( B2 != zero_zero_real )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_2017_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_2018_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% divide_self
tff(fact_2019_divide__self__if,axiom,
    ! [A: complex] :
      ( ( ( A = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = zero_zero_complex ) )
      & ( ( A != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ) ).

% divide_self_if
tff(fact_2020_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(divide_divide_real(A),A) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ) ).

% divide_self_if
tff(fact_2021_divide__eq__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( aa_real_real(divide_divide_real(B2),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B2 ) ) ) ).

% divide_eq_eq_1
tff(fact_2022_eq__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(B2),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B2 ) ) ) ).

% eq_divide_eq_1
tff(fact_2023_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(divide_divide_real(one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_2024_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(divide_divide_real(one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_2025_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2026_power__inject__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_nat_real(power_power_real(A),M) = aa_nat_real(power_power_real(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2027_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( ( aa_nat_int(power_power_int(A),M) = aa_nat_int(power_power_int(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2028_max__0__1_I2_J,axiom,
    aa_real_real(ord_max_real(one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_2029_max__0__1_I2_J,axiom,
    aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_2030_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_2031_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_2032_max__0__1_I1_J,axiom,
    aa_real_real(ord_max_real(zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_2033_max__0__1_I1_J,axiom,
    aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_2034_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_2035_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_2036_less__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),one_one_nat))
    <=> ( N = zero_zero_nat ) ) ).

% less_one
tff(fact_2037_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% zero_le_divide_1_iff
tff(fact_2038_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% divide_le_0_1_iff
tff(fact_2039_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% zero_less_divide_1_iff
tff(fact_2040_less__divide__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B2),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ) ).

% less_divide_eq_1_pos
tff(fact_2041_less__divide__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B2),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) ) ) ).

% less_divide_eq_1_neg
tff(fact_2042_divide__less__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B2),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) ) ) ).

% divide_less_eq_1_pos
tff(fact_2043_divide__less__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B2),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ) ).

% divide_less_eq_1_neg
tff(fact_2044_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(one_one_real),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% divide_less_0_1_iff
tff(fact_2045_power__strict__increasing__iff,axiom,
    ! [B2: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2046_power__strict__increasing__iff,axiom,
    ! [B2: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(B2),X)),aa_nat_real(power_power_real(B2),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2047_power__strict__increasing__iff,axiom,
    ! [B2: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B2),X)),aa_nat_int(power_power_int(B2),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2048_divide__le__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B2),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ) ).

% divide_le_eq_1_neg
tff(fact_2049_divide__le__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B2),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ) ).

% divide_le_eq_1_pos
tff(fact_2050_le__divide__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B2),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ) ).

% le_divide_eq_1_neg
tff(fact_2051_le__divide__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B2),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ) ).

% le_divide_eq_1_pos
tff(fact_2052_one__add__one,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),one_one_complex) = numera6690914467698888265omplex(bit0(one)) ).

% one_add_one
tff(fact_2053_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% one_add_one
tff(fact_2054_one__add__one,axiom,
    aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(bit0(one)) ).

% one_add_one
tff(fact_2055_one__add__one,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = numeral_numeral_int(bit0(one)) ).

% one_add_one
tff(fact_2056_one__add__one,axiom,
    aa_real_real(plus_plus_real(one_one_real),one_one_real) = numeral_numeral_real(bit0(one)) ).

% one_add_one
tff(fact_2057_power__strict__decreasing__iff,axiom,
    ! [B2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2058_power__strict__decreasing__iff,axiom,
    ! [B2: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(B2),M)),aa_nat_real(power_power_real(B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2059_power__strict__decreasing__iff,axiom,
    ! [B2: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B2),M)),aa_nat_int(power_power_int(B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2060_power__increasing__iff,axiom,
    ! [B2: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(B2),X)),aa_nat_real(power_power_real(B2),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2061_power__increasing__iff,axiom,
    ! [B2: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2062_power__increasing__iff,axiom,
    ! [B2: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B2),X)),aa_nat_int(power_power_int(B2),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2063_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% Suc_1
tff(fact_2064_one__plus__numeral,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_2065_one__plus__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_2066_one__plus__numeral,axiom,
    ! [N: num] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(N)) = numera1916890842035813515d_enat(plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_2067_one__plus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),numeral_numeral_int(N)) = numeral_numeral_int(plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_2068_one__plus__numeral,axiom,
    ! [N: num] : aa_real_real(plus_plus_real(one_one_real),numeral_numeral_real(N)) = numeral_numeral_real(plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_2069_numeral__plus__one,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),one_one_complex) = numera6690914467698888265omplex(plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_2070_numeral__plus__one,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat) = aa_num_nat(numeral_numeral_nat,plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_2071_numeral__plus__one,axiom,
    ! [N: num] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N)),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_2072_numeral__plus__one,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(N)),one_one_int) = numeral_numeral_int(plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_2073_numeral__plus__one,axiom,
    ! [N: num] : aa_real_real(plus_plus_real(numeral_numeral_real(N)),one_one_real) = numeral_numeral_real(plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_2074_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(N)),one_on7984719198319812577d_enat))
    <=> pp(aa_num_bool(ord_less_eq_num(N),one)) ) ).

% numeral_le_one_iff
tff(fact_2075_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),one_one_real))
    <=> pp(aa_num_bool(ord_less_eq_num(N),one)) ) ).

% numeral_le_one_iff
tff(fact_2076_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat))
    <=> pp(aa_num_bool(ord_less_eq_num(N),one)) ) ).

% numeral_le_one_iff
tff(fact_2077_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(N)),one_one_int))
    <=> pp(aa_num_bool(ord_less_eq_num(N),one)) ) ).

% numeral_le_one_iff
tff(fact_2078_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(ord_less_num(one),N)) ) ).

% one_less_numeral_iff
tff(fact_2079_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(ord_less_num(one),N)) ) ).

% one_less_numeral_iff
tff(fact_2080_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),numeral_numeral_int(N)))
    <=> pp(aa_num_bool(ord_less_num(one),N)) ) ).

% one_less_numeral_iff
tff(fact_2081_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),numeral_numeral_real(N)))
    <=> pp(aa_num_bool(ord_less_num(one),N)) ) ).

% one_less_numeral_iff
tff(fact_2082_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_2083_bits__1__div__2,axiom,
    aa_int_int(divide_divide_int(one_one_int),numeral_numeral_int(bit0(one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_2084_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_2085_one__div__two__eq__zero,axiom,
    aa_int_int(divide_divide_int(one_one_int),numeral_numeral_int(bit0(one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_2086_power__decreasing__iff,axiom,
    ! [B2: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(B2),M)),aa_nat_real(power_power_real(B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2087_power__decreasing__iff,axiom,
    ! [B2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2088_power__decreasing__iff,axiom,
    ! [B2: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B2),M)),aa_nat_int(power_power_int(B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2089_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_2090_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_2091_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
    <=> ( X = one_one_complex ) ) ).

% one_reorient
tff(fact_2092_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_2093_le__numeral__extra_I4_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),one_one_real)) ).

% le_numeral_extra(4)
tff(fact_2094_le__numeral__extra_I4_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),one_one_nat)) ).

% le_numeral_extra(4)
tff(fact_2095_le__numeral__extra_I4_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),one_one_int)) ).

% le_numeral_extra(4)
tff(fact_2096_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_2097_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_2098_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_2099_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_2100_zero__neq__one,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_neq_one
tff(fact_2101_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),one_one_nat)) ).

% less_numeral_extra(4)
tff(fact_2102_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,one_on7984719198319812577d_enat),one_on7984719198319812577d_enat)) ).

% less_numeral_extra(4)
tff(fact_2103_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),one_one_real)) ).

% less_numeral_extra(4)
tff(fact_2104_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),one_one_int)) ).

% less_numeral_extra(4)
tff(fact_2105_div__add__self1,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != 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,B2),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_2106_div__add__self1,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B2)),one_one_int) ) ) ).

% div_add_self1
tff(fact_2107_div__add__self2,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != 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),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_2108_div__add__self2,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B2)),one_one_int) ) ) ).

% div_add_self2
tff(fact_2109_not__one__le__zero,axiom,
    ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat)) ).

% not_one_le_zero
tff(fact_2110_not__one__le__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).

% not_one_le_zero
tff(fact_2111_not__one__le__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).

% not_one_le_zero
tff(fact_2112_not__one__le__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).

% not_one_le_zero
tff(fact_2113_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2114_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2115_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2116_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2117_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat)) ).

% zero_less_one_class.zero_le_one
tff(fact_2118_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% zero_less_one_class.zero_le_one
tff(fact_2119_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one_class.zero_le_one
tff(fact_2120_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% zero_less_one_class.zero_le_one
tff(fact_2121_not__one__less__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).

% not_one_less_zero
tff(fact_2122_not__one__less__zero,axiom,
    ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat)) ).

% not_one_less_zero
tff(fact_2123_not__one__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),zero_zero_real)) ).

% not_one_less_zero
tff(fact_2124_not__one__less__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),zero_zero_int)) ).

% not_one_less_zero
tff(fact_2125_zero__less__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one
tff(fact_2126_zero__less__one,axiom,
    pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat)) ).

% zero_less_one
tff(fact_2127_zero__less__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% zero_less_one
tff(fact_2128_zero__less__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% zero_less_one
tff(fact_2129_less__numeral__extra_I1_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% less_numeral_extra(1)
tff(fact_2130_less__numeral__extra_I1_J,axiom,
    pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat)) ).

% less_numeral_extra(1)
tff(fact_2131_less__numeral__extra_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% less_numeral_extra(1)
tff(fact_2132_less__numeral__extra_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% less_numeral_extra(1)
tff(fact_2133_one__le__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(N))) ).

% one_le_numeral
tff(fact_2134_one__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),numeral_numeral_real(N))) ).

% one_le_numeral
tff(fact_2135_one__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% one_le_numeral
tff(fact_2136_one__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),numeral_numeral_int(N))) ).

% one_le_numeral
tff(fact_2137_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat)) ).

% not_numeral_less_one
tff(fact_2138_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(N)),one_on7984719198319812577d_enat)) ).

% not_numeral_less_one
tff(fact_2139_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,numeral_numeral_int(N)),one_one_int)) ).

% not_numeral_less_one
tff(fact_2140_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(N)),one_one_real)) ).

% not_numeral_less_one
tff(fact_2141_add__mono1,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),one_one_nat))) ) ).

% add_mono1
tff(fact_2142_add__mono1,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),one_on7984719198319812577d_enat)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B2),one_on7984719198319812577d_enat))) ) ).

% add_mono1
tff(fact_2143_add__mono1,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_real_real(plus_plus_real(B2),one_one_real))) ) ).

% add_mono1
tff(fact_2144_add__mono1,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int))) ) ).

% add_mono1
tff(fact_2145_less__add__one,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))) ).

% less_add_one
tff(fact_2146_less__add__one,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(plus_plus_real(A),one_one_real))) ).

% less_add_one
tff(fact_2147_less__add__one,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))) ).

% less_add_one
tff(fact_2148_right__inverse__eq,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = one_one_complex )
      <=> ( A = B2 ) ) ) ).

% right_inverse_eq
tff(fact_2149_right__inverse__eq,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(A),B2) = one_one_real )
      <=> ( A = B2 ) ) ) ).

% right_inverse_eq
tff(fact_2150_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_2151_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_2152_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(X)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(X)),one_on7984719198319812577d_enat) ).

% one_plus_numeral_commute
tff(fact_2153_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),numeral_numeral_int(X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(X)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_2154_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_real_real(plus_plus_real(one_one_real),numeral_numeral_real(X)) = aa_real_real(plus_plus_real(numeral_numeral_real(X)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_2155_one__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(power_power_real(A),N))) ) ).

% one_le_power
tff(fact_2156_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% one_le_power
tff(fact_2157_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ).

% one_le_power
tff(fact_2158_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = M )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_2159_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ) ) ).

% div_less_dividend
tff(fact_2160_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_2161_power__0,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2162_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2163_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2164_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_2165_Suc__eq__plus1,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).

% Suc_eq_plus1
tff(fact_2166_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_2167_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).

% Suc_eq_plus1_left
tff(fact_2168_zero__less__two,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% zero_less_two
tff(fact_2169_zero__less__two,axiom,
    pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),one_on7984719198319812577d_enat))) ).

% zero_less_two
tff(fact_2170_zero__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),one_one_real))) ).

% zero_less_two
tff(fact_2171_zero__less__two,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int))) ).

% zero_less_two
tff(fact_2172_less__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B2),A)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) ) ) ) ).

% less_divide_eq_1
tff(fact_2173_divide__less__eq__1,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B2),A)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_2174_power__le__one,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),one_one_real)) ) ) ).

% power_le_one
tff(fact_2175_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat)) ) ) ).

% power_le_one
tff(fact_2176_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),one_one_int)) ) ) ).

% power_le_one
tff(fact_2177_less__half__sum,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B2)),aa_real_real(plus_plus_real(one_one_real),one_one_real)))) ) ).

% less_half_sum
tff(fact_2178_gt__half__sum,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B2)),aa_real_real(plus_plus_real(one_one_real),one_one_real))),B2)) ) ).

% gt_half_sum
tff(fact_2179_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_Extended_enat(power_8040749407984259932d_enat(zero_z5237406670263579293d_enat),N) = one_on7984719198319812577d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_Extended_enat(power_8040749407984259932d_enat(zero_z5237406670263579293d_enat),N) = zero_z5237406670263579293d_enat ) ) ) ).

% power_0_left
tff(fact_2180_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% power_0_left
tff(fact_2181_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(power_power_real(zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(power_power_real(zero_zero_real),N) = zero_zero_real ) ) ) ).

% power_0_left
tff(fact_2182_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% power_0_left
tff(fact_2183_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% power_0_left
tff(fact_2184_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_2185_power__gt1,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(power_power_real(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_2186_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_2187_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2188_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2189_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2190_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5))) ) ) ).

% power_strict_increasing
tff(fact_2191_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(A),N5))) ) ) ).

% power_strict_increasing
tff(fact_2192_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N5))) ) ) ).

% power_strict_increasing
tff(fact_2193_power__increasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(A),N5))) ) ) ).

% power_increasing
tff(fact_2194_power__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5))) ) ) ).

% power_increasing
tff(fact_2195_power__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N5))) ) ) ).

% power_increasing
tff(fact_2196_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_2197_divide__le__eq__1,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B2),A)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_2198_le__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B2),A)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ) ) ).

% le_divide_eq_1
tff(fact_2199_power__Suc__le__self,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2200_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2201_power__Suc__le__self,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2202_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),one_one_nat)) ) ) ).

% power_Suc_less_one
tff(fact_2203_power__Suc__less__one,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,N))),one_one_real)) ) ) ).

% power_Suc_less_one
tff(fact_2204_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),one_one_int)) ) ) ).

% power_Suc_less_one
tff(fact_2205_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2206_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N5)),aa_nat_real(power_power_real(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2207_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N5)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2208_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N5)),aa_nat_real(power_power_real(A),N))) ) ) ) ).

% power_decreasing
tff(fact_2209_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).

% power_decreasing
tff(fact_2210_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N5)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_decreasing
tff(fact_2211_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2212_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2213_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2214_self__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(power_power_real(A),N))) ) ) ).

% self_le_power
tff(fact_2215_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% self_le_power
tff(fact_2216_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(A),N))) ) ) ).

% self_le_power
tff(fact_2217_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% one_less_power
tff(fact_2218_one__less__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(power_power_real(A),N))) ) ) ).

% one_less_power
tff(fact_2219_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ) ).

% one_less_power
tff(fact_2220_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% nat_1_add_1
tff(fact_2221_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),K))) ) ).

% div_le_mono
tff(fact_2222_div__le__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ).

% div_le_dividend
tff(fact_2223_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K))
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_2224_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),K))
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_2225_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2226_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),N))) ).

% Suc_div_le_mono
tff(fact_2227_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M))) ) ) ).

% div_le_mono2
tff(fact_2228_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% div_greater_zero_iff
tff(fact_2229_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2230_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2231_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2232_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2233_nat__induct2,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct2
tff(fact_2234_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(H),L),D) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),D))),L) ).

% bit_concat_def
tff(fact_2235_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))),X)),N) = X ) ) ).

% low_inv
tff(fact_2236_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))),X)),N) = Y ) ) ).

% high_inv
tff(fact_2237_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
     => ( ( ? [A3: bool,B3: bool] : A1 = vEBT_Leaf(A3,B3)
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X2: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(vEBT_invar_vebt(X2),N2)) )
                 => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M3))
                   => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3) )
                     => ( ( M3 = N2 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                         => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_1))
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                                 => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_1)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X2: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_invar_vebt(X2),N2)) )
                   => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M3))
                     => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3) )
                       => ( ( M3 = aa_nat_nat(suc,N2) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                           => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_1))
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                                   => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_1)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X2: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_invar_vebt(X2),N2)) )
                     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M3))
                       => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3) )
                         => ( ( M3 = N2 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                             => ( ! [I5: nat] :
                                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3)))
                                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I5)),X_13))
                                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I5)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X2: vEBT_VEBT] :
                                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_1)) ) )
                                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg2)))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I5: nat] :
                                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3)))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I5 )
                                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                & ! [X2: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N2) = I5 )
                                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N2))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X2))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X2: vEBT_VEBT] :
                            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                           => pp(aa_nat_bool(vEBT_invar_vebt(X2),N2)) )
                       => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M3))
                         => ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3) )
                           => ( ( M3 = aa_nat_nat(suc,N2) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                               => ( ! [I5: nat] :
                                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3)))
                                     => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I5)),X_13))
                                      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I5)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X2: vEBT_VEBT] :
                                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                                         => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_1)) ) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg2)))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I5: nat] :
                                                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3)))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I5 )
                                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                  & ! [X2: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N2) = I5 )
                                                        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N2))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X2))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_2238_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
    <=> ( ( ? [A6: bool,B6: bool] : A1 = vEBT_Leaf(A6,B6)
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X5),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N4))
            & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X5),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N4)))
            & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,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_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X5),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N4))
            & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ! [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N4)))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N4)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I3 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N4))) )
                    & ! [X5: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N4) = I3 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N4))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X5))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma3)) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X5),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N4)))
            & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,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)) )
            & ! [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,N4))))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,N4))))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I3 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N4))) )
                    & ! [X5: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N4) = I3 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N4))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X5))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_2239_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(1)
tff(fact_2240_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(2)
tff(fact_2241_pos2,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% pos2
tff(fact_2242_Leaf__0__not,axiom,
    ! [A: bool,B2: bool] : ~ pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B2)),zero_zero_nat)) ).

% Leaf_0_not
tff(fact_2243_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( N = one_one_nat )
       => ? [A3: bool,B3: bool] : T = vEBT_Leaf(A3,B3) ) ) ).

% deg_1_Leafy
tff(fact_2244_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
     => ? [A3: bool,B3: bool] : T = vEBT_Leaf(A3,B3) ) ).

% deg_1_Leaf
tff(fact_2245_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
    <=> ? [A6: bool,B6: bool] : T = vEBT_Leaf(A6,B6) ) ).

% deg1Leaf
tff(fact_2246_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B2: 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,B2),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
tff(fact_2247_mult__cancel__right,axiom,
    ! [A: int,C: int,B2: 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,B2),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
tff(fact_2248_mult__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( aa_real_real(times_times_real(A),C) = aa_real_real(times_times_real(B2),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
tff(fact_2249_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B2),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
tff(fact_2250_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B2: 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),B2) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
tff(fact_2251_mult__cancel__left,axiom,
    ! [C: int,A: int,B2: 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),B2) )
    <=> ( ( C = zero_zero_int )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
tff(fact_2252_mult__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( aa_real_real(times_times_real(C),A) = aa_real_real(times_times_real(C),B2) )
    <=> ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
tff(fact_2253_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B2) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
tff(fact_2254_mult__eq__0__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_2255_mult__eq__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_2256_mult__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(times_times_real(A),B2) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_2257_mult__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B2) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_2258_mult__eq__0__iff,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B2) = zero_z5237406670263579293d_enat )
    <=> ( ( A = zero_z5237406670263579293d_enat )
        | ( B2 = zero_z5237406670263579293d_enat ) ) ) ).

% mult_eq_0_iff
tff(fact_2259_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_2260_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_2261_mult__zero__right,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_2262_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_2263_mult__zero__right,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% mult_zero_right
tff(fact_2264_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_2265_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_2266_mult__zero__left,axiom,
    ! [A: real] : aa_real_real(times_times_real(zero_zero_real),A) = zero_zero_real ).

% mult_zero_left
tff(fact_2267_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_2268_mult__zero__left,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(zero_z5237406670263579293d_enat),A) = zero_z5237406670263579293d_enat ).

% mult_zero_left
tff(fact_2269_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_2270_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_2271_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),one_one_real) = A ).

% mult.right_neutral
tff(fact_2272_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_2273_mult_Oright__neutral,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),one_on7984719198319812577d_enat) = A ).

% mult.right_neutral
tff(fact_2274_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_2275_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_2276_mult__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(one_one_real),A) = A ).

% mult_1
tff(fact_2277_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_2278_mult__1,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(one_on7984719198319812577d_enat),A) = A ).

% mult_1
tff(fact_2279_times__divide__eq__left,axiom,
    ! [B2: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B2),C)),A) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B2),A)),C) ).

% times_divide_eq_left
tff(fact_2280_times__divide__eq__left,axiom,
    ! [B2: real,C: real,A: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(B2),C)),A) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(B2),A)),C) ).

% times_divide_eq_left
tff(fact_2281_divide__divide__eq__left,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B2)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(B2),C)) ).

% divide_divide_eq_left
tff(fact_2282_divide__divide__eq__left,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A),B2)),C) = aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(B2),C)) ).

% divide_divide_eq_left
tff(fact_2283_divide__divide__eq__right,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(divide1717551699836669952omplex(B2),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),B2) ).

% divide_divide_eq_right
tff(fact_2284_divide__divide__eq__right,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(divide_divide_real(A),aa_real_real(divide_divide_real(B2),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),B2) ).

% divide_divide_eq_right
tff(fact_2285_times__divide__eq__right,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(divide1717551699836669952omplex(B2),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B2)),C) ).

% times_divide_eq_right
tff(fact_2286_times__divide__eq__right,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(divide_divide_real(B2),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B2)),C) ).

% times_divide_eq_right
tff(fact_2287_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_2288_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_2289_mult__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_2290_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_2291_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_2292_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_2293_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_2294_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(times_times_real(A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_2295_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_2296_mult__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C) )
    <=> ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_2297_mult__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( ( C = aa_real_real(times_times_real(B2),C) )
    <=> ( ( C = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_2298_mult__cancel__right1,axiom,
    ! [C: complex,B2: complex] :
      ( ( C = aa_complex_complex(times_times_complex(B2),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_2299_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_2300_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(times_times_real(C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_2301_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_2302_mult__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) )
    <=> ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_2303_mult__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( ( C = aa_real_real(times_times_real(C),B2) )
    <=> ( ( C = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_2304_mult__cancel__left1,axiom,
    ! [C: complex,B2: complex] :
      ( ( C = aa_complex_complex(times_times_complex(C),B2) )
    <=> ( ( C = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_2305_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2306_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y),Y)) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2307_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(C),B2)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2308_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(C),B2)) = aa_real_real(divide_divide_real(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2309_nonzero__mult__div__cancel__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2310_nonzero__mult__div__cancel__right,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != 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),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2311_nonzero__mult__div__cancel__right,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2312_nonzero__mult__div__cancel__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2313_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B2),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2314_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C)) = aa_real_real(divide_divide_real(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2315_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(B2),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2316_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(B2),C)) = aa_real_real(divide_divide_real(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2317_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2318_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B2: 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),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2319_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2320_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2321_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B2)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2322_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2)) = aa_real_real(divide_divide_real(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2323_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( ( C = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B2)) = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B2)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2324_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ( C = zero_zero_real )
       => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2)) = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2)) = aa_real_real(divide_divide_real(A),B2) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2325_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B2: 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),B2)) = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) ) ) ) ).

% div_mult_mult1_if
tff(fact_2326_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ( C = zero_zero_int )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) = aa_int_int(divide_divide_int(A),B2) ) ) ) ).

% div_mult_mult1_if
tff(fact_2327_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B2: 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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) ) ) ).

% div_mult_mult2
tff(fact_2328_div__mult__mult2,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(divide_divide_int(A),B2) ) ) ).

% div_mult_mult2
tff(fact_2329_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B2: 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),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) ) ) ).

% div_mult_mult1
tff(fact_2330_div__mult__mult1,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) = aa_int_int(divide_divide_int(A),B2) ) ) ).

% div_mult_mult1
tff(fact_2331_distrib__right__numeral,axiom,
    ! [A: complex,B2: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B2)),numera6690914467698888265omplex(V)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B2),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_2332_distrib__right__numeral,axiom,
    ! [A: nat,B2: 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),B2)),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,B2),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_2333_distrib__right__numeral,axiom,
    ! [A: extended_enat,B2: extended_enat,V: num] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)),numera1916890842035813515d_enat(V)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),numera1916890842035813515d_enat(V))),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),numera1916890842035813515d_enat(V))) ).

% distrib_right_numeral
tff(fact_2334_distrib__right__numeral,axiom,
    ! [A: int,B2: 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),B2)),numeral_numeral_int(V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(V))),aa_int_int(aa_int_fun_int_int(times_times_int,B2),numeral_numeral_int(V))) ).

% distrib_right_numeral
tff(fact_2335_distrib__right__numeral,axiom,
    ! [A: real,B2: real,V: num] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B2)),numeral_numeral_real(V)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),numeral_numeral_real(V))),aa_real_real(times_times_real(B2),numeral_numeral_real(V))) ).

% distrib_right_numeral
tff(fact_2336_distrib__left__numeral,axiom,
    ! [V: num,B2: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B2),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B2)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% distrib_left_numeral
tff(fact_2337_distrib__left__numeral,axiom,
    ! [V: num,B2: 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,B2),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)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C)) ).

% distrib_left_numeral
tff(fact_2338_distrib__left__numeral,axiom,
    ! [V: num,B2: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B2),C)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V)),B2)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V)),C)) ).

% distrib_left_numeral
tff(fact_2339_distrib__left__numeral,axiom,
    ! [V: num,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),C)) ).

% distrib_left_numeral
tff(fact_2340_distrib__left__numeral,axiom,
    ! [V: num,B2: real,C: real] : aa_real_real(times_times_real(numeral_numeral_real(V)),aa_real_real(plus_plus_real(B2),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(numeral_numeral_real(V)),B2)),aa_real_real(times_times_real(numeral_numeral_real(V)),C)) ).

% distrib_left_numeral
tff(fact_2341_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_2342_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_2343_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% mult_less_cancel2
tff(fact_2344_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_2345_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc_right
tff(fact_2346_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(divide_divide_real(B2),numeral_numeral_real(W2))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),numeral_numeral_real(W2))),B2)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2347_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B2),numeral_numeral_real(W2))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(times_times_real(A),numeral_numeral_real(W2)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2348_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: complex,W2: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),numera6690914467698888265omplex(W2)) = A )
    <=> ( ( ( numera6690914467698888265omplex(W2) != zero_zero_complex )
         => ( B2 = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)) ) )
        & ( ( numera6690914467698888265omplex(W2) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2349_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( ( aa_real_real(divide_divide_real(B2),numeral_numeral_real(W2)) = A )
    <=> ( ( ( numeral_numeral_real(W2) != zero_zero_real )
         => ( B2 = aa_real_real(times_times_real(A),numeral_numeral_real(W2)) ) )
        & ( ( numeral_numeral_real(W2) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2350_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B2: complex,W2: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B2),numera6690914467698888265omplex(W2)) )
    <=> ( ( ( numera6690914467698888265omplex(W2) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)) = B2 ) )
        & ( ( numera6690914467698888265omplex(W2) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2351_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( ( A = aa_real_real(divide_divide_real(B2),numeral_numeral_real(W2)) )
    <=> ( ( ( numeral_numeral_real(W2) != zero_zero_real )
         => ( aa_real_real(times_times_real(A),numeral_numeral_real(W2)) = B2 ) )
        & ( ( numeral_numeral_real(W2) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2352_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(B2),numeral_numeral_real(W2))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),numeral_numeral_real(W2))),B2)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2353_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B2),numeral_numeral_real(W2))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(times_times_real(A),numeral_numeral_real(W2)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2354_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2355_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(A),B2)) = aa_real_real(divide_divide_real(one_one_real),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2356_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(B2),aa_complex_complex(times_times_complex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2357_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(B2),aa_real_real(times_times_real(A),B2)) = aa_real_real(divide_divide_real(one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2358_div__mult__self4,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( B2 != 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,B2),C)),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ) ) ).

% div_mult_self4
tff(fact_2359_div__mult__self4,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)),A)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B2)) ) ) ).

% div_mult_self4
tff(fact_2360_div__mult__self3,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( B2 != 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),B2)),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ) ) ).

% div_mult_self3
tff(fact_2361_div__mult__self3,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)),A)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B2)) ) ) ).

% div_mult_self3
tff(fact_2362_div__mult__self2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != 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,B2),C))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ) ) ).

% div_mult_self2
tff(fact_2363_div__mult__self2,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B2)) ) ) ).

% div_mult_self2
tff(fact_2364_div__mult__self1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != 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),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ) ) ).

% div_mult_self1
tff(fact_2365_div__mult__self1,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B2)) ) ) ).

% div_mult_self1
tff(fact_2366_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ) ) ).

% one_le_mult_iff
tff(fact_2367_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% mult_le_cancel2
tff(fact_2368_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_2369_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_2370_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B2: 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),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2371_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B2: 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),B2)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2372_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(A),B2)),C) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B2),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2373_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),B2)),C) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B2),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2374_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B2)),C) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2375_mult_Oassoc,axiom,
    ! [A: nat,B2: 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),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) ).

% mult.assoc
tff(fact_2376_mult_Oassoc,axiom,
    ! [A: int,B2: 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),B2)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% mult.assoc
tff(fact_2377_mult_Oassoc,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(A),B2)),C) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B2),C)) ).

% mult.assoc
tff(fact_2378_mult_Oassoc,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),B2)),C) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B2),C)) ).

% mult.assoc
tff(fact_2379_mult_Oassoc,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B2)),C) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),C)) ).

% mult.assoc
tff(fact_2380_mult_Ocommute,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A) ).

% mult.commute
tff(fact_2381_mult_Ocommute,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),A) ).

% mult.commute
tff(fact_2382_mult_Ocommute,axiom,
    ! [A: real,B2: real] : aa_real_real(times_times_real(A),B2) = aa_real_real(times_times_real(B2),A) ).

% mult.commute
tff(fact_2383_mult_Ocommute,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(times_times_complex(A),B2) = aa_complex_complex(times_times_complex(B2),A) ).

% mult.commute
tff(fact_2384_mult_Ocommute,axiom,
    ! [A: extended_enat,B2: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B2) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),A) ).

% mult.commute
tff(fact_2385_mult_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),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,B2),C)) ).

% mult.left_commute
tff(fact_2386_mult_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B2),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,B2),C)) ).

% mult.left_commute
tff(fact_2387_mult_Oleft__commute,axiom,
    ! [B2: real,A: real,C: real] : aa_real_real(times_times_real(B2),aa_real_real(times_times_real(A),C)) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B2),C)) ).

% mult.left_commute
tff(fact_2388_mult_Oleft__commute,axiom,
    ! [B2: complex,A: complex,C: complex] : aa_complex_complex(times_times_complex(B2),aa_complex_complex(times_times_complex(A),C)) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B2),C)) ).

% mult.left_commute
tff(fact_2389_mult_Oleft__commute,axiom,
    ! [B2: extended_enat,A: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),C)) ).

% mult.left_commute
tff(fact_2390_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf(X21,X222)) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_2391_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B2: 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,B2),C) )
      <=> ( A = B2 ) ) ) ).

% mult_right_cancel
tff(fact_2392_mult__right__cancel,axiom,
    ! [C: int,A: int,B2: 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,B2),C) )
      <=> ( A = B2 ) ) ) ).

% mult_right_cancel
tff(fact_2393_mult__right__cancel,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(A),C) = aa_real_real(times_times_real(B2),C) )
      <=> ( A = B2 ) ) ) ).

% mult_right_cancel
tff(fact_2394_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B2),C) )
      <=> ( A = B2 ) ) ) ).

% mult_right_cancel
tff(fact_2395_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B2: 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),B2) )
      <=> ( A = B2 ) ) ) ).

% mult_left_cancel
tff(fact_2396_mult__left__cancel,axiom,
    ! [C: int,A: int,B2: 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),B2) )
      <=> ( A = B2 ) ) ) ).

% mult_left_cancel
tff(fact_2397_mult__left__cancel,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(C),A) = aa_real_real(times_times_real(C),B2) )
      <=> ( A = B2 ) ) ) ).

% mult_left_cancel
tff(fact_2398_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B2) )
      <=> ( A = B2 ) ) ) ).

% mult_left_cancel
tff(fact_2399_no__zero__divisors,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_2400_no__zero__divisors,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_2401_no__zero__divisors,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(times_times_real(A),B2) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_2402_no__zero__divisors,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( aa_complex_complex(times_times_complex(A),B2) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_2403_no__zero__divisors,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( A != zero_z5237406670263579293d_enat )
     => ( ( B2 != zero_z5237406670263579293d_enat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B2) != zero_z5237406670263579293d_enat ) ) ) ).

% no_zero_divisors
tff(fact_2404_divisors__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_2405_divisors__zero,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_2406_divisors__zero,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(times_times_real(A),B2) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_2407_divisors__zero,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B2) = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_2408_divisors__zero,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B2) = zero_z5237406670263579293d_enat )
     => ( ( A = zero_z5237406670263579293d_enat )
        | ( B2 = zero_z5237406670263579293d_enat ) ) ) ).

% divisors_zero
tff(fact_2409_mult__not__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B2 != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_2410_mult__not__zero,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B2 != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_2411_mult__not__zero,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(times_times_real(A),B2) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B2 != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_2412_mult__not__zero,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B2) != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B2 != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_2413_mult__not__zero,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B2) != zero_z5237406670263579293d_enat )
     => ( ( A != zero_z5237406670263579293d_enat )
        & ( B2 != zero_z5237406670263579293d_enat ) ) ) ).

% mult_not_zero
tff(fact_2414_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_2415_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_2416_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2417_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2418_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(one_on7984719198319812577d_enat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2419_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_2420_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_2421_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_2422_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_2423_mult_Ocomm__neutral,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),one_on7984719198319812577d_enat) = A ).

% mult.comm_neutral
tff(fact_2424_crossproduct__noteq,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ( A != B2 )
        & ( 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,B2),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,B2),C)) ) ) ).

% crossproduct_noteq
tff(fact_2425_crossproduct__noteq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( A != B2 )
        & ( 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,B2),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,B2),C)) ) ) ).

% crossproduct_noteq
tff(fact_2426_crossproduct__noteq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( A != B2 )
        & ( C != D ) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),D)) != aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),D)),aa_real_real(times_times_real(B2),C)) ) ) ).

% crossproduct_noteq
tff(fact_2427_crossproduct__noteq,axiom,
    ! [A: complex,B2: complex,C: complex,D: complex] :
      ( ( ( A != B2 )
        & ( C != D ) )
    <=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B2),D)) != aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),D)),aa_complex_complex(times_times_complex(B2),C)) ) ) ).

% crossproduct_noteq
tff(fact_2428_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X: nat,Z2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W2),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W2),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
tff(fact_2429_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X: int,Z2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z2)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
tff(fact_2430_crossproduct__eq,axiom,
    ! [W2: real,Y: real,X: real,Z2: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(W2),Y)),aa_real_real(times_times_real(X),Z2)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(W2),Z2)),aa_real_real(times_times_real(X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
tff(fact_2431_crossproduct__eq,axiom,
    ! [W2: complex,Y: complex,X: complex,Z2: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(W2),Y)),aa_complex_complex(times_times_complex(X),Z2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(W2),Z2)),aa_complex_complex(times_times_complex(X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
tff(fact_2432_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),E2)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),E2)),C) ).

% combine_common_factor
tff(fact_2433_combine__common__factor,axiom,
    ! [A: int,E2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),E2)),C) ).

% combine_common_factor
tff(fact_2434_combine__common__factor,axiom,
    ! [A: real,E2: real,B2: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E2)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B2),E2)),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B2)),E2)),C) ).

% combine_common_factor
tff(fact_2435_combine__common__factor,axiom,
    ! [A: complex,E2: complex,B2: complex,C: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),E2)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(B2),E2)),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B2)),E2)),C) ).

% combine_common_factor
tff(fact_2436_combine__common__factor,axiom,
    ! [A: extended_enat,E2: extended_enat,B2: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),E2)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),E2)),C)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)),E2)),C) ).

% combine_common_factor
tff(fact_2437_distrib__right,axiom,
    ! [A: nat,B2: 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),B2)),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,B2),C)) ).

% distrib_right
tff(fact_2438_distrib__right,axiom,
    ! [A: int,B2: 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),B2)),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,B2),C)) ).

% distrib_right
tff(fact_2439_distrib__right,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B2)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C)) ).

% distrib_right
tff(fact_2440_distrib__right,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B2)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B2),C)) ).

% distrib_right
tff(fact_2441_distrib__right,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)),C) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),C)) ).

% distrib_right
tff(fact_2442_distrib__left,axiom,
    ! [A: nat,B2: 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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_2443_distrib__left,axiom,
    ! [A: int,B2: 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,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% distrib_left
tff(fact_2444_distrib__left,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(plus_plus_real(B2),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),B2)),aa_real_real(times_times_real(A),C)) ).

% distrib_left
tff(fact_2445_distrib__left,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(plus_plus_complex(B2),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),B2)),aa_complex_complex(times_times_complex(A),C)) ).

% distrib_left
tff(fact_2446_distrib__left,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B2),C)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B2)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)) ).

% distrib_left
tff(fact_2447_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B2: 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),B2)),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,B2),C)) ).

% comm_semiring_class.distrib
tff(fact_2448_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B2: 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),B2)),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,B2),C)) ).

% comm_semiring_class.distrib
tff(fact_2449_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B2)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C)) ).

% comm_semiring_class.distrib
tff(fact_2450_comm__semiring__class_Odistrib,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B2)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B2),C)) ).

% comm_semiring_class.distrib
tff(fact_2451_comm__semiring__class_Odistrib,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)),C) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),C)) ).

% comm_semiring_class.distrib
tff(fact_2452_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B2: 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,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2453_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(plus_plus_real(B2),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),B2)),aa_real_real(times_times_real(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2454_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(plus_plus_complex(B2),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),B2)),aa_complex_complex(times_times_complex(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2455_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B2: 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),B2)),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,B2),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2456_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B2)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2457_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B2)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B2),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2458_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z2: complex,W2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z2),W2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),Z2)),aa_complex_complex(times_times_complex(Y),W2)) ).

% times_divide_times_eq
tff(fact_2459_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z2: real,W2: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(Z2),W2)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(Y),W2)) ).

% times_divide_times_eq
tff(fact_2460_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z2: complex,W2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z2),W2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),W2)),aa_complex_complex(times_times_complex(Y),Z2)) ).

% divide_divide_times_eq
tff(fact_2461_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z2: real,W2: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(Z2),W2)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(X),W2)),aa_real_real(times_times_real(Y),Z2)) ).

% divide_divide_times_eq
tff(fact_2462_divide__divide__eq__left_H,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B2)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(C),B2)) ).

% divide_divide_eq_left'
tff(fact_2463_divide__divide__eq__left_H,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A),B2)),C) = aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(C),B2)) ).

% divide_divide_eq_left'
tff(fact_2464_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_2465_mult__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).

% mult_0
tff(fact_2466_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ).

% mult_le_mono2
tff(fact_2467_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ).

% mult_le_mono1
tff(fact_2468_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L))) ) ) ).

% mult_le_mono
tff(fact_2469_le__square,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_square
tff(fact_2470_le__cube,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)))) ).

% le_cube
tff(fact_2471_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).

% add_mult_distrib2
tff(fact_2472_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).

% add_mult_distrib
tff(fact_2473_nat__mult__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).

% nat_mult_1
tff(fact_2474_nat__mult__1__right,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).

% nat_mult_1_right
tff(fact_2475_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Leaf(Uu,Uv)),Uw)) ).

% VEBT_internal.membermima.simps(1)
tff(fact_2476_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fFalse,fFalse))) ).

% VEBT_internal.minNull.simps(1)
tff(fact_2477_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fTrue,Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_2478_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(Uu,fTrue))) ).

% VEBT_internal.minNull.simps(3)
tff(fact_2479_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ).

% nat_mult_max_left
tff(fact_2480_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ).

% nat_mult_max_right
tff(fact_2481_mult__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,C),D))
       => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),B2))
         => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),C))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),D))) ) ) ) ) ).

% mult_mono
tff(fact_2482_mult__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),D))) ) ) ) ) ).

% mult_mono
tff(fact_2483_mult__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D))) ) ) ) ) ).

% mult_mono
tff(fact_2484_mult__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))) ) ) ) ) ).

% mult_mono
tff(fact_2485_mult__mono_H,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,C),D))
       => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),A))
         => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),C))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2486_mult__mono_H,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2487_mult__mono_H,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2488_mult__mono_H,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2489_zero__le__square,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),A))) ).

% zero_le_square
tff(fact_2490_zero__le__square,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A))) ).

% zero_le_square
tff(fact_2491_split__mult__pos__le,axiom,
    ! [A: real,B2: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B2))) ) ).

% split_mult_pos_le
tff(fact_2492_split__mult__pos__le,axiom,
    ! [A: int,B2: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))) ) ).

% split_mult_pos_le
tff(fact_2493_mult__left__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2))) ) ) ).

% mult_left_mono_neg
tff(fact_2494_mult__left__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))) ) ) ).

% mult_left_mono_neg
tff(fact_2495_mult__nonpos__nonpos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B2))) ) ) ).

% mult_nonpos_nonpos
tff(fact_2496_mult__nonpos__nonpos,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))) ) ) ).

% mult_nonpos_nonpos
tff(fact_2497_mult__left__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),C))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(C),A)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(C),B2))) ) ) ).

% mult_left_mono
tff(fact_2498_mult__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2))) ) ) ).

% mult_left_mono
tff(fact_2499_mult__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))) ) ) ).

% mult_left_mono
tff(fact_2500_mult__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))) ) ) ).

% mult_left_mono
tff(fact_2501_mult__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2502_mult__right__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2503_mult__right__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),C))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),C))) ) ) ).

% mult_right_mono
tff(fact_2504_mult__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C))) ) ) ).

% mult_right_mono
tff(fact_2505_mult__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C))) ) ) ).

% mult_right_mono
tff(fact_2506_mult__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))) ) ) ).

% mult_right_mono
tff(fact_2507_mult__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),B2)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) ) ) ) ).

% mult_le_0_iff
tff(fact_2508_mult__le__0__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) ) ) ) ).

% mult_le_0_iff
tff(fact_2509_split__mult__neg__le,axiom,
    ! [A: real,B2: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),B2)),zero_zero_real)) ) ).

% split_mult_neg_le
tff(fact_2510_split__mult__neg__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),zero_zero_nat)) ) ).

% split_mult_neg_le
tff(fact_2511_split__mult__neg__le,axiom,
    ! [A: int,B2: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int)) ) ).

% split_mult_neg_le
tff(fact_2512_mult__nonneg__nonneg,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B2))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2513_mult__nonneg__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2514_mult__nonneg__nonneg,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2515_mult__nonneg__nonpos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),B2)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2516_mult__nonneg__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2517_mult__nonneg__nonpos,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2518_mult__nonpos__nonneg,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),B2)),zero_zero_real)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2519_mult__nonpos__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),zero_zero_nat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2520_mult__nonpos__nonneg,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2521_mult__nonneg__nonpos2,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(B2),A)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2522_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2523_mult__nonneg__nonpos2,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2524_zero__le__mult__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2525_zero__le__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2526_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),C))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(C),A)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(C),B2))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2527_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2528_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2529_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2530_mult__neg__neg,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B2))) ) ) ).

% mult_neg_neg
tff(fact_2531_mult__neg__neg,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))) ) ) ).

% mult_neg_neg
tff(fact_2532_not__square__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),A)),zero_zero_real)) ).

% not_square_less_zero
tff(fact_2533_not__square__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int)) ).

% not_square_less_zero
tff(fact_2534_mult__less__0__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),B2)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) ) ) ) ).

% mult_less_0_iff
tff(fact_2535_mult__less__0__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ) ) ).

% mult_less_0_iff
tff(fact_2536_mult__neg__pos,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),zero_zero_nat)) ) ) ).

% mult_neg_pos
tff(fact_2537_mult__neg__pos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),B2)),zero_zero_real)) ) ) ).

% mult_neg_pos
tff(fact_2538_mult__neg__pos,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int)) ) ) ).

% mult_neg_pos
tff(fact_2539_mult__pos__neg,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),zero_zero_nat)) ) ) ).

% mult_pos_neg
tff(fact_2540_mult__pos__neg,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),B2)),zero_zero_real)) ) ) ).

% mult_pos_neg
tff(fact_2541_mult__pos__neg,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int)) ) ) ).

% mult_pos_neg
tff(fact_2542_mult__pos__pos,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2))) ) ) ).

% mult_pos_pos
tff(fact_2543_mult__pos__pos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B2))) ) ) ).

% mult_pos_pos
tff(fact_2544_mult__pos__pos,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))) ) ) ).

% mult_pos_pos
tff(fact_2545_mult__pos__neg2,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A)),zero_zero_nat)) ) ) ).

% mult_pos_neg2
tff(fact_2546_mult__pos__neg2,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(B2),A)),zero_zero_real)) ) ) ).

% mult_pos_neg2
tff(fact_2547_mult__pos__neg2,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),zero_zero_int)) ) ) ).

% mult_pos_neg2
tff(fact_2548_zero__less__mult__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real)) ) ) ) ).

% zero_less_mult_iff
tff(fact_2549_zero__less__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int)) ) ) ) ).

% zero_less_mult_iff
tff(fact_2550_zero__less__mult__pos,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2)) ) ) ).

% zero_less_mult_pos
tff(fact_2551_zero__less__mult__pos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) ) ) ).

% zero_less_mult_pos
tff(fact_2552_zero__less__mult__pos,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ) ).

% zero_less_mult_pos
tff(fact_2553_zero__less__mult__pos2,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2)) ) ) ).

% zero_less_mult_pos2
tff(fact_2554_zero__less__mult__pos2,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(times_times_real(B2),A)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) ) ) ).

% zero_less_mult_pos2
tff(fact_2555_zero__less__mult__pos2,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ) ).

% zero_less_mult_pos2
tff(fact_2556_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2557_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2558_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2559_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2560_mult__strict__left__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2561_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2562_mult__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))) ) ) ).

% mult_strict_left_mono
tff(fact_2563_mult__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2))) ) ) ).

% mult_strict_left_mono
tff(fact_2564_mult__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))) ) ) ).

% mult_strict_left_mono
tff(fact_2565_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2566_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2567_mult__strict__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2568_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2569_mult__strict__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C))) ) ) ).

% mult_strict_right_mono
tff(fact_2570_mult__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C))) ) ) ).

% mult_strict_right_mono
tff(fact_2571_mult__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))) ) ) ).

% mult_strict_right_mono
tff(fact_2572_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2573_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2574_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2575_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2576_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2577_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B2: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B2 )
          & ( 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,B2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2578_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B2: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B2 )
          & ( 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,B2),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2579_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B2: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( aa_real_real(plus_plus_real(A),aa_real_real(times_times_real(R2),C)) != aa_real_real(plus_plus_real(B2),aa_real_real(times_times_real(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2580_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B2: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R2),C)) != aa_complex_complex(plus_plus_complex(B2),aa_complex_complex(times_times_complex(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2581_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% less_1_mult
tff(fact_2582_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(times_times_real(M),N))) ) ) ).

% less_1_mult
tff(fact_2583_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),N))) ) ) ).

% less_1_mult
tff(fact_2584_frac__eq__eq,axiom,
    ! [Y: complex,Z2: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(divide1717551699836669952omplex(W2),Z2) )
        <=> ( aa_complex_complex(times_times_complex(X),Z2) = aa_complex_complex(times_times_complex(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2585_frac__eq__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( aa_real_real(divide_divide_real(X),Y) = aa_real_real(divide_divide_real(W2),Z2) )
        <=> ( aa_real_real(times_times_real(X),Z2) = aa_real_real(times_times_real(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2586_divide__eq__eq,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C) = A )
    <=> ( ( ( C != zero_zero_complex )
         => ( B2 = aa_complex_complex(times_times_complex(A),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
tff(fact_2587_divide__eq__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( aa_real_real(divide_divide_real(B2),C) = A )
    <=> ( ( ( C != zero_zero_real )
         => ( B2 = aa_real_real(times_times_real(A),C) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
tff(fact_2588_eq__divide__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B2),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C) = B2 ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
tff(fact_2589_eq__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A = aa_real_real(divide_divide_real(B2),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(times_times_real(A),C) = B2 ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
tff(fact_2590_divide__eq__imp,axiom,
    ! [C: complex,B2: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B2 = aa_complex_complex(times_times_complex(A),C) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B2),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2591_divide__eq__imp,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B2 = aa_real_real(times_times_real(A),C) )
       => ( aa_real_real(divide_divide_real(B2),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2592_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = B2 )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B2),C) ) ) ) ).

% eq_divide_imp
tff(fact_2593_eq__divide__imp,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(A),C) = B2 )
       => ( A = aa_real_real(divide_divide_real(B2),C) ) ) ) ).

% eq_divide_imp
tff(fact_2594_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B2: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C) = A )
      <=> ( B2 = aa_complex_complex(times_times_complex(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2595_nonzero__divide__eq__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(B2),C) = A )
      <=> ( B2 = aa_real_real(times_times_real(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2596_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B2),C) )
      <=> ( aa_complex_complex(times_times_complex(A),C) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2597_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(divide_divide_real(B2),C) )
      <=> ( aa_real_real(times_times_real(A),C) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2598_power__Suc,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_Suc
tff(fact_2599_power__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).

% power_Suc
tff(fact_2600_power__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,N)) = aa_real_real(times_times_real(A),aa_nat_real(power_power_real(A),N)) ).

% power_Suc
tff(fact_2601_power__Suc,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),N)) ).

% power_Suc
tff(fact_2602_power__Suc,axiom,
    ! [A: extended_enat,N: nat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(A),aa_nat_nat(suc,N)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_nat_Extended_enat(power_8040749407984259932d_enat(A),N)) ).

% power_Suc
tff(fact_2603_power__Suc2,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) ).

% power_Suc2
tff(fact_2604_power__Suc2,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) ).

% power_Suc2
tff(fact_2605_power__Suc2,axiom,
    ! [A: real,N: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,N)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(A),N)),A) ).

% power_Suc2
tff(fact_2606_power__Suc2,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),A) ).

% power_Suc2
tff(fact_2607_power__Suc2,axiom,
    ! [A: extended_enat,N: nat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(A),aa_nat_nat(suc,N)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(power_8040749407984259932d_enat(A),N)),A) ).

% power_Suc2
tff(fact_2608_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_2609_power__add,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_add
tff(fact_2610_power__add,axiom,
    ! [A: int,M: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ).

% power_add
tff(fact_2611_power__add,axiom,
    ! [A: real,M: nat,N: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)) ).

% power_add
tff(fact_2612_power__add,axiom,
    ! [A: complex,M: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N)) ).

% power_add
tff(fact_2613_power__add,axiom,
    ! [A: extended_enat,M: nat,N: nat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(power_8040749407984259932d_enat(A),M)),aa_nat_Extended_enat(power_8040749407984259932d_enat(A),N)) ).

% power_add
tff(fact_2614_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ) ).

% mult_less_mono1
tff(fact_2615_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ) ).

% mult_less_mono2
tff(fact_2616_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_2617_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_2618_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc
tff(fact_2619_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Leaf(Uu,Uv)),D))
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_2620_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2621_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),I)) ) ).

% less_mult_imp_div_less
tff(fact_2622_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)),M)) ).

% div_times_less_eq_dividend
tff(fact_2623_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))),M)) ).

% times_div_less_eq_dividend
tff(fact_2624_power__odd__eq,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = 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),N)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_2625_power__odd__eq,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),N)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_2626_power__odd__eq,axiom,
    ! [A: real,N: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_real_real(times_times_real(A),aa_nat_real(power_power_real(aa_nat_real(power_power_real(A),N)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_2627_power__odd__eq,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),N)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_2628_power__odd__eq,axiom,
    ! [A: extended_enat,N: nat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_nat_Extended_enat(power_8040749407984259932d_enat(aa_nat_Extended_enat(power_8040749407984259932d_enat(A),N)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_2629_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) ).

% Suc_double_not_eq_double
tff(fact_2630_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% double_not_eq_Suc_double
tff(fact_2631_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_2632_invar__vebt_Ointros_I1_J,axiom,
    ! [A: bool,B2: bool] : pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B2)),aa_nat_nat(suc,zero_zero_nat))) ).

% invar_vebt.intros(1)
tff(fact_2633_mult__less__le__imp__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2634_mult__less__le__imp__less,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2635_mult__less__le__imp__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2636_mult__le__less__imp__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2637_mult__le__less__imp__less,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2638_mult__le__less__imp__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2639_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ) ).

% mult_right_le_imp_le
tff(fact_2640_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2)) ) ) ).

% mult_right_le_imp_le
tff(fact_2641_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ) ).

% mult_right_le_imp_le
tff(fact_2642_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ) ).

% mult_left_le_imp_le
tff(fact_2643_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2)) ) ) ).

% mult_left_le_imp_le
tff(fact_2644_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ) ).

% mult_left_le_imp_le
tff(fact_2645_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2646_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2647_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2648_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2649_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2650_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2651_mult__strict__mono_H,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2652_mult__strict__mono_H,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2653_mult__strict__mono_H,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2654_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ) ).

% mult_right_less_imp_less
tff(fact_2655_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2)) ) ) ).

% mult_right_less_imp_less
tff(fact_2656_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) ) ) ).

% mult_right_less_imp_less
tff(fact_2657_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2658_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2659_mult__strict__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2660_mult__strict__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2661_mult__strict__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2662_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ) ).

% mult_left_less_imp_less
tff(fact_2663_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2)) ) ) ).

% mult_left_less_imp_less
tff(fact_2664_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) ) ) ).

% mult_left_less_imp_less
tff(fact_2665_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2666_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2667_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2668_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2669_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2670_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2671_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2672_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2673_mult__le__one,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),one_on7984719198319812577d_enat))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),B2))
       => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),one_on7984719198319812577d_enat))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B2)),one_on7984719198319812577d_enat)) ) ) ) ).

% mult_le_one
tff(fact_2674_mult__le__one,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),B2)),one_one_real)) ) ) ) ).

% mult_le_one
tff(fact_2675_mult__le__one,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),one_one_nat)) ) ) ) ).

% mult_le_one
tff(fact_2676_mult__le__one,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),one_one_int)) ) ) ) ).

% mult_le_one
tff(fact_2677_mult__left__le,axiom,
    ! [C: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,C),one_on7984719198319812577d_enat))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),A))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2678_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2679_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2680_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2681_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y),Y))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2682_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2683_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2684_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2685_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: bool,B2: bool] :
      ( ( ( X = zero_zero_nat )
       => ( vEBT_vebt_insert(vEBT_Leaf(A,B2),X) = vEBT_Leaf(fTrue,B2) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B2),X) = vEBT_Leaf(A,fTrue) ) )
          & ( ( X != one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B2),X) = vEBT_Leaf(A,B2) ) ) ) ) ) ).

% vebt_insert.simps(1)
tff(fact_2686_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y),Y))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2687_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2688_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y),Y))),zero_zero_real)) ).

% not_sum_squares_lt_zero
tff(fact_2689_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)) ).

% not_sum_squares_lt_zero
tff(fact_2690_divide__less__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B2),C)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(times_times_real(A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),B2)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_2691_less__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(B2),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),B2)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(times_times_real(A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_2692_neg__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B2),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),B2)) ) ) ).

% neg_divide_less_eq
tff(fact_2693_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(B2),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(times_times_real(A),C))) ) ) ).

% neg_less_divide_eq
tff(fact_2694_pos__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B2),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(times_times_real(A),C))) ) ) ).

% pos_divide_less_eq
tff(fact_2695_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(B2),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),B2)) ) ) ).

% pos_less_divide_eq
tff(fact_2696_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(times_times_real(Z2),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(X),Y)),Z2)) ) ) ).

% mult_imp_div_pos_less
tff(fact_2697_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(Z2),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_2698_divide__strict__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B2))) ) ) ) ).

% divide_strict_left_mono
tff(fact_2699_divide__strict__left__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B2))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2700_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(2)
tff(fact_2701_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: complex,C: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C) = numera6690914467698888265omplex(W2) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B2 = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C) ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2702_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W2: num] :
      ( ( aa_real_real(divide_divide_real(B2),C) = numeral_numeral_real(W2) )
    <=> ( ( ( C != zero_zero_real )
         => ( B2 = aa_real_real(times_times_real(numeral_numeral_real(W2)),C) ) )
        & ( ( C = zero_zero_real )
         => ( numeral_numeral_real(W2) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2703_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W2) = aa_complex_complex(divide1717551699836669952omplex(B2),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C) = B2 ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2704_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: real,C: real] :
      ( ( numeral_numeral_real(W2) = aa_real_real(divide_divide_real(B2),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(times_times_real(numeral_numeral_real(W2)),C) = B2 ) )
        & ( ( C = zero_zero_real )
         => ( numeral_numeral_real(W2) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2705_vebt__member_Osimps_I1_J,axiom,
    ! [A: bool,B2: bool,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Leaf(A,B2)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B2) )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_2706_divide__add__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z2)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Y),Z2))),Z2) ) ) ).

% divide_add_eq_iff
tff(fact_2707_divide__add__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X),Z2)),Y) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(Y),Z2))),Z2) ) ) ).

% divide_add_eq_iff
tff(fact_2708_add__divide__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z2)),Y)),Z2) ) ) ).

% add_divide_eq_iff
tff(fact_2709_add__divide__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(Y),Z2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),Z2)),Y)),Z2) ) ) ).

% add_divide_eq_iff
tff(fact_2710_add__num__frac,axiom,
    ! [Y: complex,Z2: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z2),aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z2),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2711_add__num__frac,axiom,
    ! [Y: real,Z2: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(plus_plus_real(Z2),aa_real_real(divide_divide_real(X),Y)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(Z2),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2712_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z2: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),Z2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z2),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2713_add__frac__num,axiom,
    ! [Y: real,X: real,Z2: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X),Y)),Z2) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(Z2),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2714_add__frac__eq,axiom,
    ! [Y: complex,Z2: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z2)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z2)) ) ) ) ).

% add_frac_eq
tff(fact_2715_add__frac__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(W2),Z2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(W2),Y))),aa_real_real(times_times_real(Y),Z2)) ) ) ) ).

% add_frac_eq
tff(fact_2716_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: complex,A: complex,B2: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B2),Z2)) = A ) )
      & ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B2),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Z2)),B2)),Z2) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2717_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: real,A: real,B2: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(plus_plus_real(A),aa_real_real(divide_divide_real(B2),Z2)) = A ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(plus_plus_real(A),aa_real_real(divide_divide_real(B2),Z2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),Z2)),B2)),Z2) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2718_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: complex,A: complex,B2: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z2)),B2) = B2 ) )
      & ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z2)),B2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B2),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2719_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: real,A: real,B2: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(A),Z2)),B2) = B2 ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(A),Z2)),B2) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),aa_real_real(times_times_real(B2),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2720_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).

% power_gt1_lemma
tff(fact_2721_power__gt1__lemma,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(times_times_real(A),aa_nat_real(power_power_real(A),N)))) ) ).

% power_gt1_lemma
tff(fact_2722_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_gt1_lemma
tff(fact_2723_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).

% power_less_power_Suc
tff(fact_2724_power__less__power__Suc,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),aa_real_real(times_times_real(A),aa_nat_real(power_power_real(A),N)))) ) ).

% power_less_power_Suc
tff(fact_2725_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_less_power_Suc
tff(fact_2726_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: bool,B2: bool,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Leaf(A,B2)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B2) )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_2727_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% one_less_mult
tff(fact_2728_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% n_less_m_mult_n
tff(fact_2729_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M))) ) ) ).

% n_less_n_mult_m
tff(fact_2730_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_2731_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q3)),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3))) ) ) ).

% div_less_iff_less_mult
tff(fact_2732_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ( X != vEBT_Leaf(fFalse,fFalse) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_2733_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ? [R3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
          & ( aa_nat_real(power_power_real(R3),aa_nat_nat(suc,N)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_2734_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(Z),X)),Y)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_2735_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2736_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2737_mult__less__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(times_times_real(B2),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),one_one_real)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2738_mult__less__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),one_one_int)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2739_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2740_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2741_mult__less__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(times_times_real(C),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),one_one_real)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2742_mult__less__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),one_one_int)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2743_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2744_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2745_mult__le__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(times_times_real(B2),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),one_one_real)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2746_mult__le__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),one_one_int)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2747_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2748_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2749_mult__le__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(times_times_real(C),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),one_one_real)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2750_mult__le__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),one_one_int)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2751_divide__left__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B2))) ) ) ) ).

% divide_left_mono_neg
tff(fact_2752_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(Z2),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_2753_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(times_times_real(Z2),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y)),Z2)) ) ) ).

% mult_imp_div_pos_le
tff(fact_2754_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(divide_divide_real(B2),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B2)) ) ) ).

% pos_le_divide_eq
tff(fact_2755_pos__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B2),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(times_times_real(A),C))) ) ) ).

% pos_divide_le_eq
tff(fact_2756_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(divide_divide_real(B2),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(times_times_real(A),C))) ) ) ).

% neg_le_divide_eq
tff(fact_2757_neg__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B2),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B2)) ) ) ).

% neg_divide_le_eq
tff(fact_2758_divide__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B2))) ) ) ) ).

% divide_left_mono
tff(fact_2759_le__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(divide_divide_real(B2),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B2)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(times_times_real(A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_2760_divide__le__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B2),C)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(times_times_real(A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B2)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_2761_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(plus_plus_real(U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(U),X)),aa_real_real(times_times_real(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2762_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2763_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B2),C)),numeral_numeral_real(W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(times_times_real(numeral_numeral_real(W2)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(numeral_numeral_real(W2)),C)),B2)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),numeral_numeral_real(W2))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2764_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(W2)),aa_real_real(divide_divide_real(B2),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(numeral_numeral_real(W2)),C)),B2)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(times_times_real(numeral_numeral_real(W2)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(W2)),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2765_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% power_Suc_less
tff(fact_2766_power__Suc__less,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),aa_nat_real(power_power_real(A),N))),aa_nat_real(power_power_real(A),N))) ) ) ).

% power_Suc_less
tff(fact_2767_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N))),aa_nat_int(power_power_int(A),N))) ) ) ).

% power_Suc_less
tff(fact_2768_mult__2,axiom,
    ! [Z2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Z2) = aa_complex_complex(plus_plus_complex(Z2),Z2) ).

% mult_2
tff(fact_2769_mult__2,axiom,
    ! [Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Z2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),Z2) ).

% mult_2
tff(fact_2770_mult__2,axiom,
    ! [Z2: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),Z2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Z2),Z2) ).

% mult_2
tff(fact_2771_mult__2,axiom,
    ! [Z2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Z2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),Z2) ).

% mult_2
tff(fact_2772_mult__2,axiom,
    ! [Z2: real] : aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Z2) = aa_real_real(plus_plus_real(Z2),Z2) ).

% mult_2
tff(fact_2773_mult__2__right,axiom,
    ! [Z2: complex] : aa_complex_complex(times_times_complex(Z2),numera6690914467698888265omplex(bit0(one))) = aa_complex_complex(plus_plus_complex(Z2),Z2) ).

% mult_2_right
tff(fact_2774_mult__2__right,axiom,
    ! [Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z2),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),Z2) ).

% mult_2_right
tff(fact_2775_mult__2__right,axiom,
    ! [Z2: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(Z2),numera1916890842035813515d_enat(bit0(one))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Z2),Z2) ).

% mult_2_right
tff(fact_2776_mult__2__right,axiom,
    ! [Z2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z2),numeral_numeral_int(bit0(one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),Z2) ).

% mult_2_right
tff(fact_2777_mult__2__right,axiom,
    ! [Z2: real] : aa_real_real(times_times_real(Z2),numeral_numeral_real(bit0(one))) = aa_real_real(plus_plus_real(Z2),Z2) ).

% mult_2_right
tff(fact_2778_left__add__twice,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(A),B2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),A)),B2) ).

% left_add_twice
tff(fact_2779_left__add__twice,axiom,
    ! [A: nat,B2: 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),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)),B2) ).

% left_add_twice
tff(fact_2780_left__add__twice,axiom,
    ! [A: extended_enat,B2: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),A)),B2) ).

% left_add_twice
tff(fact_2781_left__add__twice,axiom,
    ! [A: int,B2: 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),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A)),B2) ).

% left_add_twice
tff(fact_2782_left__add__twice,axiom,
    ! [A: real,B2: real] : aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(A),B2)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),A)),B2) ).

% left_add_twice
tff(fact_2783_div__nat__eqI,axiom,
    ! [N: nat,Q3: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q3))))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_2784_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Q3)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_2785_split__div,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ( ( N != zero_zero_nat )
         => ! [I3: nat,J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I3)),J2) )
               => pp(aa_nat_bool(P,I3)) ) ) ) ) ) ).

% split_div
tff(fact_2786_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_2787_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))))) ) ).

% dividend_less_times_div
tff(fact_2788_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y) )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ~ pp(Y) )
       => ( ( ? [Uv2: bool] : X = vEBT_Leaf(fTrue,Uv2)
           => pp(Y) )
         => ( ( ? [Uu2: bool] : X = vEBT_Leaf(Uu2,fTrue)
             => pp(Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ pp(Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_2789_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(plus_plus_real(U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(U),X)),aa_real_real(times_times_real(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2790_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2791_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B2),C)),numeral_numeral_real(W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(times_times_real(numeral_numeral_real(W2)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(numeral_numeral_real(W2)),C)),B2)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),numeral_numeral_real(W2))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2792_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(W2)),aa_real_real(divide_divide_real(B2),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(numeral_numeral_real(W2)),C)),B2)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(times_times_real(numeral_numeral_real(W2)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(W2)),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2793_sum__squares__bound,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),X)),Y)),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% sum_squares_bound
tff(fact_2794_split__div_H,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
          & pp(aa_nat_bool(P,zero_zero_nat)) )
        | ? [Q5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q5)),M))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q5))))
            & pp(aa_nat_bool(P,Q5)) ) ) ) ).

% split_div'
tff(fact_2795_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X4)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X4)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X4)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),X4) ) ) ) ) ).

% vebt_member.cases
tff(fact_2796_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S3),X4)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3),X4)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),X4)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),X4) ) ) ) ) ).

% vebt_insert.cases
tff(fact_2797_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),X4)
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),X4)
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),X4) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_2798_power2__sum,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_2799_power2__sum,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_2800_power2__sum,axiom,
    ! [X: extended_enat,Y: extended_enat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(power_8040749407984259932d_enat(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_Extended_enat(power_8040749407984259932d_enat(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_2801_power2__sum,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_2802_power2__sum,axiom,
    ! [X: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(plus_plus_real(X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_2803_zero__le__even__power_H,axiom,
    ! [A: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ).

% zero_le_even_power'
tff(fact_2804_zero__le__even__power_H,axiom,
    ! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ).

% zero_le_even_power'
tff(fact_2805_nat__bit__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))) ) )
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => pp(aa_nat_bool(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2)))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_2806_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( aa_nat_real(power_power_real(U),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(times_times_real(X),Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),Y)),numeral_numeral_real(bit0(one))))) ) ) ) ).

% arith_geo_mean
tff(fact_2807_triangle__def,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% triangle_def
tff(fact_2808_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ( aa_nat_real(power_power_real(R3),N) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_2809_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ? [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X4))
            & ( aa_nat_real(power_power_real(X4),N) = A )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4))
                  & ( aa_nat_real(power_power_real(Y4),N) = A ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_2810_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2811_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2812_odd__power__less__zero,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),zero_zero_real)) ) ).

% odd_power_less_zero
tff(fact_2813_odd__power__less__zero,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),zero_zero_int)) ) ).

% odd_power_less_zero
tff(fact_2814_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2815_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_2816_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_2817_set__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% set_bit_0
tff(fact_2818_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,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))))) ).

% set_bit_0
tff(fact_2819_unset__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% unset_bit_0
tff(fact_2820_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,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))) ).

% unset_bit_0
tff(fact_2821_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_2822_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_2823_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2824_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B2)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2825_discrete,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B2)) ) ).

% discrete
tff(fact_2826_discrete,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B2)) ) ).

% discrete
tff(fact_2827_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
    <=> ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_2828_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_2829_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_2830_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_2831_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2832_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
       => ( aa_int_int(divide_divide_int(A),B2) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2833_div__positive,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2))) ) ) ).

% div_positive
tff(fact_2834_div__positive,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B2))) ) ) ).

% div_positive
tff(fact_2835_mult__le__cancel__iff1,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(Y),Z2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_2836_mult__le__cancel__iff1,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_2837_mult__le__cancel__iff2,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(Z2),X)),aa_real_real(times_times_real(Z2),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_2838_mult__le__cancel__iff2,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z2),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z2),Y)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_2839_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2)))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_2840_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( unique6319869463603278526ux_int(product_Pair_int_int(Q3,R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_2841_low__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% low_def
tff(fact_2842_even__succ__div__exp,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_2843_even__succ__div__exp,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_2844_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,N)),aa_nat_set_nat(nat_set_decode,X)))
    <=> pp(aa_set_nat_bool(member_nat(N),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ).

% set_decode_Suc
tff(fact_2845_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y = vEBT_Leaf(fTrue,B3) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y = vEBT_Leaf(A3,fTrue) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y = vEBT_Leaf(A3,B3) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,zero_zero_nat,Ts2,S3) )
             => ( Y != vEBT_Node(Info2,zero_zero_nat,Ts2,S3) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3) )
               => ( Y != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                   => ( Y != if_VEBT_VEBT(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)),aa_bool_bool(fNot,fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2)) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_2846_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : size_s7466405169056248089T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)) ).

% length_product
tff(fact_2847_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int] : size_s3661962791536183091BT_int(produc7292646706713671643BT_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_int_nat(size_size_list_int,Ys)) ).

% length_product
tff(fact_2848_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] : size_s6152045936467909847BT_nat(produc7295137177222721919BT_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_nat_nat(size_size_list_nat,Ys)) ).

% length_product
tff(fact_2849_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,aa_list_int_nat(size_size_list_int,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)) ).

% length_product
tff(fact_2850_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,aa_list_int_nat(size_size_list_int,Xs)),aa_list_int_nat(size_size_list_int,Ys)) ).

% length_product
tff(fact_2851_length__product,axiom,
    ! [Xs: list_int,Ys: list_nat] : size_s7647898544948552527nt_nat(product_int_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_int_nat(size_size_list_int,Xs)),aa_list_nat_nat(size_size_list_nat,Ys)) ).

% length_product
tff(fact_2852_length__product,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)) ).

% length_product
tff(fact_2853_length__product,axiom,
    ! [Xs: list_nat,Ys: list_int] : size_s2970893825323803983at_int(product_nat_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_int_nat(size_size_list_int,Ys)) ).

% length_product
tff(fact_2854_length__product,axiom,
    ! [Xs: list_nat,Ys: list_nat] : size_s5460976970255530739at_nat(product_nat_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_nat_nat(size_size_list_nat,Ys)) ).

% length_product
tff(fact_2855_mult__less__iff1,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(Y),Z2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_2856_mult__less__iff1,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_2857_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_2858_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),one_one_nat))
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_2859_finite__Collect__disjI,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_af(P),Q))))
    <=> ( pp(aa_set_real_bool(finite_finite_real,collect_real(P)))
        & pp(aa_set_real_bool(finite_finite_real,collect_real(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_2860_finite__Collect__disjI,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ag(P),Q)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(P))
        & finite8100373058378681591st_nat(collect_list_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_2861_finite__Collect__disjI,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_ah(P),Q))))
    <=> ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
        & pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_2862_finite__Collect__disjI,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_ai(P),Q))))
    <=> ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
        & pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_2863_finite__Collect__disjI,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_aj(P),Q))))
    <=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
        & pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_2864_finite__Collect__disjI,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ak(P),Q))))
    <=> ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
        & pp(aa_set_int_bool(finite_finite_int,collect_int(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_2865_finite__Collect__disjI,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,Q: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_al(P),Q))))
    <=> ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(P)))
        & pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_2866_finite__Collect__conjI,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ( pp(aa_set_real_bool(finite_finite_real,collect_real(P)))
        | pp(aa_set_real_bool(finite_finite_real,collect_real(Q))) )
     => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_am(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_2867_finite__Collect__conjI,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ( finite8100373058378681591st_nat(collect_list_nat(P))
        | finite8100373058378681591st_nat(collect_list_nat(Q)) )
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_an(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_2868_finite__Collect__conjI,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
        | pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(Q))) )
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_ao(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_2869_finite__Collect__conjI,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
        | pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q))) )
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_ap(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_2870_finite__Collect__conjI,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
        | pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(Q))) )
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_aq(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_2871_finite__Collect__conjI,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
        | pp(aa_set_int_bool(finite_finite_int,collect_int(Q))) )
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ar(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_2872_finite__Collect__conjI,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,Q: fun_Ex8563670519792538868t_bool] :
      ( ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(P)))
        | pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(Q))) )
     => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_as(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_2873_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_2874_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_2875_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_2876_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,zero_zero_complex),A))
    <=> ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
tff(fact_2877_dvd__0__left__iff,axiom,
    ! [A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,zero_z5237406670263579293d_enat),A))
    <=> ( A = zero_z5237406670263579293d_enat ) ) ).

% dvd_0_left_iff
tff(fact_2878_dvd__0__right,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).

% dvd_0_right
tff(fact_2879_dvd__0__right,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),zero_zero_real)) ).

% dvd_0_right
tff(fact_2880_dvd__0__right,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),zero_zero_int)) ).

% dvd_0_right
tff(fact_2881_dvd__0__right,axiom,
    ! [A: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),zero_zero_complex)) ).

% dvd_0_right
tff(fact_2882_dvd__0__right,axiom,
    ! [A: extended_enat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,A),zero_z5237406670263579293d_enat)) ).

% dvd_0_right
tff(fact_2883_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2)) ) ).

% dvd_add_triv_right_iff
tff(fact_2884_dvd__add__triv__right__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2)) ) ).

% dvd_add_triv_right_iff
tff(fact_2885_dvd__add__triv__right__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(plus_plus_real(B2),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2)) ) ).

% dvd_add_triv_right_iff
tff(fact_2886_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2)) ) ).

% dvd_add_triv_left_iff
tff(fact_2887_dvd__add__triv__left__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2)) ) ).

% dvd_add_triv_left_iff
tff(fact_2888_dvd__add__triv__left__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(plus_plus_real(A),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2)) ) ).

% dvd_add_triv_left_iff
tff(fact_2889_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_2890_dvd__1__left,axiom,
    ! [K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K)) ).

% dvd_1_left
tff(fact_2891_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_2892_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_2893_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_2894_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_2895_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_2896_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_2897_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_2898_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_2899_div__dvd__div,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),C)) ) ) ) ).

% div_dvd_div
tff(fact_2900_div__dvd__div,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(B2),A)),aa_int_int(divide_divide_int(C),A)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),C)) ) ) ) ).

% div_dvd_div
tff(fact_2901_mod__add__self2,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% mod_add_self2
tff(fact_2902_mod__add__self2,axiom,
    ! [A: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2) = aa_int_int(modulo_modulo_int(A),B2) ).

% mod_add_self2
tff(fact_2903_mod__add__self1,axiom,
    ! [B2: 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,B2),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% mod_add_self1
tff(fact_2904_mod__add__self1,axiom,
    ! [B2: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)),B2) = aa_int_int(modulo_modulo_int(A),B2) ).

% mod_add_self1
tff(fact_2905_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( ( K = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_2906_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) ) ).

% mod_less
tff(fact_2907_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_at(A2))) ) ).

% finite_Collect_subsets
tff(fact_2908_finite__Collect__subsets,axiom,
    ! [A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => finite5468666774076196335d_enat(collec2260605976452661553d_enat(aTP_Lamm_au(A2))) ) ).

% finite_Collect_subsets
tff(fact_2909_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_av,A2)))) ) ).

% finite_Collect_subsets
tff(fact_2910_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_aw(A2))) ) ).

% finite_Collect_subsets
tff(fact_2911_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,K)))) ).

% finite_Collect_less_nat
tff(fact_2912_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ay,K)))) ).

% finite_Collect_le_nat
tff(fact_2913_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_2914_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_2915_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_2916_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_2917_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
    <=> ( ( C = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2)) ) ) ).

% dvd_mult_cancel_right
tff(fact_2918_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C)))
    <=> ( ( C = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2)) ) ) ).

% dvd_mult_cancel_right
tff(fact_2919_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B2),C)))
    <=> ( ( C = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B2)) ) ) ).

% dvd_mult_cancel_right
tff(fact_2920_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
    <=> ( ( C = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2)) ) ) ).

% dvd_mult_cancel_left
tff(fact_2921_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B2)))
    <=> ( ( C = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2)) ) ) ).

% dvd_mult_cancel_left
tff(fact_2922_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B2)))
    <=> ( ( C = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B2)) ) ) ).

% dvd_mult_cancel_left
tff(fact_2923_unit__prod,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),one_one_nat)) ) ) ).

% unit_prod
tff(fact_2924_unit__prod,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),one_one_int)) ) ) ).

% unit_prod
tff(fact_2925_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2926_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(times_times_int,C),A))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2927_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(plus_plus_real(B2),aa_real_real(times_times_real(C),A))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2928_dvd__add__times__triv__right__iff,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),aa_complex_complex(plus_plus_complex(B2),aa_complex_complex(times_times_complex(C),A))))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B2)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2929_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2930_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2931_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(plus_plus_real(aa_real_real(times_times_real(C),A)),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2932_dvd__add__times__triv__left__iff,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(C),A)),B2)))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B2)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2933_mod__mult__self1__is__0,axiom,
    ! [B2: 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,B2),A)),B2) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_2934_mod__mult__self1__is__0,axiom,
    ! [B2: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),B2) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_2935_mod__mult__self2__is__0,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),B2) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_2936_mod__mult__self2__is__0,axiom,
    ! [A: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),B2) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_2937_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_2938_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_2939_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_2940_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_2941_dvd__div__mult__self,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A)),A) = B2 ) ) ).

% dvd_div_mult_self
tff(fact_2942_dvd__div__mult__self,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B2),A)),A) = B2 ) ) ).

% dvd_div_mult_self
tff(fact_2943_dvd__mult__div__cancel,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A)) = B2 ) ) ).

% dvd_mult_div_cancel
tff(fact_2944_dvd__mult__div__cancel,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B2),A)) = B2 ) ) ).

% dvd_mult_div_cancel
tff(fact_2945_unit__div,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),one_one_nat)) ) ) ).

% unit_div
tff(fact_2946_unit__div,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(A),B2)),one_one_int)) ) ) ).

% unit_div
tff(fact_2947_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat)) ) ).

% unit_div_1_unit
tff(fact_2948_unit__div__1__unit,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(one_one_int),A)),one_one_int)) ) ).

% unit_div_1_unit
tff(fact_2949_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_2950_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( aa_int_int(divide_divide_int(one_one_int),aa_int_int(divide_divide_int(one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_2951_div__add,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),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,B2),C)) ) ) ) ).

% div_add
tff(fact_2952_div__add,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B2))
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B2),C)) ) ) ) ).

% div_add
tff(fact_2953_mod__div__trivial,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),B2) = zero_zero_nat ).

% mod_div_trivial
tff(fact_2954_mod__div__trivial,axiom,
    ! [A: int,B2: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B2)),B2) = zero_zero_int ).

% mod_div_trivial
tff(fact_2955_bits__mod__div__trivial,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),B2) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_2956_bits__mod__div__trivial,axiom,
    ! [A: int,B2: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B2)),B2) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_2957_mod__mult__self4,axiom,
    ! [B2: 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,B2),C)),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% mod_mult_self4
tff(fact_2958_mod__mult__self4,axiom,
    ! [B2: int,C: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)),A)),B2) = aa_int_int(modulo_modulo_int(A),B2) ).

% mod_mult_self4
tff(fact_2959_mod__mult__self3,axiom,
    ! [C: nat,B2: 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),B2)),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% mod_mult_self3
tff(fact_2960_mod__mult__self3,axiom,
    ! [C: int,B2: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)),A)),B2) = aa_int_int(modulo_modulo_int(A),B2) ).

% mod_mult_self3
tff(fact_2961_mod__mult__self2,axiom,
    ! [A: nat,B2: 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,B2),C))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% mod_mult_self2
tff(fact_2962_mod__mult__self2,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))),B2) = aa_int_int(modulo_modulo_int(A),B2) ).

% mod_mult_self2
tff(fact_2963_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B2: 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),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% mod_mult_self1
tff(fact_2964_mod__mult__self1,axiom,
    ! [A: int,C: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))),B2) = aa_int_int(modulo_modulo_int(A),B2) ).

% mod_mult_self1
tff(fact_2965_dvd__imp__mod__0,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_2966_dvd__imp__mod__0,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
     => ( aa_int_int(modulo_modulo_int(B2),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_2967_mod__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_2968_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_2969_unit__div__mult__self,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A)),A) = B2 ) ) ).

% unit_div_mult_self
tff(fact_2970_unit__div__mult__self,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B2),A)),A) = B2 ) ) ).

% unit_div_mult_self
tff(fact_2971_unit__mult__div__div,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),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,B2),A) ) ) ).

% unit_mult_div_div
tff(fact_2972_unit__mult__div__div,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(divide_divide_int(one_one_int),A)) = aa_int_int(divide_divide_int(B2),A) ) ) ).

% unit_mult_div_div
tff(fact_2973_even__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ) ).

% even_Suc
tff(fact_2974_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ) ).

% even_Suc_Suc_iff
tff(fact_2975_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2)) ) ) ).

% pow_divides_pow_iff
tff(fact_2976_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B2),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2)) ) ) ).

% pow_divides_pow_iff
tff(fact_2977_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self4
tff(fact_2978_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self3
tff(fact_2979_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self2
tff(fact_2980_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self1
tff(fact_2981_even__add,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2)) ) ) ).

% even_add
tff(fact_2982_even__add,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),B2)) ) ) ).

% even_add
tff(fact_2983_odd__add,axiom,
    ! [A: nat,B2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)))
    <=> ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
        <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2)) ) ) ).

% odd_add
tff(fact_2984_odd__add,axiom,
    ! [A: int,B2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)))
    <=> ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),B2)) ) ) ).

% odd_add
tff(fact_2985_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ).

% odd_Suc_div_two
tff(fact_2986_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))) ) ) ).

% even_Suc_div_two
tff(fact_2987_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% mod2_Suc_Suc
tff(fact_2988_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: 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)),N))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_2989_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,X)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),X)) ) ).

% set_decode_0
tff(fact_2990_zero__le__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2991_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2992_power__less__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_2993_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_2994_power__less__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),zero_zero_real))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq
tff(fact_2995_power__less__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),zero_zero_int))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% power_less_zero_eq
tff(fact_2996_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)) ) ).

% even_plus_one_iff
tff(fact_2997_even__plus__one__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)) ) ).

% even_plus_one_iff
tff(fact_2998_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2999_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_3000_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_3001_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) != zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_3002_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_3003_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_3004_zero__less__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
          & ( A != zero_zero_real ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3005_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
          & ( A != zero_zero_int ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3006_even__succ__div__two,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) ) ) ).

% even_succ_div_two
tff(fact_3007_even__succ__div__two,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),numeral_numeral_int(bit0(one))) = aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))) ) ) ).

% even_succ_div_two
tff(fact_3008_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_3009_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),numeral_numeral_int(bit0(one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_3010_even__succ__div__2,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) ) ) ).

% even_succ_div_2
tff(fact_3011_even__succ__div__2,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),numeral_numeral_int(bit0(one))) = aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))) ) ) ).

% even_succ_div_2
tff(fact_3012_even__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_3013_even__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_nat_int(power_power_int(A),N)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_3014_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_3015_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3016_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3017_power__le__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3018_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W2)))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3019_even__succ__mod__exp,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3020_even__succ__mod__exp,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3021_strict__subset__divisors__dvd,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_az,A))),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_az,B2))))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B2),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3022_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ba,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ba,B2))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3023_strict__subset__divisors__dvd,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_bb,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_bb,B2))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3024_less__set__def,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(A2),B))
    <=> ord_le5412861186789288264t_bool(aTP_Lamm_a(A2),aTP_Lamm_a(B)) ) ).

% less_set_def
tff(fact_3025_less__set__def,axiom,
    ! [A2: set_real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A2),B))
    <=> ord_le7519368973566061808l_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B)) ) ).

% less_set_def
tff(fact_3026_less__set__def,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),B))
    <=> ord_le4523237222650602006t_bool(aTP_Lamm_ac(A2),aTP_Lamm_ac(B)) ) ).

% less_set_def
tff(fact_3027_less__set__def,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B))
    <=> ord_le6841235830588351564t_bool(aTP_Lamm_ad(A2),aTP_Lamm_ad(B)) ) ).

% less_set_def
tff(fact_3028_less__set__def,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B))
    <=> ord_le3716944864119727344t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B)) ) ).

% less_set_def
tff(fact_3029_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_3030_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B2) = zero_zero_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_3031_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3032_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
    <=> ( aa_int_int(modulo_modulo_int(B2),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3033_mod__0__imp__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A)) ) ).

% mod_0_imp_dvd
tff(fact_3034_mod__0__imp__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B2) = zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A)) ) ).

% mod_0_imp_dvd
tff(fact_3035_empty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_bc) ).

% empty_def
tff(fact_3036_empty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_bd) ).

% empty_def
tff(fact_3037_empty__def,axiom,
    bot_bo7653980558646680370d_enat = collec4429806609662206161d_enat(aTP_Lamm_be) ).

% empty_def
tff(fact_3038_empty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_bf) ).

% empty_def
tff(fact_3039_empty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_bg) ).

% empty_def
tff(fact_3040_empty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_bh) ).

% empty_def
tff(fact_3041_dvd__refl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A)) ).

% dvd_refl
tff(fact_3042_dvd__refl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),A)) ).

% dvd_refl
tff(fact_3043_dvd__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_trans
tff(fact_3044_dvd__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_trans
tff(fact_3045_dvd__mod__iff,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_3046_dvd__mod__iff,axiom,
    ! [C: int,B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_3047_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3048_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3049_not__finite__existsD,axiom,
    ! [P: fun_real_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,collect_real(P)))
     => ? [X_12: real] : pp(aa_real_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_3050_not__finite__existsD,axiom,
    ! [P: fun_list_nat_bool] :
      ( ~ finite8100373058378681591st_nat(collect_list_nat(P))
     => ? [X_12: list_nat] : pp(aa_list_nat_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_3051_not__finite__existsD,axiom,
    ! [P: fun_set_nat_bool] :
      ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
     => ? [X_12: set_nat] : pp(aa_set_nat_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_3052_not__finite__existsD,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => ? [X_12: nat] : pp(aa_nat_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_3053_not__finite__existsD,axiom,
    ! [P: fun_complex_bool] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
     => ? [X_12: complex] : pp(aa_complex_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_3054_not__finite__existsD,axiom,
    ! [P: fun_int_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
     => ? [X_12: int] : pp(aa_int_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_3055_not__finite__existsD,axiom,
    ! [P: fun_Ex8563670519792538868t_bool] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(P)))
     => ? [X_12: extended_enat] : pp(aa_Ext2772979874139974381t_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_3056_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B: set_nat,R: fun_re9121486324101951821t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B))
                  & pp(aa_nat_bool(aa_real_fun_nat_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bi(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3057_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B: set_complex,R: fun_re3010880487650781423x_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B))
                  & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(R,X4),Xa)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_bj(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3058_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B: set_int,R: fun_re8661203372455305201t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B))
                  & pp(aa_int_bool(aa_real_fun_int_bool(R,X4),Xa)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_bk(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3059_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B: set_Extended_enat,R: fun_re7357811994743134493t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => ? [Xa: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),B))
                  & pp(aa_Ext2772979874139974381t_bool(aa_rea3571252896447469150t_bool(R,X4),Xa)) ) )
         => ? [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),B))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_Ext4143593917056713754l_bool(aa_fun3666023268991729404l_bool(aTP_Lamm_bl(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3060_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B: set_nat,R: fun_nat_fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_bm(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3061_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B: set_complex,R: fun_na5941084329111710283x_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B))
                  & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(R,X4),Xa)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_bn(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3062_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B: set_int,R: fun_nat_fun_int_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B))
                  & pp(aa_int_bool(aa_nat_fun_int_bool(R,X4),Xa)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_bo(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3063_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B: set_Extended_enat,R: fun_na938974098958486209t_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => ? [Xa: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),B))
                  & pp(aa_Ext2772979874139974381t_bool(aa_nat6328455298437991426t_bool(R,X4),Xa)) ) )
         => ? [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),B))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_Ext6309394871002001526t_bool(aa_fun9145731736113502972t_bool(aTP_Lamm_bp(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3064_pigeonhole__infinite__rel,axiom,
    ! [A2: set_complex,B: set_nat,R: fun_co2841405903858895691t_bool] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B))
                  & pp(aa_nat_bool(aa_com5335841084339320244t_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B))
              & ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_bq(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3065_pigeonhole__infinite__rel,axiom,
    ! [A2: set_complex,B: set_complex,R: fun_co1378794544173401837x_bool] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B))
                  & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),Xa)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B))
              & ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_com1553036987509291606x_bool(aa_fun2898175553265993724x_bool(aTP_Lamm_br(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3066_pred__subset__eq,axiom,
    ! [R: set_Extended_enat,S2: set_Extended_enat] :
      ( ord_le1334005008332522836t_bool(aTP_Lamm_a(R),aTP_Lamm_a(S2))
    <=> pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(R),S2)) ) ).

% pred_subset_eq
tff(fact_3067_pred__subset__eq,axiom,
    ! [R: set_real,S2: set_real] :
      ( ord_le5573776112191778788l_bool(aTP_Lamm_aa(R),aTP_Lamm_aa(S2))
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(R),S2)) ) ).

% pred_subset_eq
tff(fact_3068_pred__subset__eq,axiom,
    ! [R: set_set_nat,S2: set_set_nat] :
      ( ord_le1461730252015488778t_bool(aTP_Lamm_ac(R),aTP_Lamm_ac(S2))
    <=> pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(R),S2)) ) ).

% pred_subset_eq
tff(fact_3069_pred__subset__eq,axiom,
    ! [R: set_nat,S2: set_nat] :
      ( ord_le8840051076831633728t_bool(aTP_Lamm_ad(R),aTP_Lamm_ad(S2))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,R),S2)) ) ).

% pred_subset_eq
tff(fact_3070_pred__subset__eq,axiom,
    ! [R: set_int,S2: set_int] :
      ( ord_le5715760110363009508t_bool(aTP_Lamm_ae(R),aTP_Lamm_ae(S2))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(R),S2)) ) ).

% pred_subset_eq
tff(fact_3071_less__eq__set__def,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
    <=> ord_le1334005008332522836t_bool(aTP_Lamm_a(A2),aTP_Lamm_a(B)) ) ).

% less_eq_set_def
tff(fact_3072_less__eq__set__def,axiom,
    ! [A2: set_real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
    <=> ord_le5573776112191778788l_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B)) ) ).

% less_eq_set_def
tff(fact_3073_less__eq__set__def,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B))
    <=> ord_le1461730252015488778t_bool(aTP_Lamm_ac(A2),aTP_Lamm_ac(B)) ) ).

% less_eq_set_def
tff(fact_3074_less__eq__set__def,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
    <=> ord_le8840051076831633728t_bool(aTP_Lamm_ad(A2),aTP_Lamm_ad(B)) ) ).

% less_eq_set_def
tff(fact_3075_less__eq__set__def,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
    <=> ord_le5715760110363009508t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B)) ) ).

% less_eq_set_def
tff(fact_3076_Collect__subset,axiom,
    ! [A2: set_Extended_enat,P: fun_Ex8563670519792538868t_bool] : pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_bs(A2),P))),A2)) ).

% Collect_subset
tff(fact_3077_Collect__subset,axiom,
    ! [A2: set_real,P: fun_real_bool] : pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bt(A2),P))),A2)) ).

% Collect_subset
tff(fact_3078_Collect__subset,axiom,
    ! [A2: set_list_nat,P: fun_list_nat_bool] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bu(A2),P)),A2) ).

% Collect_subset
tff(fact_3079_Collect__subset,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_bool] : pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_bv(A2),P))),A2)) ).

% Collect_subset
tff(fact_3080_Collect__subset,axiom,
    ! [A2: set_nat,P: fun_nat_bool] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bw(A2),P))),A2)) ).

% Collect_subset
tff(fact_3081_Collect__subset,axiom,
    ! [A2: set_int,P: fun_int_bool] : pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bx(A2),P))),A2)) ).

% Collect_subset
tff(fact_3082_subset__divisors__dvd,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_az,A))),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_az,B2))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2)) ) ).

% subset_divisors_dvd
tff(fact_3083_subset__divisors__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ba,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ba,B2))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2)) ) ).

% subset_divisors_dvd
tff(fact_3084_subset__divisors__dvd,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_bb,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_bb,B2))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2)) ) ).

% subset_divisors_dvd
tff(fact_3085_Collect__restrict,axiom,
    ! [X6: set_Extended_enat,P: fun_Ex8563670519792538868t_bool] : pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_bs(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3086_Collect__restrict,axiom,
    ! [X6: set_real,P: fun_real_bool] : pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bt(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3087_Collect__restrict,axiom,
    ! [X6: set_list_nat,P: fun_list_nat_bool] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bu(X6),P)),X6) ).

% Collect_restrict
tff(fact_3088_Collect__restrict,axiom,
    ! [X6: set_set_nat,P: fun_set_nat_bool] : pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_bv(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3089_Collect__restrict,axiom,
    ! [X6: set_nat,P: fun_nat_bool] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bw(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3090_Collect__restrict,axiom,
    ! [X6: set_int,P: fun_int_bool] : pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bx(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3091_prop__restrict,axiom,
    ! [X: extended_enat,Z4: set_Extended_enat,X6: set_Extended_enat,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),Z4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(Z4),collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_bs(X6),P))))
       => pp(aa_Ext2772979874139974381t_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_3092_prop__restrict,axiom,
    ! [X: real,Z4: set_real,X6: set_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(X),Z4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(Z4),collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bt(X6),P))))
       => pp(aa_real_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_3093_prop__restrict,axiom,
    ! [X: list_nat,Z4: set_list_nat,X6: set_list_nat,P: fun_list_nat_bool] :
      ( member_list_nat(X,Z4)
     => ( ord_le6045566169113846134st_nat(Z4,collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bu(X6),P)))
       => pp(aa_list_nat_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_3094_prop__restrict,axiom,
    ! [X: set_nat,Z4: set_set_nat,X6: set_set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),Z4))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(Z4),collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_bv(X6),P))))
       => pp(aa_set_nat_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_3095_prop__restrict,axiom,
    ! [X: nat,Z4: set_nat,X6: set_nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(X),Z4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Z4),collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bw(X6),P))))
       => pp(aa_nat_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_3096_prop__restrict,axiom,
    ! [X: int,Z4: set_int,X6: set_int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(X),Z4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Z4),collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bx(X6),P))))
       => pp(aa_int_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_3097_lambda__zero,axiom,
    aTP_Lamm_by = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_3098_lambda__zero,axiom,
    aTP_Lamm_bz = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_3099_lambda__zero,axiom,
    aTP_Lamm_ca = times_times_real(zero_zero_real) ).

% lambda_zero
tff(fact_3100_lambda__zero,axiom,
    aTP_Lamm_cb = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_3101_lambda__zero,axiom,
    aTP_Lamm_cc = times_7803423173614009249d_enat(zero_z5237406670263579293d_enat) ).

% lambda_zero
tff(fact_3102_lambda__one,axiom,
    aTP_Lamm_cd = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_3103_lambda__one,axiom,
    aTP_Lamm_ce = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_3104_lambda__one,axiom,
    aTP_Lamm_cf = times_times_real(one_one_real) ).

% lambda_one
tff(fact_3105_lambda__one,axiom,
    aTP_Lamm_cg = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_3106_lambda__one,axiom,
    aTP_Lamm_ch = times_7803423173614009249d_enat(one_on7984719198319812577d_enat) ).

% lambda_one
tff(fact_3107_max__def__raw,axiom,
    ! [X2: real,Xa: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Xa))
       => ( aa_real_real(ord_max_real(X2),Xa) = Xa ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Xa))
       => ( aa_real_real(ord_max_real(X2),Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_3108_max__def__raw,axiom,
    ! [X2: set_nat,Xa: set_nat] :
      ( ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X2),Xa))
       => ( ord_max_set_nat(X2,Xa) = Xa ) )
      & ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X2),Xa))
       => ( ord_max_set_nat(X2,Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_3109_max__def__raw,axiom,
    ! [X2: set_int,Xa: set_int] :
      ( ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Xa))
       => ( ord_max_set_int(X2,Xa) = Xa ) )
      & ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(X2),Xa))
       => ( ord_max_set_int(X2,Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_3110_max__def__raw,axiom,
    ! [X2: nat,Xa: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Xa) = Xa ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_3111_max__def__raw,axiom,
    ! [X2: int,Xa: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Xa))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Xa) = Xa ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Xa))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_3112_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ba,M)))) ) ).

% finite_divisors_nat
tff(fact_3113_unit__imp__mod__eq__0,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_3114_unit__imp__mod__eq__0,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( aa_int_int(modulo_modulo_int(A),B2) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_3115_bot__empty__eq2,axiom,
    ! [X2: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bot_bo6560894463593755925t_bool,X2),Xa))
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Xa),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_3116_bot__empty__eq2,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(bot_bo8694918001306130243t_bool,X2),Xa))
    <=> member373505688050248522BT_nat(produc738532404422230701BT_nat(X2,Xa),bot_bo1642239108664514429BT_nat) ) ).

% bot_empty_eq2
tff(fact_3117_bot__empty__eq2,axiom,
    ! [X2: int,Xa: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(bot_bo6207810721336151133t_bool,X2),Xa))
    <=> member5262025264175285858nt_int(product_Pair_int_int(X2,Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_3118_bot__empty__eq2,axiom,
    ! [X2: product_prod_nat_nat,Xa: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(bot_bo5325672116769796377t_bool,X2),Xa))
    <=> member8206827879206165904at_nat(produc6161850002892822231at_nat(X2,Xa),bot_bo5327735625951526323at_nat) ) ).

% bot_empty_eq2
tff(fact_3119_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_bool,I: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ci(P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_3120_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(F,N2)))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cj(F),U)))) ) ).

% finite_less_ub
tff(fact_3121_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_3122_mod__add__right__eq,axiom,
    ! [A: nat,B2: 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,B2),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),B2)),C) ).

% mod_add_right_eq
tff(fact_3123_mod__add__right__eq,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(B2),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) ).

% mod_add_right_eq
tff(fact_3124_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B2: 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)),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),C) ).

% mod_add_left_eq
tff(fact_3125_mod__add__left__eq,axiom,
    ! [A: int,C: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),B2)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) ).

% mod_add_left_eq
tff(fact_3126_mod__add__cong,axiom,
    ! [A: nat,C: nat,A5: nat,B2: 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,B2),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),B2)),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_3127_mod__add__cong,axiom,
    ! [A: int,C: int,A5: int,B2: int,B5: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A5),C) )
     => ( ( aa_int_int(modulo_modulo_int(B2),C) = aa_int_int(modulo_modulo_int(B5),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A5),B5)),C) ) ) ) ).

% mod_add_cong
tff(fact_3128_mod__add__eq,axiom,
    ! [A: nat,C: nat,B2: 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,B2),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),B2)),C) ).

% mod_add_eq
tff(fact_3129_mod__add__eq,axiom,
    ! [A: int,C: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B2),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) ).

% mod_add_eq
tff(fact_3130_dvd__0__left,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_3131_dvd__0__left,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_3132_dvd__0__left,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_3133_dvd__0__left,axiom,
    ! [A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,zero_zero_complex),A))
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
tff(fact_3134_dvd__0__left,axiom,
    ! [A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,zero_z5237406670263579293d_enat),A))
     => ( A = zero_z5237406670263579293d_enat ) ) ).

% dvd_0_left
tff(fact_3135_dvd__field__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2))
    <=> ( ( A = zero_zero_real )
       => ( B2 = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_3136_dvd__field__iff,axiom,
    ! [A: complex,B2: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B2))
    <=> ( ( A = zero_zero_complex )
       => ( B2 = zero_zero_complex ) ) ) ).

% dvd_field_iff
tff(fact_3137_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),N) ).

% mod_Suc_Suc_eq
tff(fact_3138_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% mod_Suc_eq
tff(fact_3139_dvdE,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A))
     => ~ ! [K3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K3) ) ).

% dvdE
tff(fact_3140_dvdE,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A))
     => ~ ! [K3: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B2),K3) ) ).

% dvdE
tff(fact_3141_dvdE,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B2),A))
     => ~ ! [K3: real] : A != aa_real_real(times_times_real(B2),K3) ) ).

% dvdE
tff(fact_3142_dvdE,axiom,
    ! [B2: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B2),A))
     => ~ ! [K3: complex] : A != aa_complex_complex(times_times_complex(B2),K3) ) ).

% dvdE
tff(fact_3143_dvdE,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,B2),A))
     => ~ ! [K3: extended_enat] : A != aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),K3) ) ).

% dvdE
tff(fact_3144_dvdI,axiom,
    ! [A: nat,B2: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A)) ) ).

% dvdI
tff(fact_3145_dvdI,axiom,
    ! [A: int,B2: int,K: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B2),K) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A)) ) ).

% dvdI
tff(fact_3146_dvdI,axiom,
    ! [A: real,B2: real,K: real] :
      ( ( A = aa_real_real(times_times_real(B2),K) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B2),A)) ) ).

% dvdI
tff(fact_3147_dvdI,axiom,
    ! [A: complex,B2: complex,K: complex] :
      ( ( A = aa_complex_complex(times_times_complex(B2),K) )
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B2),A)) ) ).

% dvdI
tff(fact_3148_dvdI,axiom,
    ! [A: extended_enat,B2: extended_enat,K: extended_enat] :
      ( ( A = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),K) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,B2),A)) ) ).

% dvdI
tff(fact_3149_dvd__def,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A))
    <=> ? [K2: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K2) ) ).

% dvd_def
tff(fact_3150_dvd__def,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A))
    <=> ? [K2: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B2),K2) ) ).

% dvd_def
tff(fact_3151_dvd__def,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B2),A))
    <=> ? [K2: real] : A = aa_real_real(times_times_real(B2),K2) ) ).

% dvd_def
tff(fact_3152_dvd__def,axiom,
    ! [B2: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B2),A))
    <=> ? [K2: complex] : A = aa_complex_complex(times_times_complex(B2),K2) ) ).

% dvd_def
tff(fact_3153_dvd__def,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,B2),A))
    <=> ? [K2: extended_enat] : A = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),K2) ) ).

% dvd_def
tff(fact_3154_dvd__mult,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C))) ) ).

% dvd_mult
tff(fact_3155_dvd__mult,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))) ) ).

% dvd_mult
tff(fact_3156_dvd__mult,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(times_times_real(B2),C))) ) ).

% dvd_mult
tff(fact_3157_dvd__mult,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),C))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),aa_complex_complex(times_times_complex(B2),C))) ) ).

% dvd_mult
tff(fact_3158_dvd__mult,axiom,
    ! [A: extended_enat,C: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,A),C))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),C))) ) ).

% dvd_mult
tff(fact_3159_dvd__mult2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C))) ) ).

% dvd_mult2
tff(fact_3160_dvd__mult2,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))) ) ).

% dvd_mult2
tff(fact_3161_dvd__mult2,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(times_times_real(B2),C))) ) ).

% dvd_mult2
tff(fact_3162_dvd__mult2,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B2))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),aa_complex_complex(times_times_complex(B2),C))) ) ).

% dvd_mult2
tff(fact_3163_dvd__mult2,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,A),B2))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),C))) ) ).

% dvd_mult2
tff(fact_3164_dvd__mult__left,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ).

% dvd_mult_left
tff(fact_3165_dvd__mult__left,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ).

% dvd_mult_left
tff(fact_3166_dvd__mult__left,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(times_times_real(A),B2)),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ).

% dvd_mult_left
tff(fact_3167_dvd__mult__left,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),B2)),C))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),C)) ) ).

% dvd_mult_left
tff(fact_3168_dvd__mult__left,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B2)),C))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,A),C)) ) ).

% dvd_mult_left
tff(fact_3169_dvd__triv__left,axiom,
    ! [A: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2))) ).

% dvd_triv_left
tff(fact_3170_dvd__triv__left,axiom,
    ! [A: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))) ).

% dvd_triv_left
tff(fact_3171_dvd__triv__left,axiom,
    ! [A: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(times_times_real(A),B2))) ).

% dvd_triv_left
tff(fact_3172_dvd__triv__left,axiom,
    ! [A: complex,B2: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),aa_complex_complex(times_times_complex(A),B2))) ).

% dvd_triv_left
tff(fact_3173_dvd__triv__left,axiom,
    ! [A: extended_enat,B2: extended_enat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B2))) ).

% dvd_triv_left
tff(fact_3174_mult__dvd__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D))) ) ) ).

% mult_dvd_mono
tff(fact_3175_mult__dvd__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))) ) ) ).

% mult_dvd_mono
tff(fact_3176_mult__dvd__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),D))) ) ) ).

% mult_dvd_mono
tff(fact_3177_mult__dvd__mono,axiom,
    ! [A: complex,B2: complex,C: complex,D: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B2))
     => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),D))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B2),D))) ) ) ).

% mult_dvd_mono
tff(fact_3178_mult__dvd__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,C),D))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),D))) ) ) ).

% mult_dvd_mono
tff(fact_3179_dvd__mult__right,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),C)) ) ).

% dvd_mult_right
tff(fact_3180_dvd__mult__right,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),C)) ) ).

% dvd_mult_right
tff(fact_3181_dvd__mult__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(times_times_real(A),B2)),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B2),C)) ) ).

% dvd_mult_right
tff(fact_3182_dvd__mult__right,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),B2)),C))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B2),C)) ) ).

% dvd_mult_right
tff(fact_3183_dvd__mult__right,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B2)),C))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,B2),C)) ) ).

% dvd_mult_right
tff(fact_3184_dvd__triv__right,axiom,
    ! [A: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A))) ).

% dvd_triv_right
tff(fact_3185_dvd__triv__right,axiom,
    ! [A: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A))) ).

% dvd_triv_right
tff(fact_3186_dvd__triv__right,axiom,
    ! [A: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(times_times_real(B2),A))) ).

% dvd_triv_right
tff(fact_3187_dvd__triv__right,axiom,
    ! [A: complex,B2: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),aa_complex_complex(times_times_complex(B2),A))) ).

% dvd_triv_right
tff(fact_3188_dvd__triv__right,axiom,
    ! [A: extended_enat,B2: extended_enat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B2),A))) ).

% dvd_triv_right
tff(fact_3189_one__dvd,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,one_one_nat),A)) ).

% one_dvd
tff(fact_3190_one__dvd,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,one_one_int),A)) ).

% one_dvd
tff(fact_3191_one__dvd,axiom,
    ! [A: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,one_one_complex),A)) ).

% one_dvd
tff(fact_3192_one__dvd,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,one_one_real),A)) ).

% one_dvd
tff(fact_3193_unit__imp__dvd,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A)) ) ).

% unit_imp_dvd
tff(fact_3194_unit__imp__dvd,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A)) ) ).

% unit_imp_dvd
tff(fact_3195_dvd__unit__imp__unit,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat)) ) ) ).

% dvd_unit_imp_unit
tff(fact_3196_dvd__unit__imp__unit,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int)) ) ) ).

% dvd_unit_imp_unit
tff(fact_3197_dvd__add__right__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_3198_dvd__add__right__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_3199_dvd__add__right__iff,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(plus_plus_real(B2),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_3200_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2)) ) ) ).

% dvd_add_left_iff
tff(fact_3201_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2)) ) ) ).

% dvd_add_left_iff
tff(fact_3202_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(plus_plus_real(B2),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2)) ) ) ).

% dvd_add_left_iff
tff(fact_3203_dvd__add,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C))) ) ) ).

% dvd_add
tff(fact_3204_dvd__add,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))) ) ) ).

% dvd_add
tff(fact_3205_dvd__add,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(plus_plus_real(B2),C))) ) ) ).

% dvd_add
tff(fact_3206_dvd__add,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,A),B2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,A),C))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,A),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B2),C))) ) ) ).

% dvd_add
tff(fact_3207_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B2))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C) )
        <=> ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3208_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B2))
       => ( ( aa_int_int(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B2),C) )
        <=> ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3209_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),B2))
       => ( ( aa_real_real(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B2),C) )
        <=> ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3210_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B2: 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,B2),C) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B2))
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3211_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( aa_int_int(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B2),C) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B2))
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3212_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( aa_real_real(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B2),C) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),B2))
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3213_div__div__div__same,axiom,
    ! [D: nat,B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),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,B2),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) ) ) ) ).

% div_div_div_same
tff(fact_3214_div__div__div__same,axiom,
    ! [D: int,B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A))
       => ( aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),D)),aa_int_int(divide_divide_int(B2),D)) = aa_int_int(divide_divide_int(A),B2) ) ) ) ).

% div_div_div_same
tff(fact_3215_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),M)) ).

% mod_less_eq_dividend
tff(fact_3216_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_3217_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat))
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_3218_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_3219_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_3220_gcd__nat_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).

% gcd_nat.extremum
tff(fact_3221_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,bit0(N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_code(2)
tff(fact_3222_numeral__code_I2_J,axiom,
    ! [N: num] : numera1916890842035813515d_enat(bit0(N)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N)),numera1916890842035813515d_enat(N)) ).

% numeral_code(2)
tff(fact_3223_numeral__code_I2_J,axiom,
    ! [N: num] : numeral_numeral_int(bit0(N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(N)),numeral_numeral_int(N)) ).

% numeral_code(2)
tff(fact_3224_numeral__code_I2_J,axiom,
    ! [N: num] : numeral_numeral_real(bit0(N)) = aa_real_real(plus_plus_real(numeral_numeral_real(N)),numeral_numeral_real(N)) ).

% numeral_code(2)
tff(fact_3225_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_3226_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3227_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
    <=> ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3228_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(nat_set_decode,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% subset_decode_imp_le
tff(fact_3229_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_3230_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B2)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_3231_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_3232_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),B2)),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_3233_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_3234_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B2) = A )
    <=> ( aa_int_int(divide_divide_int(A),B2) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_3235_mod__eqE,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B2),C) )
     => ~ ! [D4: int] : B2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D4)) ) ).

% mod_eqE
tff(fact_3236_div__add1__eq,axiom,
    ! [A: nat,B2: 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),B2)),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,B2),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,B2),C))),C)) ).

% div_add1_eq
tff(fact_3237_div__add1__eq,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B2),C))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B2),C))),C)) ).

% div_add1_eq
tff(fact_3238_not__is__unit__0,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),one_one_nat)) ).

% not_is_unit_0
tff(fact_3239_not__is__unit__0,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),one_one_int)) ).

% not_is_unit_0
tff(fact_3240_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))) ) ) ).

% minf(10)
tff(fact_3241_minf_I10_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),Z))
     => ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X2),S)))
      <=> ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X2),S))) ) ) ).

% minf(10)
tff(fact_3242_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S))) ) ) ).

% minf(10)
tff(fact_3243_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))) ) ) ).

% minf(10)
tff(fact_3244_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))) ) ) ).

% minf(9)
tff(fact_3245_minf_I9_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),Z))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X2),S)))
      <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X2),S))) ) ) ).

% minf(9)
tff(fact_3246_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S))) ) ) ).

% minf(9)
tff(fact_3247_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))) ) ) ).

% minf(9)
tff(fact_3248_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))) ) ) ).

% pinf(10)
tff(fact_3249_pinf_I10_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z),X2))
     => ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X2),S)))
      <=> ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X2),S))) ) ) ).

% pinf(10)
tff(fact_3250_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S))) ) ) ).

% pinf(10)
tff(fact_3251_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))) ) ) ).

% pinf(10)
tff(fact_3252_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))) ) ) ).

% pinf(9)
tff(fact_3253_pinf_I9_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Z),X2))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X2),S)))
      <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X2),S))) ) ) ).

% pinf(9)
tff(fact_3254_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S))) ) ) ).

% pinf(9)
tff(fact_3255_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))) ) ) ).

% pinf(9)
tff(fact_3256_dvd__div__eq__0__iff,axiom,
    ! [B2: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B2),A))
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3257_dvd__div__eq__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3258_dvd__div__eq__0__iff,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A))
     => ( ( aa_int_int(divide_divide_int(A),B2) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3259_dvd__div__eq__0__iff,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B2),A))
     => ( ( aa_real_real(divide_divide_real(A),B2) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3260_unit__mult__right__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3261_unit__mult__right__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B2),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3262_unit__mult__left__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3263_unit__mult__left__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),C) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3264_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),C)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3265_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),C)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3266_dvd__mult__unit__iff_H,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3267_dvd__mult__unit__iff_H,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3268_mult__unit__dvd__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% mult_unit_dvd_iff
tff(fact_3269_mult__unit__dvd__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% mult_unit_dvd_iff
tff(fact_3270_dvd__mult__unit__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_mult_unit_iff
tff(fact_3271_dvd__mult__unit__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_mult_unit_iff
tff(fact_3272_is__unit__mult__iff,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat)) ) ) ).

% is_unit_mult_iff
tff(fact_3273_is__unit__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int)) ) ) ).

% is_unit_mult_iff
tff(fact_3274_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = zero_zero_nat ) )
      & ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) != N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ) ) ) ).

% mod_Suc
tff(fact_3275_finite__set__decode,axiom,
    ! [N: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(nat_set_decode,N))) ).

% finite_set_decode
tff(fact_3276_mod__induct,axiom,
    ! [P: fun_nat_bool,N: nat,P3: nat,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),P3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),P3))
         => ( ! [N2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),P3))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),P3))) ) )
           => pp(aa_nat_bool(P,M)) ) ) ) ) ).

% mod_induct
tff(fact_3277_dvd__div__mult,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A)),C) ) ) ).

% dvd_div_mult
tff(fact_3278_dvd__div__mult,axiom,
    ! [C: int,B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B2))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B2),C)),A) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),C) ) ) ).

% dvd_div_mult
tff(fact_3279_div__mult__swap,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),C) ) ) ).

% div_mult_swap
tff(fact_3280_div__mult__swap,axiom,
    ! [C: int,B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B2))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B2),C)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C) ) ) ).

% div_mult_swap
tff(fact_3281_div__div__eq__right,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3282_div__div__eq__right,axiom,
    ! [C: int,B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(divide_divide_int(B2),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B2)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3283_dvd__div__mult2__eq,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3284_dvd__div__mult2__eq,axiom,
    ! [B2: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)),A))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B2)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3285_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C))) ) ).

% dvd_mult_imp_div
tff(fact_3286_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(B2),C))) ) ).

% dvd_mult_imp_div
tff(fact_3287_div__mult__div__if__dvd,axiom,
    ! [B2: nat,A: nat,D: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),C))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),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,B2),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3288_div__mult__div__if__dvd,axiom,
    ! [B2: int,A: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),C))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B2)),aa_int_int(divide_divide_int(C),D)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3289_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [M3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M3),zero_zero_nat))
     => ( ! [M3: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M3),N2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M3),N2)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_3290_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).

% mod_less_divisor
tff(fact_3291_dvd__div__unit__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_div_unit_iff
tff(fact_3292_dvd__div__unit__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(C),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_div_unit_iff
tff(fact_3293_div__unit__dvd__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% div_unit_dvd_iff
tff(fact_3294_div__unit__dvd__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(A),B2)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% div_unit_dvd_iff
tff(fact_3295_unit__div__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B2 = C ) ) ) ).

% unit_div_cancel
tff(fact_3296_unit__div__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( ( aa_int_int(divide_divide_int(B2),A) = aa_int_int(divide_divide_int(C),A) )
      <=> ( B2 = C ) ) ) ).

% unit_div_cancel
tff(fact_3297_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),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,B2),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3298_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B2))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B2),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3299_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),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,B2),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3300_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B2),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3301_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_3302_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q2: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q2) ) ).

% mod_eq_0D
tff(fact_3303_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M))) ) ) ).

% dvd_power_le
tff(fact_3304_dvd__power__le,axiom,
    ! [X: real,Y: real,N: nat,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(power_power_real(X),N)),aa_nat_real(power_power_real(Y),M))) ) ) ).

% dvd_power_le
tff(fact_3305_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N: nat,M: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),M))) ) ) ).

% dvd_power_le
tff(fact_3306_dvd__power__le,axiom,
    ! [X: int,Y: int,N: nat,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),M))) ) ) ).

% dvd_power_le
tff(fact_3307_power__le__dvd,axiom,
    ! [A: nat,N: nat,B2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),B2)) ) ) ).

% power_le_dvd
tff(fact_3308_power__le__dvd,axiom,
    ! [A: real,N: nat,B2: real,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(power_power_real(A),N)),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(power_power_real(A),M)),B2)) ) ) ).

% power_le_dvd
tff(fact_3309_power__le__dvd,axiom,
    ! [A: complex,N: nat,B2: complex,M: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),N)),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M)),B2)) ) ) ).

% power_le_dvd
tff(fact_3310_power__le__dvd,axiom,
    ! [A: int,N: nat,B2: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),B2)) ) ) ).

% power_le_dvd
tff(fact_3311_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% le_imp_power_dvd
tff(fact_3312_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N))) ) ).

% le_imp_power_dvd
tff(fact_3313_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N))) ) ).

% le_imp_power_dvd
tff(fact_3314_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N))) ) ).

% le_imp_power_dvd
tff(fact_3315_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_3316_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).

% dvd_pos_nat
tff(fact_3317_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ) ).

% nat_dvd_not_less
tff(fact_3318_bezout__add__nat,axiom,
    ! [A: nat,B2: nat] :
    ? [D4: nat,X4: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),B2))
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y3)),D4) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D4) ) ) ) ).

% bezout_add_nat
tff(fact_3319_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B2: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B2))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),Y3)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_3320_parity__cases,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) != zero_zero_nat ) )
     => ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_3321_parity__cases,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
       => ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) != zero_zero_int ) )
     => ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
         => ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_3322_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_nat ) ) ) ).

% mod2_eq_if
tff(fact_3323_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
       => ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
       => ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) = one_one_int ) ) ) ).

% mod2_eq_if
tff(fact_3324_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_ck(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3325_finite__lists__length__eq,axiom,
    ! [A2: set_Extended_enat,N: nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => finite1862508098717546133d_enat(collec8433460942617342167d_enat(aa_nat4017767907309459068t_bool(aTP_Lamm_cl(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3326_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_cm(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3327_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_cn(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3328_finite__lists__length__eq,axiom,
    ! [A2: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_co(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3329_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_3330_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3331_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B2))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3332_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3333_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
       => ( aa_int_int(modulo_modulo_int(A),B2) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3334_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,bit0(N))),aa_num_nat(numeral_numeral_nat,bit0(Q3))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3335_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( aa_int_int(modulo_modulo_int(numeral_numeral_int(bit0(N))),numeral_numeral_int(bit0(Q3))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(numeral_numeral_int(N)),numeral_numeral_int(Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3336_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_3337_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_int_int(modulo_modulo_int(numeral_numeral_int(N)),numeral_numeral_int(one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_3338_finite__lists__length__le,axiom,
    ! [A2: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_cp(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3339_finite__lists__length__le,axiom,
    ! [A2: set_Extended_enat,N: nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => finite1862508098717546133d_enat(collec8433460942617342167d_enat(aa_nat4017767907309459068t_bool(aTP_Lamm_cq(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3340_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_cr(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3341_finite__lists__length__le,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_cs(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3342_finite__lists__length__le,axiom,
    ! [A2: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_ct(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3343_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: 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,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_3344_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: 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,B2),aa_int_int(divide_divide_int(A),B2))),aa_int_int(modulo_modulo_int(A),B2))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_3345_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B2: 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),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_3346_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B2)),B2)),aa_int_int(modulo_modulo_int(A),B2))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_3347_mod__div__decomp,axiom,
    ! [A: nat,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_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)) ).

% mod_div_decomp
tff(fact_3348_mod__div__decomp,axiom,
    ! [A: int,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_int_int(divide_divide_int(A),B2)),B2)),aa_int_int(modulo_modulo_int(A),B2)) ).

% mod_div_decomp
tff(fact_3349_div__mult__mod__eq,axiom,
    ! [A: nat,B2: 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),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)) = A ).

% div_mult_mod_eq
tff(fact_3350_div__mult__mod__eq,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B2)),B2)),aa_int_int(modulo_modulo_int(A),B2)) = A ).

% div_mult_mod_eq
tff(fact_3351_mod__div__mult__eq,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),B2)) = A ).

% mod_div_mult_eq
tff(fact_3352_mod__div__mult__eq,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B2)),B2)) = A ).

% mod_div_mult_eq
tff(fact_3353_mod__mult__div__eq,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2))) = A ).

% mod_mult_div_eq
tff(fact_3354_mod__mult__div__eq,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(divide_divide_int(A),B2))) = A ).

% mod_mult_div_eq
tff(fact_3355_mult__div__mod__eq,axiom,
    ! [B2: 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,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)) = A ).

% mult_div_mod_eq
tff(fact_3356_mult__div__mod__eq,axiom,
    ! [B2: 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,B2),aa_int_int(divide_divide_int(A),B2))),aa_int_int(modulo_modulo_int(A),B2)) = A ).

% mult_div_mod_eq
tff(fact_3357_div__mult1__eq,axiom,
    ! [A: nat,B2: 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),B2)),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,B2),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,B2),C))),C)) ).

% div_mult1_eq
tff(fact_3358_div__mult1__eq,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B2),C))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(modulo_modulo_int(B2),C))),C)) ).

% div_mult1_eq
tff(fact_3359_unit__dvdE,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] : B2 != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) ) ) ).

% unit_dvdE
tff(fact_3360_unit__dvdE,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] : B2 != aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) ) ) ).

% unit_dvdE
tff(fact_3361_unity__coeff__ex,axiom,
    ! [P: fun_nat_bool,L: nat] :
      ( ? [X5: nat] : pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X5)))
    <=> ? [X5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),zero_zero_nat)))
          & pp(aa_nat_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_3362_unity__coeff__ex,axiom,
    ! [P: fun_int_bool,L: int] :
      ( ? [X5: int] : pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X5)))
    <=> ? [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),zero_zero_int)))
          & pp(aa_int_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_3363_unity__coeff__ex,axiom,
    ! [P: fun_real_bool,L: real] :
      ( ? [X5: real] : pp(aa_real_bool(P,aa_real_real(times_times_real(L),X5)))
    <=> ? [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),aa_real_real(plus_plus_real(X5),zero_zero_real)))
          & pp(aa_real_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_3364_unity__coeff__ex,axiom,
    ! [P: fun_complex_bool,L: complex] :
      ( ? [X5: complex] : pp(aa_complex_bool(P,aa_complex_complex(times_times_complex(L),X5)))
    <=> ? [X5: complex] :
          ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,L),aa_complex_complex(plus_plus_complex(X5),zero_zero_complex)))
          & pp(aa_complex_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_3365_unity__coeff__ex,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,L: extended_enat] :
      ( ? [X5: extended_enat] : pp(aa_Ext2772979874139974381t_bool(P,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(L),X5)))
    <=> ? [X5: extended_enat] :
          ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(dvd_dv3785147216227455552d_enat,L),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X5),zero_z5237406670263579293d_enat)))
          & pp(aa_Ext2772979874139974381t_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_3366_dvd__div__eq__mult,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A) = C )
        <=> ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3367_dvd__div__eq__mult,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
       => ( ( aa_int_int(divide_divide_int(B2),A) = C )
        <=> ( B2 = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3368_div__dvd__iff__mult,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3369_div__dvd__iff__mult,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(A),B2)),C))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3370_dvd__div__iff__mult,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B2)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3371_dvd__div__iff__mult,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( C != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(B2),C)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B2)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3372_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B2: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),D))
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),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,B2),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3373_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B2: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),D))
           => ( ( aa_int_int(divide_divide_int(B2),A) = aa_int_int(divide_divide_int(D),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B2),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3374_unit__div__eq__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3375_unit__div__eq__0__iff,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( ( aa_int_int(divide_divide_int(A),B2) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3376_is__unit__div__mult2__eq,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_nat))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3377_is__unit__div__mult2__eq,axiom,
    ! [B2: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B2)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3378_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),C) ) ) ).

% unit_div_mult_swap
tff(fact_3379_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B2),C)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C) ) ) ).

% unit_div_mult_swap
tff(fact_3380_unit__div__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),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),B2)),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)),B2) ) ) ).

% unit_div_commute
tff(fact_3381_unit__div__commute,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B2)),C) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B2) ) ) ).

% unit_div_commute
tff(fact_3382_div__mult__unit2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C) ) ) ) ).

% div_mult_unit2
tff(fact_3383_div__mult__unit2,axiom,
    ! [C: int,B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B2)),C) ) ) ) ).

% div_mult_unit2
tff(fact_3384_unit__eq__div2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B2) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = C ) ) ) ).

% unit_eq_div2
tff(fact_3385_unit__eq__div2,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( ( A = aa_int_int(divide_divide_int(C),B2) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = C ) ) ) ).

% unit_eq_div2
tff(fact_3386_unit__eq__div1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2) ) ) ) ).

% unit_eq_div1
tff(fact_3387_unit__eq__div1,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( ( aa_int_int(divide_divide_int(A),B2) = C )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) ) ) ) ).

% unit_eq_div1
tff(fact_3388_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).

% mod_le_divisor
tff(fact_3389_is__unit__power__iff,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3390_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3391_div__less__mono,axiom,
    ! [A2: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),N) = zero_zero_nat )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),N))) ) ) ) ) ).

% div_less_mono
tff(fact_3392_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q3) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ~ ! [S3: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_3393_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q3) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ~ ! [S3: nat] : N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_3394_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ? [Q2: nat] : X = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3395_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).

% dvd_imp_le
tff(fact_3396_div__mod__decomp,axiom,
    ! [A2: nat,N: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N)) ).

% div_mod_decomp
tff(fact_3397_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q3))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).

% mod_mult2_eq
tff(fact_3398_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_mult_cancel
tff(fact_3399_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_3400_bezout__add__strong__nat,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ? [D4: nat,X4: nat,Y3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),B2))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y3)),D4) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3401_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
         => ~ ! [Va: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_3402_even__zero,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),zero_zero_nat)) ).

% even_zero
tff(fact_3403_even__zero,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),zero_zero_int)) ).

% even_zero
tff(fact_3404_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ~ ( ( A != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B3),one_one_nat))
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B3 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B3) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3405_is__unitE,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ~ ( ( A != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B3),one_one_int))
               => ( ( aa_int_int(divide_divide_int(one_one_int),A) = B3 )
                 => ( ( aa_int_int(divide_divide_int(one_one_int),B3) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B3) = one_one_int )
                     => ( aa_int_int(divide_divide_int(C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3406_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),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),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3407_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) = aa_int_int(divide_divide_int(one_one_int),B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3408_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),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,B2),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3409_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)) = aa_int_int(divide_divide_int(one_one_int),B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3410_odd__even__add,axiom,
    ! [A: nat,B2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2))) ) ) ).

% odd_even_add
tff(fact_3411_odd__even__add,axiom,
    ! [A: int,B2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))) ) ) ).

% odd_even_add
tff(fact_3412_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N: nat] :
      ( ( X != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),one_one_nat))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_3413_dvd__power__iff,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),N)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),one_one_int))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_3414_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_nat ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N))) ) ).

% dvd_power
tff(fact_3415_dvd__power,axiom,
    ! [N: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_real ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_nat_real(power_power_real(X),N))) ) ).

% dvd_power
tff(fact_3416_dvd__power,axiom,
    ! [N: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_complex ) )
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),aa_nat_complex(power_power_complex(X),N))) ) ).

% dvd_power
tff(fact_3417_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_int ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_nat_int(power_power_int(X),N))) ) ).

% dvd_power
tff(fact_3418_split__mod,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,M)) )
        & ( ( N != zero_zero_nat )
         => ! [I3: nat,J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I3)),J2) )
               => pp(aa_nat_bool(P,J2)) ) ) ) ) ) ).

% split_mod
tff(fact_3419_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_3420_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_3421_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_3422_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3423_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),B2)),C))),aa_int_int(modulo_modulo_int(A),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3424_product__nth,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s5460976970255530739at_nat(Xs)),size_s5460976970255530739at_nat(Ys))))
     => ( nth_Pr6744343527793145070at_nat(produc3544356994491977349at_nat(Xs,Ys),N) = produc6161850002892822231at_nat(nth_Pr7617993195940197384at_nat(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s5460976970255530739at_nat(Ys))),nth_Pr7617993195940197384at_nat(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s5460976970255530739at_nat(Ys)))) ) ) ).

% product_nth
tff(fact_3425_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys))))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),N) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)))) ) ) ).

% product_nth
tff(fact_3426_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_int_nat(size_size_list_int,Ys))))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys),N) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_list_int_nat(size_size_list_int,Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_list_int_nat(size_size_list_int,Ys)))) ) ) ).

% product_nth
tff(fact_3427_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_nat_nat(size_size_list_nat,Ys))))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),N) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_list_nat_nat(size_size_list_nat,Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_list_nat_nat(size_size_list_nat,Ys)))) ) ) ).

% product_nth
tff(fact_3428_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_int_nat(size_size_list_int,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys))))
     => ( nth_Pr3474266648193625910T_VEBT(produc662631939642741121T_VEBT(Xs,Ys),N) = produc3329399203697025711T_VEBT(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)))) ) ) ).

% product_nth
tff(fact_3429_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_int_nat(size_size_list_int,Xs)),aa_list_int_nat(size_size_list_int,Ys))))
     => ( nth_Pr4439495888332055232nt_int(product_int_int(Xs,Ys),N) = product_Pair_int_int(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_list_int_nat(size_size_list_int,Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_list_int_nat(size_size_list_int,Ys)))) ) ) ).

% product_nth
tff(fact_3430_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_int_nat(size_size_list_int,Xs)),aa_list_nat_nat(size_size_list_nat,Ys))))
     => ( nth_Pr8617346907841251940nt_nat(product_int_nat(Xs,Ys),N) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_list_nat_nat(size_size_list_nat,Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_list_nat_nat(size_size_list_nat,Ys)))) ) ) ).

% product_nth
tff(fact_3431_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys))))
     => ( nth_Pr744662078594809490T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys),N) = produc599794634098209291T_VEBT(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)))) ) ) ).

% product_nth
tff(fact_3432_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_int_nat(size_size_list_int,Ys))))
     => ( nth_Pr3440142176431000676at_int(product_nat_int(Xs,Ys),N) = product_Pair_nat_int(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_list_int_nat(size_size_list_int,Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_list_int_nat(size_size_list_int,Ys)))) ) ) ).

% product_nth
tff(fact_3433_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_nat_nat(size_size_list_nat,Ys))))
     => ( nth_Pr7617993195940197384at_nat(product_nat_nat(Xs,Ys),N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_list_nat_nat(size_size_list_nat,Ys)))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_list_nat_nat(size_size_list_nat,Ys)))) ) ) ).

% product_nth
tff(fact_3434_power__mono__odd,axiom,
    ! [N: nat,A: real,B2: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B2),N))) ) ) ).

% power_mono_odd
tff(fact_3435_power__mono__odd,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B2),N))) ) ) ).

% power_mono_odd
tff(fact_3436_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_3437_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% odd_pos
tff(fact_3438_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% dvd_power_iff_le
tff(fact_3439_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3440_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3441_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3442_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),A)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3443_divmod__digit__0_I2_J,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3444_divmod__digit__0_I2_J,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B2))),B2))
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B2)) = aa_int_int(modulo_modulo_int(A),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3445_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_3446_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_3447_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)))
         => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_3448_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_3449_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M: nat] : aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M)) = aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_3450_oddE,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_3451_oddE,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B3)),one_one_int) ) ).

% oddE
tff(fact_3452_zero__le__power__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_3453_zero__le__power__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_3454_zero__le__odd__power,axiom,
    ! [N: nat,A: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),N)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% zero_le_odd_power
tff(fact_3455_zero__le__odd__power,axiom,
    ! [N: nat,A: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% zero_le_odd_power
tff(fact_3456_zero__le__even__power,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),N))) ) ).

% zero_le_even_power
tff(fact_3457_zero__le__even__power,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_le_even_power
tff(fact_3458_verit__le__mono__div,axiom,
    ! [A2: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),N))) ) ) ).

% verit_le_mono_div
tff(fact_3459_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd2)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)))
         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_3460_divmod__digit__0_I1_J,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3461_divmod__digit__0_I1_J,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B2))),B2))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B2))) = aa_int_int(divide_divide_int(A),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3462_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)),X))
    <=> ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                 => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)))
                     => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_3463_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc)),X))
    <=> ( ( X = Mi )
        | ( X = Ma )
        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)))
           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_3464_zero__less__power__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & ( A != zero_zero_real ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_3465_zero__less__power__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & ( A != zero_zero_int ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_3466_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
      <=> pp(Y) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => pp(Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3)
               => ( pp(Y)
                <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3467_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B3) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3)
             => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                   => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3468_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B3) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3)
               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                   => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3469_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_3470_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( ( aa_int_int(modulo_modulo_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),M)) = aa_int_int(modulo_modulo_int(X),M) )
          | ( aa_int_int(modulo_modulo_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_3471_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                     => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd)
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                     => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_3472_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% unset_bit_Suc
tff(fact_3473_unset__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))))) ).

% unset_bit_Suc
tff(fact_3474_set__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% set_bit_Suc
tff(fact_3475_set__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))))) ).

% set_bit_Suc
tff(fact_3476_power__le__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3477_power__le__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3478_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B3) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2)
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                               => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3479_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => pp(Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => pp(Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( pp(Y)
                <=> ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
                 => ( pp(Y)
                  <=> ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd)
                   => ( pp(Y)
                    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3480_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( ! [Uu2: bool,Uv2: bool] : X != vEBT_Leaf(Uu2,Uv2)
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd)
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3481_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X) = if_VEBT_VEBT(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_bool_bool(fNot,fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)) ).

% vebt_insert.simps(5)
tff(fact_3482_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B3) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2)
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3483_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => pp(Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => pp(Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => pp(Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2)
                   => ( pp(Y)
                    <=> ~ ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3484_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3485_divmod__digit__1_I1_J,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B2))))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B2)))),one_one_int) = aa_int_int(divide_divide_int(A),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3486_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_cu(N),C)))) ) ).

% finite_nth_roots
tff(fact_3487_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_set_real_bool(finite_finite_real,collect_real(aTP_Lamm_cv(N)))) ) ).

% finite_roots_unity
tff(fact_3488_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_cw,N)))) ) ).

% finite_roots_unity
tff(fact_3489_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y = vEBT_Leaf(fTrue,B3) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y = vEBT_Leaf(A3,fTrue) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y = vEBT_Leaf(A3,B3) ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero_nat,Ts2,S3) )
               => ( ( Y = vEBT_Node(Info2,zero_zero_nat,Ts2,S3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S3),Xa2)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3) )
                 => ( ( Y = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3),Xa2)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = if_VEBT_VEBT(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)),aa_bool_bool(fNot,fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_3490_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% flip_bit_Suc
tff(fact_3491_flip__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))))) ).

% flip_bit_Suc
tff(fact_3492_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_nat,Y: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8134817056309787853l_bool(aTP_Lamm_cx(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8134817056309787853l_bool(aTP_Lamm_cx(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8134817056309787853l_bool(aa_fun1579482225532230253l_bool(aTP_Lamm_cy(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3493_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_nat,Y: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1783405840085286277t_bool(aTP_Lamm_cz(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1783405840085286277t_bool(aTP_Lamm_cz(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1783405840085286277t_bool(aa_fun5267048652646918913t_bool(aTP_Lamm_da(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3494_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: fun_complex_nat,Y: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun4400319914713427657x_bool(aTP_Lamm_db(I6),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun4400319914713427657x_bool(aTP_Lamm_db(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun4400319914713427657x_bool(aa_fun3291692084277223911x_bool(aTP_Lamm_dc(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3495_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: fun_int_nat,Y: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5627573752281863885t_bool(aTP_Lamm_dd(I6),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5627573752281863885t_bool(aTP_Lamm_dd(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5627573752281863885t_bool(aa_fun3624795673971170029t_bool(aTP_Lamm_de(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3496_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_Extended_enat,X: fun_Ex7660298398204468809at_nat,Y: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun1105177683714449665t_bool(aTP_Lamm_df(I6),X))))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun1105177683714449665t_bool(aTP_Lamm_df(I6),Y))))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun1105177683714449665t_bool(aa_fun6406732541880856067t_bool(aTP_Lamm_dg(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3497_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_int,Y: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun2745684120339038577l_bool(aTP_Lamm_dh(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun2745684120339038577l_bool(aTP_Lamm_dh(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun2745684120339038577l_bool(aa_fun2217247558787703221l_bool(aTP_Lamm_di(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3498_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_int,Y: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8175233552433550377t_bool(aTP_Lamm_dj(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8175233552433550377t_bool(aTP_Lamm_dj(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8175233552433550377t_bool(aa_fun3357394936337705033t_bool(aTP_Lamm_dk(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3499_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: fun_complex_int,Y: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun4637897056364002669x_bool(aTP_Lamm_dl(I6),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun4637897056364002669x_bool(aTP_Lamm_dl(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun4637897056364002669x_bool(aa_fun3778671508893124399x_bool(aTP_Lamm_dm(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3500_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: fun_int_int,Y: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun2796029427775352177t_bool(aTP_Lamm_dn(I6),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun2796029427775352177t_bool(aTP_Lamm_dn(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun2796029427775352177t_bool(aa_fun1715141957661956149t_bool(aTP_Lamm_do(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3501_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_Extended_enat,X: fun_Ex3482447378695272101at_int,Y: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun821126609344339805t_bool(aTP_Lamm_dp(I6),X))))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun821126609344339805t_bool(aTP_Lamm_dp(I6),Y))))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun821126609344339805t_bool(aa_fun2211662805375314179t_bool(aTP_Lamm_dq(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3502_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_nat,Y: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8134817056309787853l_bool(aTP_Lamm_dr(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8134817056309787853l_bool(aTP_Lamm_dr(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8134817056309787853l_bool(aa_fun1579482225532230253l_bool(aTP_Lamm_ds(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3503_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_nat,Y: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1783405840085286277t_bool(aTP_Lamm_dt(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1783405840085286277t_bool(aTP_Lamm_dt(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1783405840085286277t_bool(aa_fun5267048652646918913t_bool(aTP_Lamm_du(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3504_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: fun_complex_nat,Y: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun4400319914713427657x_bool(aTP_Lamm_dv(I6),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun4400319914713427657x_bool(aTP_Lamm_dv(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun4400319914713427657x_bool(aa_fun3291692084277223911x_bool(aTP_Lamm_dw(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3505_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: fun_int_nat,Y: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5627573752281863885t_bool(aTP_Lamm_dx(I6),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5627573752281863885t_bool(aTP_Lamm_dx(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5627573752281863885t_bool(aa_fun3624795673971170029t_bool(aTP_Lamm_dy(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3506_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_Extended_enat,X: fun_Ex7660298398204468809at_nat,Y: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun1105177683714449665t_bool(aTP_Lamm_dz(I6),X))))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun1105177683714449665t_bool(aTP_Lamm_dz(I6),Y))))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun1105177683714449665t_bool(aa_fun6406732541880856067t_bool(aTP_Lamm_ea(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3507_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_eb(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_eb(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_ec(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3508_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ed(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ed(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_ee(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3509_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ef(I6),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ef(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_eg(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3510_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_eh(I6),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_eh(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_ei(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3511_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_Extended_enat,X: fun_Ex1451620480498224165t_real,Y: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aTP_Lamm_ej(I6),X))))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aTP_Lamm_ej(I6),Y))))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aa_fun6293361260138588547t_bool(aTP_Lamm_ek(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3512_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ pp(Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ pp(Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                     => ( ( pp(Y)
                        <=> ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                 => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                                         => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_3513_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),Xa2))
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_3514_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),Xa2))
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_3515_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M))
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_3516_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)))
    <=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3517_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),A)))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3518_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3) )
                 => ( ( pp(Y)
                    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                         => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_3519_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3),Xa2))
                 => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_3520_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3),Xa2))
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_3521_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ~ pp(Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( ( pp(Y)
                    <=> ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
                   => ( ( pp(Y)
                      <=> ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                             => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2)) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd) )
                     => ( ( pp(Y)
                        <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                             => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_3522_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2))
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2))
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),Xa2))
                       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_3523_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2))
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2))
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),Xa2))
                   => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_3524_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_3525_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_3526_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_3527_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_3528_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% flip_bit_0
tff(fact_3529_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,zero_n2684676970156552555ol_int(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))))) ).

% flip_bit_0
tff(fact_3530_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))))) ).

% signed_take_bit_Suc
tff(fact_3531_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3532_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N) = zero_zero_int )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3533_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_set_nat_nat(nat_set_encode,A2)))
      <=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) ) ) ).

% even_set_encode_iff
tff(fact_3534_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : aa_num_nat(size_num,bit0(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_3535_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_3536_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = zero_n2684676970156552555ol_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_3537_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_3538_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_3539_diff__self,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% diff_self
tff(fact_3540_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_3541_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_3542_diff__0__right,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_0_right
tff(fact_3543_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_3544_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_3545_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_3546_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_3547_diff__zero,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_zero
tff(fact_3548_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3549_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_3550_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_3551_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3552_add__diff__cancel,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2) = A ).

% add_diff_cancel
tff(fact_3553_add__diff__cancel,axiom,
    ! [A: real,B2: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B2)),B2) = A ).

% add_diff_cancel
tff(fact_3554_diff__add__cancel,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),B2) = A ).

% diff_add_cancel
tff(fact_3555_diff__add__cancel,axiom,
    ! [A: real,B2: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B2)),B2) = A ).

% diff_add_cancel
tff(fact_3556_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B2: 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),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2) ).

% add_diff_cancel_left
tff(fact_3557_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B2: 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),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) ).

% add_diff_cancel_left
tff(fact_3558_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B2: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B2)) = aa_real_real(minus_minus_real(A),B2) ).

% add_diff_cancel_left
tff(fact_3559_add__diff__cancel__left_H,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),A) = B2 ).

% add_diff_cancel_left'
tff(fact_3560_add__diff__cancel__left_H,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),A) = B2 ).

% add_diff_cancel_left'
tff(fact_3561_add__diff__cancel__left_H,axiom,
    ! [A: real,B2: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B2)),A) = B2 ).

% add_diff_cancel_left'
tff(fact_3562_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B2: 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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2) ).

% add_diff_cancel_right
tff(fact_3563_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B2: 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,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) ).

% add_diff_cancel_right
tff(fact_3564_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B2: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B2),C)) = aa_real_real(minus_minus_real(A),B2) ).

% add_diff_cancel_right
tff(fact_3565_add__diff__cancel__right_H,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),B2) = A ).

% add_diff_cancel_right'
tff(fact_3566_add__diff__cancel__right_H,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2) = A ).

% add_diff_cancel_right'
tff(fact_3567_add__diff__cancel__right_H,axiom,
    ! [A: real,B2: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B2)),B2) = A ).

% add_diff_cancel_right'
tff(fact_3568_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_Suc_Suc
tff(fact_3569_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) ).

% Suc_diff_diff
tff(fact_3570_diff__0__eq__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_3571_diff__self__eq__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_3572_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_3573_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_3574_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_n3304061248610475627l_real(P)),zero_n3304061248610475627l_real(Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_3575_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_3576_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_n2684676970156552555ol_int(P)),zero_n2684676970156552555ol_int(Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_3577_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real(fFalse) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_3578_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex(fFalse) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_3579_of__bool__eq_I1_J,axiom,
    zero_n1046097342994218471d_enat(fFalse) = zero_z5237406670263579293d_enat ).

% of_bool_eq(1)
tff(fact_3580_of__bool__eq_I1_J,axiom,
    zero_n2687167440665602831ol_nat(fFalse) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_3581_of__bool__eq_I1_J,axiom,
    zero_n2684676970156552555ol_int(fFalse) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_3582_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n3304061248610475627l_real(P) = zero_zero_real )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_3583_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n1201886186963655149omplex(P) = zero_zero_complex )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_3584_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n1046097342994218471d_enat(P) = zero_z5237406670263579293d_enat )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_3585_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2687167440665602831ol_nat(P) = zero_zero_nat )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_3586_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2684676970156552555ol_int(P) = zero_zero_int )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_3587_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_n1046097342994218471d_enat(P)),zero_n1046097342994218471d_enat(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_3588_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P)),zero_n3304061248610475627l_real(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_3589_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_3590_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_n2684676970156552555ol_int(P)),zero_n2684676970156552555ol_int(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_3591_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n1201886186963655149omplex(P) = one_one_complex )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_3592_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n3304061248610475627l_real(P) = one_one_real )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_3593_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2687167440665602831ol_nat(P) = one_one_nat )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_3594_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2684676970156552555ol_int(P) = one_one_int )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_3595_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex(fTrue) = one_one_complex ).

% of_bool_eq(2)
tff(fact_3596_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real(fTrue) = one_one_real ).

% of_bool_eq(2)
tff(fact_3597_of__bool__eq_I2_J,axiom,
    zero_n2687167440665602831ol_nat(fTrue) = one_one_nat ).

% of_bool_eq(2)
tff(fact_3598_of__bool__eq_I2_J,axiom,
    zero_n2684676970156552555ol_int(fTrue) = one_one_int ).

% of_bool_eq(2)
tff(fact_3599_signed__take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_3600_of__bool__or__iff,axiom,
    ! [P: bool,Q: bool] : zero_n2687167440665602831ol_nat(fdisj(P,Q)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)) ).

% of_bool_or_iff
tff(fact_3601_of__bool__or__iff,axiom,
    ! [P: bool,Q: bool] : zero_n2684676970156552555ol_int(fdisj(P,Q)) = aa_int_int(aa_int_fun_int_int(ord_max_int,zero_n2684676970156552555ol_int(P)),zero_n2684676970156552555ol_int(Q)) ).

% of_bool_or_iff
tff(fact_3602_diff__ge__0__iff__ge,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(A),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_3603_diff__ge__0__iff__ge,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_3604_diff__gt__0__iff__gt,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(A),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_3605_diff__gt__0__iff__gt,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_3606_le__add__diff__inverse2,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => ( aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B2)),B2) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3607_le__add__diff__inverse2,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)),B2) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3608_le__add__diff__inverse2,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),B2) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3609_le__add__diff__inverse,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => ( aa_real_real(plus_plus_real(B2),aa_real_real(minus_minus_real(A),B2)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3610_le__add__diff__inverse,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3611_le__add__diff__inverse,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3612_diff__numeral__special_I9_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),one_one_complex) = zero_zero_complex ).

% diff_numeral_special(9)
tff(fact_3613_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_3614_diff__numeral__special_I9_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_3615_diff__add__zero,axiom,
    ! [A: nat,B2: 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),B2)) = zero_zero_nat ).

% diff_add_zero
tff(fact_3616_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),numeral_numeral_int(bit0(K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),numeral_numeral_int(K))),numeral_numeral_int(bit0(one))) ).

% signed_take_bit_Suc_bit0
tff(fact_3617_div__diff,axiom,
    ! [C: int,A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B2))
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B2),C)) ) ) ) ).

% div_diff
tff(fact_3618_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_3619_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_n2687167440665602831ol_nat(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_3620_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_n2684676970156552555ol_int(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_3621_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% zero_less_diff
tff(fact_3622_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_3623_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% diff_is_0_eq
tff(fact_3624_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P)),one_one_real))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_3625_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_n2687167440665602831ol_nat(P)),one_one_nat))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_3626_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_n2684676970156552555ol_int(P)),one_one_int))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_3627_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n1201886186963655149omplex(aa_bool_bool(fNot,P)) = aa_complex_complex(minus_minus_complex(one_one_complex),zero_n1201886186963655149omplex(P)) ).

% of_bool_not_iff
tff(fact_3628_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n3304061248610475627l_real(aa_bool_bool(fNot,P)) = aa_real_real(minus_minus_real(one_one_real),zero_n3304061248610475627l_real(P)) ).

% of_bool_not_iff
tff(fact_3629_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n2684676970156552555ol_int(aa_bool_bool(fNot,P)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),zero_n2684676970156552555ol_int(P)) ).

% of_bool_not_iff
tff(fact_3630_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_3631_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_3632_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_3633_diff__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),one_one_nat) = N ).

% diff_Suc_1
tff(fact_3634_Suc__0__mod__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),N) = zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),aa_nat_nat(suc,zero_zero_nat)))) ).

% Suc_0_mod_eq
tff(fact_3635_signed__take__bit__Suc__1,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_3636_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_3637_set__encode__inverse,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_nat_set_nat(nat_set_decode,aa_set_nat_nat(nat_set_encode,A2)) = A2 ) ) ).

% set_encode_inverse
tff(fact_3638_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% Suc_pred
tff(fact_3639_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_3640_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_3641_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).

% Suc_diff_1
tff(fact_3642_even__diff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))) ) ).

% even_diff
tff(fact_3643_of__bool__half__eq__0,axiom,
    ! [B2: bool] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_n2687167440665602831ol_nat(B2)),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_3644_of__bool__half__eq__0,axiom,
    ! [B2: bool] : aa_int_int(divide_divide_int(zero_n2684676970156552555ol_int(B2)),numeral_numeral_int(bit0(one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_3645_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_3646_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ).

% even_diff_nat
tff(fact_3647_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),one_one_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3648_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)),one_one_int)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3649_bits__1__div__exp,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_3650_bits__1__div__exp,axiom,
    ! [N: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = zero_n2684676970156552555ol_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_3651_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_3652_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = zero_n2684676970156552555ol_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_3653_of__bool__eq__iff,axiom,
    ! [P3: bool,Q3: bool] :
      ( ( zero_n2687167440665602831ol_nat(P3) = zero_n2687167440665602831ol_nat(Q3) )
    <=> ( pp(P3)
      <=> pp(Q3) ) ) ).

% of_bool_eq_iff
tff(fact_3654_of__bool__eq__iff,axiom,
    ! [P3: bool,Q3: bool] :
      ( ( zero_n2684676970156552555ol_int(P3) = zero_n2684676970156552555ol_int(Q3) )
    <=> ( pp(P3)
      <=> pp(Q3) ) ) ).

% of_bool_eq_iff
tff(fact_3655_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).

% diff_commute
tff(fact_3656_diff__right__commute,axiom,
    ! [A: nat,C: nat,B2: 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)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)),C) ).

% diff_right_commute
tff(fact_3657_diff__right__commute,axiom,
    ! [A: int,C: int,B2: 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)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C) ).

% diff_right_commute
tff(fact_3658_diff__right__commute,axiom,
    ! [A: real,C: real,B2: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B2) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B2)),C) ).

% diff_right_commute
tff(fact_3659_diff__eq__diff__eq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( A = B2 )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3660_diff__eq__diff__eq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B2) = aa_real_real(minus_minus_real(C),D) )
     => ( ( A = B2 )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3661_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n3304061248610475627l_real(fconj(P,Q)) = aa_real_real(times_times_real(zero_n3304061248610475627l_real(P)),zero_n3304061248610475627l_real(Q)) ).

% of_bool_conj
tff(fact_3662_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n1201886186963655149omplex(fconj(P,Q)) = aa_complex_complex(times_times_complex(zero_n1201886186963655149omplex(P)),zero_n1201886186963655149omplex(Q)) ).

% of_bool_conj
tff(fact_3663_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n1046097342994218471d_enat(fconj(P,Q)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(zero_n1046097342994218471d_enat(P)),zero_n1046097342994218471d_enat(Q)) ).

% of_bool_conj
tff(fact_3664_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n2687167440665602831ol_nat(fconj(P,Q)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)) ).

% of_bool_conj
tff(fact_3665_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n2684676970156552555ol_int(fconj(P,Q)) = aa_int_int(aa_int_fun_int_int(times_times_int,zero_n2684676970156552555ol_int(P)),zero_n2684676970156552555ol_int(Q)) ).

% of_bool_conj
tff(fact_3666_diff__mono,axiom,
    ! [A: real,B2: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B2),D))) ) ) ).

% diff_mono
tff(fact_3667_diff__mono,axiom,
    ! [A: int,B2: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D))) ) ) ).

% diff_mono
tff(fact_3668_diff__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B2))) ) ).

% diff_left_mono
tff(fact_3669_diff__left__mono,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2))) ) ).

% diff_left_mono
tff(fact_3670_diff__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B2),C))) ) ).

% diff_right_mono
tff(fact_3671_diff__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C))) ) ).

% diff_right_mono
tff(fact_3672_diff__eq__diff__less__eq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B2) = aa_real_real(minus_minus_real(C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3673_diff__eq__diff__less__eq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3674_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( A = B2 )
    <=> ( aa_complex_complex(minus_minus_complex(A),B2) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_3675_eq__iff__diff__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_3676_eq__iff__diff__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( A = B2 )
    <=> ( aa_real_real(minus_minus_real(A),B2) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_3677_diff__strict__mono,axiom,
    ! [A: real,B2: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B2),D))) ) ) ).

% diff_strict_mono
tff(fact_3678_diff__strict__mono,axiom,
    ! [A: int,B2: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D))) ) ) ).

% diff_strict_mono
tff(fact_3679_diff__eq__diff__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B2) = aa_real_real(minus_minus_real(C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_3680_diff__eq__diff__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_3681_diff__strict__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B2))) ) ).

% diff_strict_left_mono
tff(fact_3682_diff__strict__left__mono,axiom,
    ! [B2: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2))) ) ).

% diff_strict_left_mono
tff(fact_3683_diff__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B2),C))) ) ).

% diff_strict_right_mono
tff(fact_3684_diff__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C))) ) ).

% diff_strict_right_mono
tff(fact_3685_right__diff__distrib_H,axiom,
    ! [A: nat,B2: 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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_3686_right__diff__distrib_H,axiom,
    ! [A: int,B2: 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,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib'
tff(fact_3687_right__diff__distrib_H,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(minus_minus_real(B2),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),B2)),aa_real_real(times_times_real(A),C)) ).

% right_diff_distrib'
tff(fact_3688_right__diff__distrib_H,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(minus_minus_complex(B2),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),B2)),aa_complex_complex(times_times_complex(A),C)) ).

% right_diff_distrib'
tff(fact_3689_left__diff__distrib_H,axiom,
    ! [B2: 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,B2),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_3690_left__diff__distrib_H,axiom,
    ! [B2: 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,B2),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ).

% left_diff_distrib'
tff(fact_3691_left__diff__distrib_H,axiom,
    ! [B2: real,C: real,A: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(B2),C)),A) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(B2),A)),aa_real_real(times_times_real(C),A)) ).

% left_diff_distrib'
tff(fact_3692_left__diff__distrib_H,axiom,
    ! [B2: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(B2),C)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(B2),A)),aa_complex_complex(times_times_complex(C),A)) ).

% left_diff_distrib'
tff(fact_3693_right__diff__distrib,axiom,
    ! [A: int,B2: 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,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib
tff(fact_3694_right__diff__distrib,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(minus_minus_real(B2),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),B2)),aa_real_real(times_times_real(A),C)) ).

% right_diff_distrib
tff(fact_3695_right__diff__distrib,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(minus_minus_complex(B2),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),B2)),aa_complex_complex(times_times_complex(A),C)) ).

% right_diff_distrib
tff(fact_3696_left__diff__distrib,axiom,
    ! [A: int,B2: 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),B2)),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,B2),C)) ).

% left_diff_distrib
tff(fact_3697_left__diff__distrib,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B2)),C) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B2),C)) ).

% left_diff_distrib
tff(fact_3698_left__diff__distrib,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B2)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B2),C)) ).

% left_diff_distrib
tff(fact_3699_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B2: 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),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) ) ) ).

% group_cancel.sub1
tff(fact_3700_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B2: real] :
      ( ( A2 = aa_real_real(plus_plus_real(K),A) )
     => ( aa_real_real(minus_minus_real(A2),B2) = aa_real_real(plus_plus_real(K),aa_real_real(minus_minus_real(A),B2)) ) ) ).

% group_cancel.sub1
tff(fact_3701_diff__eq__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2) ) ) ).

% diff_eq_eq
tff(fact_3702_diff__eq__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( aa_real_real(minus_minus_real(A),B2) = C )
    <=> ( A = aa_real_real(plus_plus_real(C),B2) ) ) ).

% diff_eq_eq
tff(fact_3703_eq__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = C ) ) ).

% eq_diff_eq
tff(fact_3704_eq__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( A = aa_real_real(minus_minus_real(C),B2) )
    <=> ( aa_real_real(plus_plus_real(A),B2) = C ) ) ).

% eq_diff_eq
tff(fact_3705_add__diff__eq,axiom,
    ! [A: int,B2: 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,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) ).

% add_diff_eq
tff(fact_3706_add__diff__eq,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(plus_plus_real(A),aa_real_real(minus_minus_real(B2),C)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B2)),C) ).

% add_diff_eq
tff(fact_3707_diff__diff__eq2,axiom,
    ! [A: int,B2: 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,B2),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)),B2) ).

% diff_diff_eq2
tff(fact_3708_diff__diff__eq2,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(minus_minus_real(B2),C)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),B2) ).

% diff_diff_eq2
tff(fact_3709_diff__add__eq,axiom,
    ! [A: int,B2: 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),B2)),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)),B2) ).

% diff_add_eq
tff(fact_3710_diff__add__eq,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B2)),C) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),B2) ).

% diff_add_eq
tff(fact_3711_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B2: 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,B2),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)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_3712_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(plus_plus_real(B2),C)) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_3713_add__implies__diff,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2) ) ) ).

% add_implies_diff
tff(fact_3714_add__implies__diff,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) ) ) ).

% add_implies_diff
tff(fact_3715_add__implies__diff,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( aa_real_real(plus_plus_real(C),B2) = A )
     => ( C = aa_real_real(minus_minus_real(A),B2) ) ) ).

% add_implies_diff
tff(fact_3716_diff__diff__eq,axiom,
    ! [A: nat,B2: 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),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) ).

% diff_diff_eq
tff(fact_3717_diff__diff__eq,axiom,
    ! [A: int,B2: 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),B2)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ).

% diff_diff_eq
tff(fact_3718_diff__diff__eq,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B2)),C) = aa_real_real(minus_minus_real(A),aa_real_real(plus_plus_real(B2),C)) ).

% diff_diff_eq
tff(fact_3719_add__diff__add,axiom,
    ! [A: int,C: int,B2: 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,B2),D)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D)) ).

% add_diff_add
tff(fact_3720_add__diff__add,axiom,
    ! [A: real,C: real,B2: real,D: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B2),D)) = aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B2)),aa_real_real(minus_minus_real(C),D)) ).

% add_diff_add
tff(fact_3721_diff__divide__distrib,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(A),B2)),C) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B2),C)) ).

% diff_divide_distrib
tff(fact_3722_dvd__diff,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z2))) ) ) ).

% dvd_diff
tff(fact_3723_dvd__diff,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Z2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_real_real(minus_minus_real(Y),Z2))) ) ) ).

% dvd_diff
tff(fact_3724_zero__induct__lemma,axiom,
    ! [P: fun_nat_bool,K: nat,I: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I))) ) ) ).

% zero_induct_lemma
tff(fact_3725_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).

% minus_nat.diff_0
tff(fact_3726_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_3727_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ) ).

% diff_less_mono2
tff(fact_3728_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N)),K)) ) ).

% less_imp_diff_less
tff(fact_3729_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_3730_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% le_diff_iff
tff(fact_3731_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_3732_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),L))) ) ).

% diff_le_mono
tff(fact_3733_diff__le__self,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ).

% diff_le_self
tff(fact_3734_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A)) ) ) ) ).

% le_diff_iff'
tff(fact_3735_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ).

% diff_le_mono2
tff(fact_3736_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% Nat.diff_cancel
tff(fact_3737_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_cancel2
tff(fact_3738_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M ).

% diff_add_inverse
tff(fact_3739_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).

% diff_add_inverse2
tff(fact_3740_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).

% diff_mult_distrib2
tff(fact_3741_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).

% diff_mult_distrib
tff(fact_3742_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z2) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z2)) ).

% max_diff_distrib_left
tff(fact_3743_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z2: real] : aa_real_real(minus_minus_real(aa_real_real(ord_max_real(X),Y)),Z2) = aa_real_real(ord_max_real(aa_real_real(minus_minus_real(X),Z2)),aa_real_real(minus_minus_real(Y),Z2)) ).

% max_diff_distrib_left
tff(fact_3744_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% dvd_diff_nat
tff(fact_3745_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real(P))) ).

% zero_less_eq_of_bool
tff(fact_3746_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_n2687167440665602831ol_nat(P))) ).

% zero_less_eq_of_bool
tff(fact_3747_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_n2684676970156552555ol_int(P))) ).

% zero_less_eq_of_bool
tff(fact_3748_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_n3304061248610475627l_real(P)),one_one_real)) ).

% of_bool_less_eq_one
tff(fact_3749_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_n2687167440665602831ol_nat(P)),one_one_nat)) ).

% of_bool_less_eq_one
tff(fact_3750_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_n2684676970156552555ol_int(P)),one_one_int)) ).

% of_bool_less_eq_one
tff(fact_3751_split__of__bool__asm,axiom,
    ! [P: fun_real_bool,P3: bool] :
      ( pp(aa_real_bool(P,zero_n3304061248610475627l_real(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_real_bool(P,one_one_real)) )
          | ( ~ pp(P3)
            & ~ pp(aa_real_bool(P,zero_zero_real)) ) ) ) ).

% split_of_bool_asm
tff(fact_3752_split__of__bool__asm,axiom,
    ! [P: fun_complex_bool,P3: bool] :
      ( pp(aa_complex_bool(P,zero_n1201886186963655149omplex(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_complex_bool(P,one_one_complex)) )
          | ( ~ pp(P3)
            & ~ pp(aa_complex_bool(P,zero_zero_complex)) ) ) ) ).

% split_of_bool_asm
tff(fact_3753_split__of__bool__asm,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,P3: bool] :
      ( pp(aa_Ext2772979874139974381t_bool(P,zero_n1046097342994218471d_enat(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_Ext2772979874139974381t_bool(P,one_on7984719198319812577d_enat)) )
          | ( ~ pp(P3)
            & ~ pp(aa_Ext2772979874139974381t_bool(P,zero_z5237406670263579293d_enat)) ) ) ) ).

% split_of_bool_asm
tff(fact_3754_split__of__bool__asm,axiom,
    ! [P: fun_nat_bool,P3: bool] :
      ( pp(aa_nat_bool(P,zero_n2687167440665602831ol_nat(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_nat_bool(P,one_one_nat)) )
          | ( ~ pp(P3)
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_of_bool_asm
tff(fact_3755_split__of__bool__asm,axiom,
    ! [P: fun_int_bool,P3: bool] :
      ( pp(aa_int_bool(P,zero_n2684676970156552555ol_int(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_int_bool(P,one_one_int)) )
          | ( ~ pp(P3)
            & ~ pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% split_of_bool_asm
tff(fact_3756_split__of__bool,axiom,
    ! [P: fun_real_bool,P3: bool] :
      ( pp(aa_real_bool(P,zero_n3304061248610475627l_real(P3)))
    <=> ( ( pp(P3)
         => pp(aa_real_bool(P,one_one_real)) )
        & ( ~ pp(P3)
         => pp(aa_real_bool(P,zero_zero_real)) ) ) ) ).

% split_of_bool
tff(fact_3757_split__of__bool,axiom,
    ! [P: fun_complex_bool,P3: bool] :
      ( pp(aa_complex_bool(P,zero_n1201886186963655149omplex(P3)))
    <=> ( ( pp(P3)
         => pp(aa_complex_bool(P,one_one_complex)) )
        & ( ~ pp(P3)
         => pp(aa_complex_bool(P,zero_zero_complex)) ) ) ) ).

% split_of_bool
tff(fact_3758_split__of__bool,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,P3: bool] :
      ( pp(aa_Ext2772979874139974381t_bool(P,zero_n1046097342994218471d_enat(P3)))
    <=> ( ( pp(P3)
         => pp(aa_Ext2772979874139974381t_bool(P,one_on7984719198319812577d_enat)) )
        & ( ~ pp(P3)
         => pp(aa_Ext2772979874139974381t_bool(P,zero_z5237406670263579293d_enat)) ) ) ) ).

% split_of_bool
tff(fact_3759_split__of__bool,axiom,
    ! [P: fun_nat_bool,P3: bool] :
      ( pp(aa_nat_bool(P,zero_n2687167440665602831ol_nat(P3)))
    <=> ( ( pp(P3)
         => pp(aa_nat_bool(P,one_one_nat)) )
        & ( ~ pp(P3)
         => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_of_bool
tff(fact_3760_split__of__bool,axiom,
    ! [P: fun_int_bool,P3: bool] :
      ( pp(aa_int_bool(P,zero_n2684676970156552555ol_int(P3)))
    <=> ( ( pp(P3)
         => pp(aa_int_bool(P,one_one_int)) )
        & ( ~ pp(P3)
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% split_of_bool
tff(fact_3761_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n3304061248610475627l_real(P3) = one_one_real ) )
      & ( ~ pp(P3)
       => ( zero_n3304061248610475627l_real(P3) = zero_zero_real ) ) ) ).

% of_bool_def
tff(fact_3762_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n1201886186963655149omplex(P3) = one_one_complex ) )
      & ( ~ pp(P3)
       => ( zero_n1201886186963655149omplex(P3) = zero_zero_complex ) ) ) ).

% of_bool_def
tff(fact_3763_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n1046097342994218471d_enat(P3) = one_on7984719198319812577d_enat ) )
      & ( ~ pp(P3)
       => ( zero_n1046097342994218471d_enat(P3) = zero_z5237406670263579293d_enat ) ) ) ).

% of_bool_def
tff(fact_3764_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n2687167440665602831ol_nat(P3) = one_one_nat ) )
      & ( ~ pp(P3)
       => ( zero_n2687167440665602831ol_nat(P3) = zero_zero_nat ) ) ) ).

% of_bool_def
tff(fact_3765_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n2684676970156552555ol_int(P3) = one_one_int ) )
      & ( ~ pp(P3)
       => ( zero_n2684676970156552555ol_int(P3) = zero_zero_int ) ) ) ).

% of_bool_def
tff(fact_3766_le__iff__diff__le__0,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),B2)),zero_zero_real)) ) ).

% le_iff_diff_le_0
tff(fact_3767_le__iff__diff__le__0,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),zero_zero_int)) ) ).

% le_iff_diff_le_0
tff(fact_3768_less__iff__diff__less__0,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),B2)),zero_zero_real)) ) ).

% less_iff_diff_less_0
tff(fact_3769_less__iff__diff__less__0,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),zero_zero_int)) ) ).

% less_iff_diff_less_0
tff(fact_3770_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A) = C )
        <=> ( B2 = 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_3771_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)) = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_3772_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),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)),B2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_3773_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_3774_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_3775_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_3776_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_3777_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B2)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_3778_le__add__diff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)),A))) ) ).

% le_add_diff
tff(fact_3779_diff__add,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)),A) = B2 ) ) ).

% diff_add
tff(fact_3780_le__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(minus_minus_real(C),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),B2)),C)) ) ).

% le_diff_eq
tff(fact_3781_le__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C)) ) ).

% le_diff_eq
tff(fact_3782_diff__le__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),B2)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(plus_plus_real(C),B2))) ) ).

% diff_le_eq
tff(fact_3783_diff__le__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))) ) ).

% diff_le_eq
tff(fact_3784_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(I),K)),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(plus_plus_real(J),K)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(I),K)),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(plus_plus_real(J),K)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3785_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3786_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3787_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(I),K)),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),aa_real_real(minus_minus_real(N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3788_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3789_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3790_less__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(minus_minus_real(C),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),B2)),C)) ) ).

% less_diff_eq
tff(fact_3791_less__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C)) ) ).

% less_diff_eq
tff(fact_3792_diff__less__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),B2)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(plus_plus_real(C),B2))) ) ).

% diff_less_eq
tff(fact_3793_diff__less__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))) ) ).

% diff_less_eq
tff(fact_3794_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3795_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ( aa_real_real(plus_plus_real(B2),aa_real_real(minus_minus_real(A),B2)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3796_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3797_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3798_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E2)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(B2),E2)),D) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B2)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3799_eq__add__iff1,axiom,
    ! [A: complex,E2: complex,C: complex,B2: complex,D: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),E2)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(B2),E2)),D) )
    <=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B2)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3800_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3801_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E2)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(B2),E2)),D) )
    <=> ( C = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B2),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3802_eq__add__iff2,axiom,
    ! [A: complex,E2: complex,C: complex,B2: complex,D: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),E2)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(B2),E2)),D) )
    <=> ( C = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(B2),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3803_square__diff__square__factored,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ).

% square_diff_square_factored
tff(fact_3804_square__diff__square__factored,axiom,
    ! [X: real,Y: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y),Y)) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(X),Y)),aa_real_real(minus_minus_real(X),Y)) ).

% square_diff_square_factored
tff(fact_3805_square__diff__square__factored,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),X)),aa_complex_complex(times_times_complex(Y),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X),Y)),aa_complex_complex(minus_minus_complex(X),Y)) ).

% square_diff_square_factored
tff(fact_3806_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A)),B2)) ).

% mult_diff_mult
tff(fact_3807_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B2: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),Y)),aa_real_real(times_times_real(A),B2)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),aa_real_real(minus_minus_real(Y),B2))),aa_real_real(times_times_real(aa_real_real(minus_minus_real(X),A)),B2)) ).

% mult_diff_mult
tff(fact_3808_mult__diff__mult,axiom,
    ! [X: complex,Y: complex,A: complex,B2: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Y)),aa_complex_complex(times_times_complex(A),B2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),aa_complex_complex(minus_minus_complex(Y),B2))),aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),A)),B2)) ).

% mult_diff_mult
tff(fact_3809_dvd__minus__mod,axiom,
    ! [B2: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)))) ).

% dvd_minus_mod
tff(fact_3810_dvd__minus__mod,axiom,
    ! [B2: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(A),B2)))) ).

% dvd_minus_mod
tff(fact_3811_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(suc,M))) ).

% diff_less_Suc
tff(fact_3812_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ).

% Suc_diff_Suc
tff(fact_3813_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ) ) ).

% diff_less
tff(fact_3814_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

% Suc_diff_le
tff(fact_3815_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ) ).

% less_diff_iff
tff(fact_3816_diff__less__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C))) ) ) ).

% diff_less_mono
tff(fact_3817_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_3818_set__encode__eq,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( ( aa_set_nat_nat(nat_set_encode,A2) = aa_set_nat_nat(nat_set_encode,B) )
        <=> ( A2 = B ) ) ) ) ).

% set_encode_eq
tff(fact_3819_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_3820_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ).

% less_diff_conv
tff(fact_3821_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_3822_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_3823_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_3824_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_3825_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ).

% le_diff_conv
tff(fact_3826_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N) ).

% diff_Suc_eq_diff_pred
tff(fact_3827_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_minus_self
tff(fact_3828_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_3829_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N)) ) ) ) ).

% dvd_diffD1
tff(fact_3830_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M)) ) ) ) ).

% dvd_diffD
tff(fact_3831_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% mod_geq
tff(fact_3832_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ) ).

% mod_if
tff(fact_3833_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% le_mod_geq
tff(fact_3834_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),M) ).

% nat_minus_add_max
tff(fact_3835_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B2),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B2),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3836_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3837_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B2),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B2)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3838_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3839_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B2),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B2),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_3840_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_3841_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B2),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B2)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_3842_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_3843_divide__diff__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z2)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(Y),Z2))),Z2) ) ) ).

% divide_diff_eq_iff
tff(fact_3844_divide__diff__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(X),Z2)),Y) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(Y),Z2))),Z2) ) ) ).

% divide_diff_eq_iff
tff(fact_3845_diff__divide__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z2)),Y)),Z2) ) ) ).

% diff_divide_eq_iff
tff(fact_3846_diff__divide__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(minus_minus_real(X),aa_real_real(divide_divide_real(Y),Z2)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),Z2)),Y)),Z2) ) ) ).

% diff_divide_eq_iff
tff(fact_3847_diff__frac__eq,axiom,
    ! [Y: complex,Z2: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z2)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z2)) ) ) ) ).

% diff_frac_eq
tff(fact_3848_diff__frac__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(W2),Z2)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(W2),Y))),aa_real_real(times_times_real(Y),Z2)) ) ) ) ).

% diff_frac_eq
tff(fact_3849_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: complex,A: complex,B2: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B2),Z2)) = A ) )
      & ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B2),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z2)),B2)),Z2) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_3850_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: real,A: real,B2: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(minus_minus_real(A),aa_real_real(divide_divide_real(B2),Z2)) = A ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(minus_minus_real(A),aa_real_real(divide_divide_real(B2),Z2)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),Z2)),B2)),Z2) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_3851_square__diff__one__factored,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).

% square_diff_one_factored
tff(fact_3852_square__diff__one__factored,axiom,
    ! [X: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),X)),one_one_real) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(X),one_one_real)),aa_real_real(minus_minus_real(X),one_one_real)) ).

% square_diff_one_factored
tff(fact_3853_square__diff__one__factored,axiom,
    ! [X: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),X)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ).

% square_diff_one_factored
tff(fact_3854_inf__period_I4_J,axiom,
    ! [D: int,D5: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D5))
     => ! [X2: int,K4: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T)))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),T))) ) ) ).

% inf_period(4)
tff(fact_3855_inf__period_I4_J,axiom,
    ! [D: real,D5: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D5))
     => ! [X2: real,K4: real] :
          ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),T)))
        <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X2),aa_real_real(times_times_real(K4),D5))),T))) ) ) ).

% inf_period(4)
tff(fact_3856_inf__period_I4_J,axiom,
    ! [D: complex,D5: complex,T: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,D),D5))
     => ! [X2: complex,K4: complex] :
          ( ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(X2),T)))
        <=> ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(X2),aa_complex_complex(times_times_complex(K4),D5))),T))) ) ) ).

% inf_period(4)
tff(fact_3857_inf__period_I3_J,axiom,
    ! [D: int,D5: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D5))
     => ! [X2: int,K4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),T))) ) ) ).

% inf_period(3)
tff(fact_3858_inf__period_I3_J,axiom,
    ! [D: real,D5: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D5))
     => ! [X2: real,K4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),T)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X2),aa_real_real(times_times_real(K4),D5))),T))) ) ) ).

% inf_period(3)
tff(fact_3859_inf__period_I3_J,axiom,
    ! [D: complex,D5: complex,T: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,D),D5))
     => ! [X2: complex,K4: complex] :
          ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(X2),T)))
        <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(X2),aa_complex_complex(times_times_complex(K4),D5))),T))) ) ) ).

% inf_period(3)
tff(fact_3860_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% minus_mult_div_eq_mod
tff(fact_3861_minus__mult__div__eq__mod,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(divide_divide_int(A),B2))) = aa_int_int(modulo_modulo_int(A),B2) ).

% minus_mult_div_eq_mod
tff(fact_3862_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B2: 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),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ).

% minus_mod_eq_mult_div
tff(fact_3863_minus__mod__eq__mult__div,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(A),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(divide_divide_int(A),B2)) ).

% minus_mod_eq_mult_div
tff(fact_3864_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B2: 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),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_3865_minus__mod__eq__div__mult,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(A),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_3866_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B2: 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),B2)),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% minus_div_mult_eq_mod
tff(fact_3867_minus__div__mult__eq__mod,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B2)),B2)) = aa_int_int(modulo_modulo_int(A),B2) ).

% minus_div_mult_eq_mod
tff(fact_3868_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(K),M)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_3869_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,I))),N)) ) ).

% diff_Suc_less
tff(fact_3870_nat__diff__split__asm,axiom,
    ! [P: fun_nat_bool,A: nat,B2: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)))
    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) )
          | ? [D6: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D6) )
              & ~ pp(aa_nat_bool(P,D6)) ) ) ) ).

% nat_diff_split_asm
tff(fact_3871_nat__diff__split,axiom,
    ! [P: fun_nat_bool,A: nat,B2: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ! [D6: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D6) )
           => pp(aa_nat_bool(P,D6)) ) ) ) ).

% nat_diff_split
tff(fact_3872_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ) ).

% less_diff_conv2
tff(fact_3873_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_3874_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_3875_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_3876_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_3877_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_3878_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_3879_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q3) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_3880_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_nat(nat_set_encode,A2) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_3881_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),zero_zero_nat)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_3882_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] : aa_int_int(divide_divide_int(aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,zero_n2684676970156552555ol_int(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M)),zero_zero_int)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_3883_frac__le__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(W2),Z2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(W2),Y))),aa_real_real(times_times_real(Y),Z2))),zero_zero_real)) ) ) ) ).

% frac_le_eq
tff(fact_3884_frac__less__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(W2),Z2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(W2),Y))),aa_real_real(times_times_real(Y),Z2))),zero_zero_real)) ) ) ) ).

% frac_less_eq
tff(fact_3885_power__diff,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N)) ) ) ) ).

% power_diff
tff(fact_3886_power__diff,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_diff
tff(fact_3887_power__diff,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ) ) ) ).

% power_diff
tff(fact_3888_power__diff,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(divide_divide_real(aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)) ) ) ) ).

% power_diff
tff(fact_3889_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_3890_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_3891_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% div_geq
tff(fact_3892_div__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% div_if
tff(fact_3893_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

% add_eq_if
tff(fact_3894_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_3895_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_3896_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

% mult_eq_if
tff(fact_3897_dvd__minus__add,axiom,
    ! [Q3: nat,N: nat,R2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q3),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),Q3)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)),Q3)))) ) ) ) ).

% dvd_minus_add
tff(fact_3898_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R2)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_3899_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),V))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(U),aa_real_real(divide_divide_real(aa_real_real(times_times_real(R2),aa_real_real(minus_minus_real(V),U))),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_3900_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3901_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N) != zero_zero_int )
     => ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3902_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_3903_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_3904_power__eq__if,axiom,
    ! [M: nat,P3: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_3905_power__eq__if,axiom,
    ! [M: nat,P3: int] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = aa_int_int(aa_int_fun_int_int(times_times_int,P3),aa_nat_int(power_power_int(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_3906_power__eq__if,axiom,
    ! [M: nat,P3: real] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_real(power_power_real(P3),M) = one_one_real ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_real(power_power_real(P3),M) = aa_real_real(times_times_real(P3),aa_nat_real(power_power_real(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_3907_power__eq__if,axiom,
    ! [M: nat,P3: complex] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = aa_complex_complex(times_times_complex(P3),aa_nat_complex(power_power_complex(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_3908_power__eq__if,axiom,
    ! [M: nat,P3: extended_enat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_Extended_enat(power_8040749407984259932d_enat(P3),M) = one_on7984719198319812577d_enat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_Extended_enat(power_8040749407984259932d_enat(P3),M) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(P3),aa_nat_Extended_enat(power_8040749407984259932d_enat(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_3909_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) ) ) ).

% power_minus_mult
tff(fact_3910_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_int(power_power_int(A),N) ) ) ).

% power_minus_mult
tff(fact_3911_power__minus__mult,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(times_times_real(aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_real(power_power_real(A),N) ) ) ).

% power_minus_mult
tff(fact_3912_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),N) ) ) ).

% power_minus_mult
tff(fact_3913_power__minus__mult,axiom,
    ! [N: nat,A: extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(power_8040749407984259932d_enat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_Extended_enat(power_8040749407984259932d_enat(A),N) ) ) ).

% power_minus_mult
tff(fact_3914_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))) ) ).

% diff_le_diff_pow
tff(fact_3915_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_3916_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_3917_bits__induct,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ! [A3: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),aa_num_nat(numeral_numeral_nat,bit0(one))) = A3 )
         => pp(aa_nat_bool(P,A3)) )
     => ( ! [A3: nat,B3: bool] :
            ( pp(aa_nat_bool(P,A3))
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(B3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A3))),aa_num_nat(numeral_numeral_nat,bit0(one))) = A3 )
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(B3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A3)))) ) )
       => pp(aa_nat_bool(P,A)) ) ) ).

% bits_induct
tff(fact_3918_bits__induct,axiom,
    ! [P: fun_int_bool,A: int] :
      ( ! [A3: int] :
          ( ( aa_int_int(divide_divide_int(A3),numeral_numeral_int(bit0(one))) = A3 )
         => pp(aa_int_bool(P,A3)) )
     => ( ! [A3: int,B3: bool] :
            ( pp(aa_int_bool(P,A3))
           => ( ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_n2684676970156552555ol_int(B3)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A3))),numeral_numeral_int(bit0(one))) = A3 )
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_n2684676970156552555ol_int(B3)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A3)))) ) )
       => pp(aa_int_bool(P,A)) ) ) ).

% bits_induct
tff(fact_3919_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) ).

% exp_mod_exp
tff(fact_3920_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,zero_n2684676970156552555ol_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M)) ).

% exp_mod_exp
tff(fact_3921_power2__diff,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X)),Y)) ).

% power2_diff
tff(fact_3922_power2__diff,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),X)),Y)) ).

% power2_diff
tff(fact_3923_power2__diff,axiom,
    ! [X: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(minus_minus_real(X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),X)),Y)) ).

% power2_diff
tff(fact_3924_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_3925_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_3926_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3927_divmod__digit__1_I2_J,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B2))))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B2))),B2) = aa_int_int(modulo_modulo_int(A),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3928_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_3929_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M)),one_one_int)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_3930_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% even_mod_4_div_2
tff(fact_3931_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% even_mask_div_iff
tff(fact_3932_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M)),one_one_int)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N))))
    <=> ( ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N) = zero_zero_int )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% even_mask_div_iff
tff(fact_3933_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q3: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(L)),R2))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q3,R2)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Q3)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),numeral_numeral_int(L))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(L)),R2))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q3,R2)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Q3),R2) ) ) ) ).

% divmod_step_eq
tff(fact_3934_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),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,bit0(one))),Q3)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),aa_num_nat(numeral_numeral_nat,L))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Q3)),R2) ) ) ) ).

% divmod_step_eq
tff(fact_3935_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,vEBT_VEBT_set_vebt(T)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),one_one_nat)))) ) ).

% inrange
tff(fact_3936_artanh__def,axiom,
    ! [X: real] : aa_real_real(artanh_real,X) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(one_one_real),X)),aa_real_real(minus_minus_real(one_one_real),X)))),numeral_numeral_real(bit0(one))) ).

% artanh_def
tff(fact_3937_signed__take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_3938_diff__shunt__var,axiom,
    ! [X: set_Extended_enat,Y: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat(X,Y) = bot_bo7653980558646680370d_enat )
    <=> pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(X),Y)) ) ).

% diff_shunt_var
tff(fact_3939_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( minus_minus_set_real(X,Y) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(X),Y)) ) ).

% diff_shunt_var
tff(fact_3940_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(X,Y) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y)) ) ).

% diff_shunt_var
tff(fact_3941_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( minus_minus_set_int(X,Y) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% diff_shunt_var
tff(fact_3942_take__bit__rec,axiom,
    ! [N: nat,A: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ) ).

% take_bit_rec
tff(fact_3943_take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = zero_zero_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))))),numeral_numeral_int(bit0(one)))),aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))) ) ) ) ).

% take_bit_rec
tff(fact_3944_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% odd_mod_4_div_2
tff(fact_3945_add_Oinverse__inverse,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A)) = A ).

% add.inverse_inverse
tff(fact_3946_add_Oinverse__inverse,axiom,
    ! [A: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,A)) = A ).

% add.inverse_inverse
tff(fact_3947_neg__equal__iff__equal,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B2) )
    <=> ( A = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_3948_neg__equal__iff__equal,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B2) )
    <=> ( A = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_3949_Diff__cancel,axiom,
    ! [A2: set_Extended_enat] : minus_925952699566721837d_enat(A2,A2) = bot_bo7653980558646680370d_enat ).

% Diff_cancel
tff(fact_3950_Diff__cancel,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,A2) = bot_bot_set_real ).

% Diff_cancel
tff(fact_3951_Diff__cancel,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,A2) = bot_bot_set_int ).

% Diff_cancel
tff(fact_3952_Diff__cancel,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,A2) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_3953_empty__Diff,axiom,
    ! [A2: set_Extended_enat] : minus_925952699566721837d_enat(bot_bo7653980558646680370d_enat,A2) = bot_bo7653980558646680370d_enat ).

% empty_Diff
tff(fact_3954_empty__Diff,axiom,
    ! [A2: set_real] : minus_minus_set_real(bot_bot_set_real,A2) = bot_bot_set_real ).

% empty_Diff
tff(fact_3955_empty__Diff,axiom,
    ! [A2: set_int] : minus_minus_set_int(bot_bot_set_int,A2) = bot_bot_set_int ).

% empty_Diff
tff(fact_3956_empty__Diff,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A2) = bot_bot_set_nat ).

% empty_Diff
tff(fact_3957_Diff__empty,axiom,
    ! [A2: set_Extended_enat] : minus_925952699566721837d_enat(A2,bot_bo7653980558646680370d_enat) = A2 ).

% Diff_empty
tff(fact_3958_Diff__empty,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,bot_bot_set_real) = A2 ).

% Diff_empty
tff(fact_3959_Diff__empty,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,bot_bot_set_int) = A2 ).

% Diff_empty
tff(fact_3960_Diff__empty,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,bot_bot_set_nat) = A2 ).

% Diff_empty
tff(fact_3961_finite__Diff2,axiom,
    ! [B: set_complex,A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B)))
      <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% finite_Diff2
tff(fact_3962_finite__Diff2,axiom,
    ! [B: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,B)))
      <=> pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% finite_Diff2
tff(fact_3963_finite__Diff2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A2,B)))
      <=> pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2)) ) ) ).

% finite_Diff2
tff(fact_3964_finite__Diff2,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,B)))
      <=> pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% finite_Diff2
tff(fact_3965_finite__Diff,axiom,
    ! [A2: set_complex,B: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B))) ) ).

% finite_Diff
tff(fact_3966_finite__Diff,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,B))) ) ).

% finite_Diff
tff(fact_3967_finite__Diff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A2,B))) ) ).

% finite_Diff
tff(fact_3968_finite__Diff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,B))) ) ).

% finite_Diff
tff(fact_3969_Compl__subset__Compl__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,uminus5710092332889474511et_nat(A2)),uminus5710092332889474511et_nat(B)))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A2)) ) ).

% Compl_subset_Compl_iff
tff(fact_3970_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(A2)),uminus1532241313380277803et_int(B)))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(B),A2)) ) ).

% Compl_subset_Compl_iff
tff(fact_3971_Compl__anti__mono,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,uminus5710092332889474511et_nat(B)),uminus5710092332889474511et_nat(A2))) ) ).

% Compl_anti_mono
tff(fact_3972_Compl__anti__mono,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(B)),uminus1532241313380277803et_int(A2))) ) ).

% Compl_anti_mono
tff(fact_3973_zle__diff1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z2),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ).

% zle_diff1_eq
tff(fact_3974_zle__add1__eq__le,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z2)) ) ).

% zle_add1_eq_le
tff(fact_3975_finite__interval__int2,axiom,
    ! [A: int,B2: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_el(A),B2)))) ).

% finite_interval_int2
tff(fact_3976_finite__interval__int3,axiom,
    ! [A: int,B2: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_em(A),B2)))) ).

% finite_interval_int3
tff(fact_3977_finite__interval__int4,axiom,
    ! [A: int,B2: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_en(A),B2)))) ).

% finite_interval_int4
tff(fact_3978_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,uminus5710092332889474511et_nat(X)),uminus5710092332889474511et_nat(Y)))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),X)) ) ).

% compl_le_compl_iff
tff(fact_3979_compl__le__compl__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(X)),uminus1532241313380277803et_int(Y)))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Y),X)) ) ).

% compl_le_compl_iff
tff(fact_3980_neg__le__iff__le,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ).

% neg_le_iff_le
tff(fact_3981_neg__le__iff__le,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ).

% neg_le_iff_le
tff(fact_3982_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_3983_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_3984_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_3985_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_3986_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
tff(fact_3987_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_3988_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
tff(fact_3989_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex = aa_complex_complex(uminus1482373934393186551omplex,A) )
    <=> ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_3990_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,A) )
    <=> ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_3991_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(uminus_uminus_real,A) )
    <=> ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_3992_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_3993_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_3994_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_3995_neg__less__iff__less,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) ) ).

% neg_less_iff_less
tff(fact_3996_neg__less__iff__less,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ).

% neg_less_iff_less
tff(fact_3997_mult__minus__left,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B2) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B2)) ).

% mult_minus_left
tff(fact_3998_mult__minus__left,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B2) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ).

% mult_minus_left
tff(fact_3999_mult__minus__left,axiom,
    ! [A: real,B2: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),B2) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(A),B2)) ).

% mult_minus_left
tff(fact_4000_minus__mult__minus,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(times_times_complex(A),B2) ).

% minus_mult_minus
tff(fact_4001_minus__mult__minus,axiom,
    ! [A: int,B2: 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,B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) ).

% minus_mult_minus
tff(fact_4002_minus__mult__minus,axiom,
    ! [A: real,B2: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(times_times_real(A),B2) ).

% minus_mult_minus
tff(fact_4003_mult__minus__right,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B2)) ).

% mult_minus_right
tff(fact_4004_mult__minus__right,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ).

% mult_minus_right
tff(fact_4005_mult__minus__right,axiom,
    ! [A: real,B2: real] : aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(A),B2)) ).

% mult_minus_right
tff(fact_4006_minus__add__distrib,axiom,
    ! [A: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B2)) ).

% minus_add_distrib
tff(fact_4007_minus__add__distrib,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B2)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B2)) ).

% minus_add_distrib
tff(fact_4008_minus__add__cancel,axiom,
    ! [A: int,B2: 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),B2)) = B2 ).

% minus_add_cancel
tff(fact_4009_minus__add__cancel,axiom,
    ! [A: real,B2: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(plus_plus_real(A),B2)) = B2 ).

% minus_add_cancel
tff(fact_4010_add__minus__cancel,axiom,
    ! [A: int,B2: 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)),B2)) = B2 ).

% add_minus_cancel
tff(fact_4011_add__minus__cancel,axiom,
    ! [A: real,B2: real] : aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),B2)) = B2 ).

% add_minus_cancel
tff(fact_4012_minus__diff__eq,axiom,
    ! [A: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A) ).

% minus_diff_eq
tff(fact_4013_minus__diff__eq,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(minus_minus_real(A),B2)) = aa_real_real(minus_minus_real(B2),A) ).

% minus_diff_eq
tff(fact_4014_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(uminus_uminus_int,Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).

% dvd_minus_iff
tff(fact_4015_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_real_real(uminus_uminus_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).

% dvd_minus_iff
tff(fact_4016_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(uminus_uminus_int,X)),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).

% minus_dvd_iff
tff(fact_4017_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(uminus_uminus_real,X)),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).

% minus_dvd_iff
tff(fact_4018_Diff__eq__empty__iff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat(A2,B) = bot_bo7653980558646680370d_enat )
    <=> pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B)) ) ).

% Diff_eq_empty_iff
tff(fact_4019_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B: set_real] :
      ( ( minus_minus_set_real(A2,B) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(A2),B)) ) ).

% Diff_eq_empty_iff
tff(fact_4020_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( minus_minus_set_nat(A2,B) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B)) ) ).

% Diff_eq_empty_iff
tff(fact_4021_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( minus_minus_set_int(A2,B) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(A2),B)) ) ).

% Diff_eq_empty_iff
tff(fact_4022_atLeastAtMost__iff,axiom,
    ! [I: extended_enat,L: extended_enat,U: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),set_or5403411693681687835d_enat(L,U)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,L),I))
        & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4023_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),set_or4548717258645045905et_nat(L,U)))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,L),I))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4024_atLeastAtMost__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( member_set_int(I,set_or370866239135849197et_int(L,U))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(L),I))
        & pp(aa_set_int_bool(ord_less_eq_set_int(I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4025_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4026_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I),set_or1266510415728281911st_int(L,U)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),I))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4027_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( pp(aa_set_real_bool(member_real(I),set_or1222579329274155063t_real(L,U)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),I))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4028_Icc__eq__Icc,axiom,
    ! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
      ( ( set_or4548717258645045905et_nat(L,H) = set_or4548717258645045905et_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,L),H))
          & ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4029_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L2: set_int,H2: set_int] :
      ( ( set_or370866239135849197et_int(L,H) = set_or370866239135849197et_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(L),H))
          & ~ pp(aa_set_int_bool(ord_less_eq_set_int(L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4030_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(L),H) = aa_nat_set_nat(set_or1269000886237332187st_nat(L2),H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),H))
          & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4031_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),H))
          & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4032_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L2: real,H2: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
          & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4033_take__bit__of__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),zero_zero_nat) = zero_zero_nat ).

% take_bit_of_0
tff(fact_4034_take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),zero_zero_int) = zero_zero_int ).

% take_bit_of_0
tff(fact_4035_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U))) ).

% finite_atLeastAtMost
tff(fact_4036_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% neg_0_le_iff_le
tff(fact_4037_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% neg_0_le_iff_le
tff(fact_4038_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_le_0_iff_le
tff(fact_4039_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_le_0_iff_le
tff(fact_4040_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% less_eq_neg_nonpos
tff(fact_4041_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% less_eq_neg_nonpos
tff(fact_4042_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_less_eq_nonneg
tff(fact_4043_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_less_eq_nonneg
tff(fact_4044_less__neg__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% less_neg_neg
tff(fact_4045_less__neg__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% less_neg_neg
tff(fact_4046_neg__less__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_pos
tff(fact_4047_neg__less__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_pos
tff(fact_4048_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% neg_0_less_iff_less
tff(fact_4049_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% neg_0_less_iff_less
tff(fact_4050_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_0_iff_less
tff(fact_4051_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_0_iff_less
tff(fact_4052_ab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_left_minus
tff(fact_4053_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_4054_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_4055_add_Oright__inverse,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,A)) = zero_zero_complex ).

% add.right_inverse
tff(fact_4056_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_4057_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_4058_verit__minus__simplify_I3_J,axiom,
    ! [B2: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),B2) = aa_complex_complex(uminus1482373934393186551omplex,B2) ).

% verit_minus_simplify(3)
tff(fact_4059_verit__minus__simplify_I3_J,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B2) = aa_int_int(uminus_uminus_int,B2) ).

% verit_minus_simplify(3)
tff(fact_4060_verit__minus__simplify_I3_J,axiom,
    ! [B2: real] : aa_real_real(minus_minus_real(zero_zero_real),B2) = aa_real_real(uminus_uminus_real,B2) ).

% verit_minus_simplify(3)
tff(fact_4061_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_4062_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_4063_diff__0,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_4064_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(M)),numeral_numeral_int(N))) ).

% add_neg_numeral_simps(3)
tff(fact_4065_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(numeral_numeral_real(M)),numeral_numeral_real(N))) ).

% add_neg_numeral_simps(3)
tff(fact_4066_uminus__add__conv__diff,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A) ).

% uminus_add_conv_diff
tff(fact_4067_uminus__add__conv__diff,axiom,
    ! [A: real,B2: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),B2) = aa_real_real(minus_minus_real(B2),A) ).

% uminus_add_conv_diff
tff(fact_4068_diff__minus__eq__add,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) ).

% diff_minus_eq_add
tff(fact_4069_diff__minus__eq__add,axiom,
    ! [A: real,B2: real] : aa_real_real(minus_minus_real(A),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(plus_plus_real(A),B2) ).

% diff_minus_eq_add
tff(fact_4070_divide__minus1,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% divide_minus1
tff(fact_4071_divide__minus1,axiom,
    ! [X: real] : aa_real_real(divide_divide_real(X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,X) ).

% divide_minus1
tff(fact_4072_atLeastatMost__empty__iff,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( set_or5403411693681687835d_enat(A,B2) = bot_bo7653980558646680370d_enat )
    <=> ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_4073_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B2) = bot_bot_set_set_nat )
    <=> ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_4074_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( set_or370866239135849197et_int(A,B2) = bot_bot_set_set_int )
    <=> ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_4075_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2) = bot_bot_set_nat )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_4076_atLeastatMost__empty__iff,axiom,
    ! [A: int,B2: int] :
      ( ( set_or1266510415728281911st_int(A,B2) = bot_bot_set_int )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_4077_atLeastatMost__empty__iff,axiom,
    ! [A: real,B2: real] :
      ( ( set_or1222579329274155063t_real(A,B2) = bot_bot_set_real )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_4078_atLeastatMost__empty__iff2,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( bot_bo7653980558646680370d_enat = set_or5403411693681687835d_enat(A,B2) )
    <=> ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4079_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B2) )
    <=> ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4080_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(A,B2) )
    <=> ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4081_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B2: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2) )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4082_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B2: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B2) )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4083_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B2: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B2) )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4084_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A,B2)),set_or4548717258645045905et_nat(C,D)))
    <=> ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
        | ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),A))
          & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4085_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B2: set_int,C: set_int,D: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A,B2),set_or370866239135849197et_int(C,D))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
        | ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A))
          & pp(aa_set_int_bool(ord_less_eq_set_int(B2),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4086_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D)))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4087_atLeastatMost__subset__iff,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B2)),set_or1266510415728281911st_int(C,D)))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4088_atLeastatMost__subset__iff,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(A,B2)),set_or1222579329274155063t_real(C,D)))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4089_atLeastatMost__empty,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),A))
     => ( set_or5403411693681687835d_enat(A,B2) = bot_bo7653980558646680370d_enat ) ) ).

% atLeastatMost_empty
tff(fact_4090_atLeastatMost__empty,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_4091_atLeastatMost__empty,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A))
     => ( set_or1266510415728281911st_int(A,B2) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_4092_atLeastatMost__empty,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A))
     => ( set_or1222579329274155063t_real(A,B2) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_4093_infinite__Icc__iff,axiom,
    ! [A: real,B2: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A,B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ).

% infinite_Icc_iff
tff(fact_4094_take__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,zero_zero_nat),A) = zero_zero_nat ).

% take_bit_0
tff(fact_4095_take__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_4096_take__bit__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),one_one_nat) = one_one_nat ).

% take_bit_Suc_1
tff(fact_4097_take__bit__Suc__1,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),one_one_int) = one_one_int ).

% take_bit_Suc_1
tff(fact_4098_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_4099_add__neg__numeral__special_I7_J,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% add_neg_numeral_special(7)
tff(fact_4100_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_4101_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(plus_plus_real(one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_4102_add__neg__numeral__special_I8_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = zero_zero_complex ).

% add_neg_numeral_special(8)
tff(fact_4103_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_4104_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_4105_diff__numeral__special_I12_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% diff_numeral_special(12)
tff(fact_4106_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_4107_diff__numeral__special_I12_J,axiom,
    aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_4108_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_4109_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
       => ( aa_real_real(ord_max_real(numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
       => ( aa_real_real(ord_max_real(numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = numeral_numeral_real(U) ) ) ) ).

% max_number_of(2)
tff(fact_4110_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(U)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,numeral_numeral_int(U)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))) = aa_int_int(uminus_uminus_int,numeral_numeral_int(V)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(U)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,numeral_numeral_int(U)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))) = numeral_numeral_int(U) ) ) ) ).

% max_number_of(2)
tff(fact_4111_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)))
       => ( aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)) = numeral_numeral_real(V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)))
       => ( aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(U)) ) ) ) ).

% max_number_of(3)
tff(fact_4112_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(U))),numeral_numeral_int(V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(U))),numeral_numeral_int(V)) = numeral_numeral_int(V) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(U))),numeral_numeral_int(V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(U))),numeral_numeral_int(V)) = aa_int_int(uminus_uminus_int,numeral_numeral_int(U)) ) ) ) ).

% max_number_of(3)
tff(fact_4113_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
       => ( aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
       => ( aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(U)) ) ) ) ).

% max_number_of(4)
tff(fact_4114_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(U))),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(U))),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))) = aa_int_int(uminus_uminus_int,numeral_numeral_int(V)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(U))),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(U))),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))) = aa_int_int(uminus_uminus_int,numeral_numeral_int(U)) ) ) ) ).

% max_number_of(4)
tff(fact_4115_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = zero_zero_nat )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4116_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),one_one_int) = zero_zero_int )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4117_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(W2))),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(plus_plus_num(V,W2)))),Y) ).

% semiring_norm(168)
tff(fact_4118_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),Y)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(plus_plus_num(V,W2)))),Y) ).

% semiring_norm(168)
tff(fact_4119_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))))
    <=> pp(aa_num_bool(ord_less_eq_num(N),M)) ) ).

% neg_numeral_le_iff
tff(fact_4120_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))))
    <=> pp(aa_num_bool(ord_less_eq_num(N),M)) ) ).

% neg_numeral_le_iff
tff(fact_4121_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))))
    <=> pp(aa_num_bool(ord_less_num(N),M)) ) ).

% neg_numeral_less_iff
tff(fact_4122_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))))
    <=> pp(aa_num_bool(ord_less_num(N),M)) ) ).

% neg_numeral_less_iff
tff(fact_4123_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_Suc_0
tff(fact_4124_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4125_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,numeral_numeral_int(M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4126_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(divide_divide_real(B2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4127_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))),B2)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4128_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: complex,W2: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = A )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex )
         => ( B2 = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4129_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( ( aa_real_real(divide_divide_real(B2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))) = A )
    <=> ( ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) != zero_zero_real )
         => ( B2 = aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))) ) )
        & ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4130_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B2: complex,W2: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = B2 ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4131_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( ( A = aa_real_real(divide_divide_real(B2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))) )
    <=> ( ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) != zero_zero_real )
         => ( aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))) = B2 ) )
        & ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4132_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),aa_int_int(uminus_uminus_int,one_one_int)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4133_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4134_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(B2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4135_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))),B2)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4136_take__bit__of__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_4137_take__bit__of__1,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),one_one_int) = zero_n2684676970156552555ol_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_4138_add__neg__numeral__special_I9_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_4139_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_4140_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_4141_even__take__bit__eq,axiom,
    ! [N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)))
    <=> ( ( N = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)) ) ) ).

% even_take_bit_eq
tff(fact_4142_even__take__bit__eq,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)))
    <=> ( ( N = zero_zero_nat )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)) ) ) ).

% even_take_bit_eq
tff(fact_4143_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_4144_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% div_Suc_eq_div_add3
tff(fact_4145_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_4146_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_4147_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(uminus_uminus_int,numeral_numeral_int(K)))),numeral_numeral_int(bit0(one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_4148_take__bit__Suc__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,zero_zero_nat)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% take_bit_Suc_0
tff(fact_4149_take__bit__Suc__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) ).

% take_bit_Suc_0
tff(fact_4150_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))) ).

% signed_take_bit_0
tff(fact_4151_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,numeral_numeral_int(aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(K))),one_one_int))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_4152_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),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,N),numeral_numeral_int(K))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_4153_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% take_bit_of_exp
tff(fact_4154_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,zero_n2684676970156552555ol_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) ).

% take_bit_of_exp
tff(fact_4155_take__bit__of__2,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% take_bit_of_2
tff(fact_4156_take__bit__of__2,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),numeral_numeral_int(bit0(one))) = aa_int_int(aa_int_fun_int_int(times_times_int,zero_n2684676970156552555ol_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))),numeral_numeral_int(bit0(one))) ).

% take_bit_of_2
tff(fact_4157_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_4158_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_4159_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)),W2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W2)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W2)) ).

% int_distrib(3)
tff(fact_4160_int__distrib_I4_J,axiom,
    ! [W2: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z22)) ).

% int_distrib(4)
tff(fact_4161_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K))
     => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)))
       => ( ! [I4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I4),K))
             => ( pp(aa_int_bool(P,I4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I4),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_less_induct
tff(fact_4162_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),N)))
     => ( ( K != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N)) ) ) ).

% zdvd_mult_cancel
tff(fact_4163_zdvd__imp__le,axiom,
    ! [Z2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z2),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),N)) ) ) ).

% zdvd_imp_le
tff(fact_4164_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M)) ) ) ).

% zdvd_not_zless
tff(fact_4165_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M))
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_4166_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),aa_int_int(aa_int_fun_int_int(times_times_int,K),M))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N)) ) ).

% zdvd_reduce
tff(fact_4167_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),D))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),T)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),T))) ) ) ).

% zdvd_period
tff(fact_4168_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_bb,I)))) ) ).

% finite_divisors_int
tff(fact_4169_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,uminus5710092332889474511et_nat(Y)),X))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,uminus5710092332889474511et_nat(X)),Y)) ) ).

% compl_le_swap2
tff(fact_4170_compl__le__swap2,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),X))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(X)),Y)) ) ).

% compl_le_swap2
tff(fact_4171_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),uminus5710092332889474511et_nat(X)))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),uminus5710092332889474511et_nat(Y))) ) ).

% compl_le_swap1
tff(fact_4172_compl__le__swap1,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),uminus1532241313380277803et_int(X)))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),uminus1532241313380277803et_int(Y))) ) ).

% compl_le_swap1
tff(fact_4173_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(X))) ) ).

% compl_mono
tff(fact_4174_compl__mono,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),uminus1532241313380277803et_int(X))) ) ).

% compl_mono
tff(fact_4175_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J))) ) ) ).

% zmult_zless_mono2
tff(fact_4176_add1__zle__eq,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ).

% add1_zle_eq
tff(fact_4177_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I))
     => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)))
       => ( ! [I4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I4))
             => ( pp(aa_int_bool(P,I4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I4),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_gr_induct
tff(fact_4178_le__imp__0__less,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z2))) ) ).

% le_imp_0_less
tff(fact_4179_zless__add1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2))
        | ( W2 = Z2 ) ) ) ).

% zless_add1_eq
tff(fact_4180_odd__less__0__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z2)),Z2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),zero_zero_int)) ) ).

% odd_less_0_iff
tff(fact_4181_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_4182_zless__imp__add1__zle,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z2)) ) ).

% zless_imp_add1_zle
tff(fact_4183_int__one__le__iff__zero__less,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2)) ) ).

% int_one_le_iff_zero_less
tff(fact_4184_odd__nonzero,axiom,
    ! [Z2: 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),Z2)),Z2) != zero_zero_int ).

% odd_nonzero
tff(fact_4185_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_4186_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_4187_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_4188_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_4189_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)),W2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W2)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W2)) ).

% int_distrib(1)
tff(fact_4190_int__distrib_I2_J,axiom,
    ! [W2: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z22)) ).

% int_distrib(2)
tff(fact_4191_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_4192_equation__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B2) )
    <=> ( B2 = aa_int_int(uminus_uminus_int,A) ) ) ).

% equation_minus_iff
tff(fact_4193_equation__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B2) )
    <=> ( B2 = aa_real_real(uminus_uminus_real,A) ) ) ).

% equation_minus_iff
tff(fact_4194_minus__equation__iff,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B2 )
    <=> ( aa_int_int(uminus_uminus_int,B2) = A ) ) ).

% minus_equation_iff
tff(fact_4195_minus__equation__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B2 )
    <=> ( aa_real_real(uminus_uminus_real,B2) = A ) ) ).

% minus_equation_iff
tff(fact_4196_take__bit__add,axiom,
    ! [N: nat,A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)) ).

% take_bit_add
tff(fact_4197_take__bit__add,axiom,
    ! [N: nat,A: int,B2: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),B2))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ).

% take_bit_add
tff(fact_4198_take__bit__tightened,axiom,
    ! [N: nat,A: nat,B2: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),B2) ) ) ) ).

% take_bit_tightened
tff(fact_4199_take__bit__tightened,axiom,
    ! [N: nat,A: int,B2: int,M: nat] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),A) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),B2) ) ) ) ).

% take_bit_tightened
tff(fact_4200_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q3))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_4201_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_4202_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_4203_le__imp__neg__le,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A))) ) ).

% le_imp_neg_le
tff(fact_4204_le__imp__neg__le,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A))) ) ).

% le_imp_neg_le
tff(fact_4205_minus__le__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),A)) ) ).

% minus_le_iff
tff(fact_4206_minus__le__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B2)),A)) ) ).

% minus_le_iff
tff(fact_4207_le__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(uminus_uminus_real,A))) ) ).

% le_minus_iff
tff(fact_4208_le__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(uminus_uminus_int,A))) ) ).

% le_minus_iff
tff(fact_4209_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_4210_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_4211_minus__less__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B2)),A)) ) ).

% minus_less_iff
tff(fact_4212_minus__less__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B2)),A)) ) ).

% minus_less_iff
tff(fact_4213_less__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),aa_int_int(uminus_uminus_int,A))) ) ).

% less_minus_iff
tff(fact_4214_less__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(uminus_uminus_real,A))) ) ).

% less_minus_iff
tff(fact_4215_square__eq__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(A),A) = aa_complex_complex(times_times_complex(B2),B2) )
    <=> ( ( A = B2 )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,B2) ) ) ) ).

% square_eq_iff
tff(fact_4216_square__eq__iff,axiom,
    ! [A: int,B2: 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,B2),B2) )
    <=> ( ( A = B2 )
        | ( A = aa_int_int(uminus_uminus_int,B2) ) ) ) ).

% square_eq_iff
tff(fact_4217_square__eq__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(times_times_real(A),A) = aa_real_real(times_times_real(B2),B2) )
    <=> ( ( A = B2 )
        | ( A = aa_real_real(uminus_uminus_real,B2) ) ) ) ).

% square_eq_iff
tff(fact_4218_minus__mult__commute,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B2) = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% minus_mult_commute
tff(fact_4219_minus__mult__commute,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B2)) ).

% minus_mult_commute
tff(fact_4220_minus__mult__commute,axiom,
    ! [A: real,B2: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),B2) = aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,B2)) ).

% minus_mult_commute
tff(fact_4221_is__num__normalize_I8_J,axiom,
    ! [A: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A)) ).

% is_num_normalize(8)
tff(fact_4222_is__num__normalize_I8_J,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B2)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_4223_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A)) ).

% add.inverse_distrib_swap
tff(fact_4224_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B2)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_4225_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_4226_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2 = aa_real_real(plus_plus_real(K),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_4227_minus__diff__commute,axiom,
    ! [B2: int,A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B2)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),B2) ).

% minus_diff_commute
tff(fact_4228_minus__diff__commute,axiom,
    ! [B2: real,A: real] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,B2)),A) = aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),B2) ).

% minus_diff_commute
tff(fact_4229_minus__divide__right,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B2)) = aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B2)) ).

% minus_divide_right
tff(fact_4230_minus__divide__divide,axiom,
    ! [A: real,B2: real] : aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(divide_divide_real(A),B2) ).

% minus_divide_divide
tff(fact_4231_minus__divide__left,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B2)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),B2) ).

% minus_divide_left
tff(fact_4232_Diff__infinite__finite,axiom,
    ! [T3: set_complex,S2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,T3))) ) ) ).

% Diff_infinite_finite
tff(fact_4233_Diff__infinite__finite,axiom,
    ! [T3: set_int,S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,T3))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
       => ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(S2,T3))) ) ) ).

% Diff_infinite_finite
tff(fact_4234_Diff__infinite__finite,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
       => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(S2,T3))) ) ) ).

% Diff_infinite_finite
tff(fact_4235_Diff__infinite__finite,axiom,
    ! [T3: set_nat,S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(S2,T3))) ) ) ).

% Diff_infinite_finite
tff(fact_4236_double__diff,axiom,
    ! [A2: set_nat,B: set_nat,C4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),C4))
       => ( minus_minus_set_nat(B,minus_minus_set_nat(C4,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_4237_double__diff,axiom,
    ! [A2: set_int,B: set_int,C4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C4))
       => ( minus_minus_set_int(B,minus_minus_set_int(C4,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_4238_Diff__subset,axiom,
    ! [A2: set_nat,B: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A2,B)),A2)) ).

% Diff_subset
tff(fact_4239_Diff__subset,axiom,
    ! [A2: set_int,B: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,B)),A2)) ).

% Diff_subset
tff(fact_4240_Diff__mono,axiom,
    ! [A2: set_nat,C4: set_nat,D5: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,D5),B))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A2,B)),minus_minus_set_nat(C4,D5))) ) ) ).

% Diff_mono
tff(fact_4241_Diff__mono,axiom,
    ! [A2: set_int,C4: set_int,D5: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(D5),B))
       => pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,B)),minus_minus_set_int(C4,D5))) ) ) ).

% Diff_mono
tff(fact_4242_psubset__imp__ex__mem,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(A2),B))
     => ? [B3: extended_enat] : pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_4243_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A2),B))
     => ? [B3: real] : pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_4244_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),B))
     => ? [B3: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(B3),minus_2163939370556025621et_nat(B,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_4245_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B))
     => ? [B3: int] : pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_4246_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B))
     => ? [B3: nat] : pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_4247_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),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,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_4248_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,numeral_numeral_int(K)))),numeral_numeral_int(bit0(one))) ).

% take_bit_Suc_minus_bit0
tff(fact_4249_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_4250_infinite__Icc,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
     => ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A,B2))) ) ).

% infinite_Icc
tff(fact_4251_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),A) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),B2) )
    <=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),A) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),B2) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_4252_signed__take__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)) = aa_int_int(if_fun_int_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M),aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_nat_fun_int_int(bit_ri631733984087533419it_int,M)),A) ).

% signed_take_bit_take_bit
tff(fact_4253_all__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
         => pp(aa_nat_bool(P,M2)) )
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
         => pp(aa_nat_bool(P,X5)) ) ) ).

% all_nat_less
tff(fact_4254_ex__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
          & pp(aa_nat_bool(P,M2)) )
    <=> ? [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
          & pp(aa_nat_bool(P,X5)) ) ) ).

% ex_nat_less
tff(fact_4255_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4256_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(M)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4257_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N))) ).

% neg_numeral_le_numeral
tff(fact_4258_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),numeral_numeral_int(N))) ).

% neg_numeral_le_numeral
tff(fact_4259_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% zero_neq_neg_numeral
tff(fact_4260_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_int_int(uminus_uminus_int,numeral_numeral_int(N)) ).

% zero_neq_neg_numeral
tff(fact_4261_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) ).

% zero_neq_neg_numeral
tff(fact_4262_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,numeral_numeral_int(M)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N)))) ).

% not_numeral_less_neg_numeral
tff(fact_4263_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).

% not_numeral_less_neg_numeral
tff(fact_4264_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),numeral_numeral_int(N))) ).

% neg_numeral_less_numeral
tff(fact_4265_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N))) ).

% neg_numeral_less_numeral
tff(fact_4266_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(4)
tff(fact_4267_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(4)
tff(fact_4268_le__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% le_minus_one_simps(2)
tff(fact_4269_le__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% le_minus_one_simps(2)
tff(fact_4270_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_4271_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_4272_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_4273_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B2 )
    <=> ( aa_complex_complex(plus_plus_complex(A),B2) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4274_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B2 )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4275_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B2 )
    <=> ( aa_real_real(plus_plus_real(A),B2) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4276_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B2) )
    <=> ( aa_complex_complex(plus_plus_complex(A),B2) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4277_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B2) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4278_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B2) )
    <=> ( aa_real_real(plus_plus_real(A),B2) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4279_add_Oinverse__unique,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B2) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = B2 ) ) ).

% add.inverse_unique
tff(fact_4280_add_Oinverse__unique,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B2 ) ) ).

% add.inverse_unique
tff(fact_4281_add_Oinverse__unique,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(plus_plus_real(A),B2) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B2 ) ) ).

% add.inverse_unique
tff(fact_4282_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_4283_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_4284_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_4285_add__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B2) = zero_zero_complex )
    <=> ( B2 = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% add_eq_0_iff
tff(fact_4286_add__eq__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = zero_zero_int )
    <=> ( B2 = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_4287_add__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(plus_plus_real(A),B2) = zero_zero_real )
    <=> ( B2 = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_4288_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(4)
tff(fact_4289_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(4)
tff(fact_4290_less__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% less_minus_one_simps(2)
tff(fact_4291_less__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% less_minus_one_simps(2)
tff(fact_4292_nonzero__minus__divide__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_4293_nonzero__minus__divide__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B2)) = aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_4294_nonzero__minus__divide__divide,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_4295_nonzero__minus__divide__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(divide_divide_real(A),B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_4296_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(times_times_complex(X),X) = one_one_complex )
    <=> ( ( X = one_one_complex )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_4297_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),X) = one_one_int )
    <=> ( ( X = one_one_int )
        | ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_4298_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(times_times_real(X),X) = one_one_real )
    <=> ( ( X = one_one_real )
        | ( X = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_4299_group__cancel_Osub2,axiom,
    ! [B: int,K: int,B2: int,A: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B2) )
     => ( 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(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) ) ) ).

% group_cancel.sub2
tff(fact_4300_group__cancel_Osub2,axiom,
    ! [B: real,K: real,B2: real,A: real] :
      ( ( B = aa_real_real(plus_plus_real(K),B2) )
     => ( aa_real_real(minus_minus_real(A),B) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K)),aa_real_real(minus_minus_real(A),B2)) ) ) ).

% group_cancel.sub2
tff(fact_4301_diff__conv__add__uminus,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B2)) ).

% diff_conv_add_uminus
tff(fact_4302_diff__conv__add__uminus,axiom,
    ! [A: real,B2: real] : aa_real_real(minus_minus_real(A),B2) = aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,B2)) ).

% diff_conv_add_uminus
tff(fact_4303_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4304_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B2: real] : aa_real_real(minus_minus_real(A),B2) = aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4305_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_4306_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A)) = aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_4307_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_4308_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),A)) = aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_4309_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_4310_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_4311_dvd__div__neg,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B2)) ) ) ).

% dvd_div_neg
tff(fact_4312_dvd__div__neg,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B2),A))
     => ( aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B2)) ) ) ).

% dvd_div_neg
tff(fact_4313_dvd__neg__div,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B2) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B2)) ) ) ).

% dvd_neg_div
tff(fact_4314_dvd__neg__div,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B2),A))
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),B2) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B2)) ) ) ).

% dvd_neg_div
tff(fact_4315_subset__Compl__self__eq,axiom,
    ! [A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),uminus417252749190364093d_enat(A2)))
    <=> ( A2 = bot_bo7653980558646680370d_enat ) ) ).

% subset_Compl_self_eq
tff(fact_4316_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),uminus612125837232591019t_real(A2)))
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_4317_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),uminus5710092332889474511et_nat(A2)))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_4318_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(A2)))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_4319_atLeastatMost__psubset__iff,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(set_or5403411693681687835d_enat(A,B2)),set_or5403411693681687835d_enat(C,D)))
    <=> ( ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A),B2))
          | ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,C),A))
            & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B2),D))
            & ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,C),A))
              | pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B2),D)) ) ) )
        & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4320_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(set_or4548717258645045905et_nat(A,B2)),set_or4548717258645045905et_nat(C,D)))
    <=> ( ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B2))
          | ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),A))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),D))
            & ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),A))
              | pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B2),D)) ) ) )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4321_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B2: set_int,C: set_int,D: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(A,B2),set_or370866239135849197et_int(C,D))
    <=> ( ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B2))
          | ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A))
            & pp(aa_set_int_bool(ord_less_eq_set_int(B2),D))
            & ( pp(aa_set_int_bool(ord_less_set_int(C),A))
              | pp(aa_set_int_bool(ord_less_set_int(B2),D)) ) ) )
        & pp(aa_set_int_bool(ord_less_eq_set_int(C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4322_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D)))
    <=> ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),D))
            & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
              | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),D)) ) ) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4323_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_or1266510415728281911st_int(A,B2)),set_or1266510415728281911st_int(C,D)))
    <=> ( ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
          | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),D))
            & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
              | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),D)) ) ) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4324_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or1222579329274155063t_real(A,B2)),set_or1222579329274155063t_real(C,D)))
    <=> ( ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
          | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),D))
            & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
              | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),D)) ) ) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4325_take__bit__signed__take__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_4326_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))),zero_zero_real)) ).

% neg_numeral_le_zero
tff(fact_4327_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(N))),zero_zero_int)) ).

% neg_numeral_le_zero
tff(fact_4328_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).

% not_zero_le_neg_numeral
tff(fact_4329_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(N)))) ).

% not_zero_le_neg_numeral
tff(fact_4330_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(N))),zero_zero_int)) ).

% neg_numeral_less_zero
tff(fact_4331_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))),zero_zero_real)) ).

% neg_numeral_less_zero
tff(fact_4332_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(N)))) ).

% not_zero_less_neg_numeral
tff(fact_4333_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).

% not_zero_less_neg_numeral
tff(fact_4334_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(3)
tff(fact_4335_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(3)
tff(fact_4336_le__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% le_minus_one_simps(1)
tff(fact_4337_le__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% le_minus_one_simps(1)
tff(fact_4338_numeral__Bit1,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_Bit1
tff(fact_4339_numeral__Bit1,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_Bit1
tff(fact_4340_numeral__Bit1,axiom,
    ! [N: num] : numera1916890842035813515d_enat(aa_num_num(bit1,N)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N)),numera1916890842035813515d_enat(N))),one_on7984719198319812577d_enat) ).

% numeral_Bit1
tff(fact_4341_numeral__Bit1,axiom,
    ! [N: num] : numeral_numeral_int(aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(N)),numeral_numeral_int(N))),one_one_int) ).

% numeral_Bit1
tff(fact_4342_numeral__Bit1,axiom,
    ! [N: num] : numeral_numeral_real(aa_num_num(bit1,N)) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(numeral_numeral_real(N)),numeral_numeral_real(N))),one_one_real) ).

% numeral_Bit1
tff(fact_4343_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(3)
tff(fact_4344_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(3)
tff(fact_4345_less__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% less_minus_one_simps(1)
tff(fact_4346_less__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% less_minus_one_simps(1)
tff(fact_4347_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(M)))) ).

% not_one_le_neg_numeral
tff(fact_4348_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(M)))) ).

% not_one_le_neg_numeral
tff(fact_4349_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_le_neg_one
tff(fact_4350_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(M)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_le_neg_one
tff(fact_4351_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real))) ).

% neg_numeral_le_neg_one
tff(fact_4352_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),aa_int_int(uminus_uminus_int,one_one_int))) ).

% neg_numeral_le_neg_one
tff(fact_4353_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(M))) ).

% neg_one_le_numeral
tff(fact_4354_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),numeral_numeral_int(M))) ).

% neg_one_le_numeral
tff(fact_4355_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real)) ).

% neg_numeral_le_one
tff(fact_4356_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),one_one_int)) ).

% neg_numeral_le_one
tff(fact_4357_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,numeral_numeral_int(M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_4358_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_4359_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(M)))) ).

% not_one_less_neg_numeral
tff(fact_4360_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(M)))) ).

% not_one_less_neg_numeral
tff(fact_4361_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,numeral_numeral_int(M)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_less_neg_one
tff(fact_4362_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_less_neg_one
tff(fact_4363_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),numeral_numeral_int(M))) ).

% neg_one_less_numeral
tff(fact_4364_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(M))) ).

% neg_one_less_numeral
tff(fact_4365_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),one_one_int)) ).

% neg_numeral_less_one
tff(fact_4366_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real)) ).

% neg_numeral_less_one
tff(fact_4367_eq__minus__divide__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B2),C)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B2) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_4368_eq__minus__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C)) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(times_times_real(A),C) = aa_real_real(uminus_uminus_real,B2) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_4369_minus__divide__eq__eq,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B2),C)) = A )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,B2) = aa_complex_complex(times_times_complex(A),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_4370_minus__divide__eq__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C)) = A )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,B2) = aa_real_real(times_times_real(A),C) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_4371_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: complex,A: complex,C: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4372_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( B2 != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B2)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(times_times_real(C),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4373_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) )
      <=> ( aa_complex_complex(times_times_complex(C),B2) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4374_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B2)) )
      <=> ( aa_real_real(times_times_real(C),B2) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4375_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B2 != zero_zero_complex )
        & ( A = aa_complex_complex(uminus1482373934393186551omplex,B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_4376_divide__eq__minus__1__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(divide_divide_real(A),B2) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B2 != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_4377_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,bit0(N))) ).

% eval_nat_numeral(3)
tff(fact_4378_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)),K))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(suc,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_4379_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(suc,N)))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_4380_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_4381_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),numeral_numeral_int(aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),numeral_numeral_int(K))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_4382_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(suc,N))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_4383_numeral__code_I3_J,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_code(3)
tff(fact_4384_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_code(3)
tff(fact_4385_numeral__code_I3_J,axiom,
    ! [N: num] : numera1916890842035813515d_enat(aa_num_num(bit1,N)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N)),numera1916890842035813515d_enat(N))),one_on7984719198319812577d_enat) ).

% numeral_code(3)
tff(fact_4386_numeral__code_I3_J,axiom,
    ! [N: num] : numeral_numeral_int(aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(N)),numeral_numeral_int(N))),one_one_int) ).

% numeral_code(3)
tff(fact_4387_numeral__code_I3_J,axiom,
    ! [N: num] : numeral_numeral_real(aa_num_num(bit1,N)) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(numeral_numeral_real(N)),numeral_numeral_real(N))),one_one_real) ).

% numeral_code(3)
tff(fact_4388_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N5)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_4389_pos__minus__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4390_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B2))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4391_neg__minus__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B2))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4392_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4393_minus__divide__less__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_4394_less__minus__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_4395_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: complex,C: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B2 = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C) ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4396_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W2: num] :
      ( ( aa_real_real(divide_divide_real(B2),C) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) )
    <=> ( ( ( C != zero_zero_real )
         => ( B2 = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4397_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B2: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = aa_complex_complex(divide1717551699836669952omplex(B2),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C) = B2 ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4398_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B2: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) = aa_real_real(divide_divide_real(B2),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C) = B2 ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4399_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,bit0(Q3))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_4400_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] : aa_int_int(modulo_modulo_int(numeral_numeral_int(aa_num_num(bit1,N))),numeral_numeral_int(bit0(Q3))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_4401_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: complex,A: complex,B2: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z2))),B2) = B2 ) )
      & ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z2))),B2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B2),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_4402_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: real,A: real,B2: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z2))),B2) = B2 ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z2))),B2) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(times_times_real(B2),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_4403_minus__divide__add__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z2))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z2))),Z2) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4404_minus__divide__add__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(X),Z2))),Y) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,X)),aa_real_real(times_times_real(Y),Z2))),Z2) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4405_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: complex,A: complex,B2: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z2))),B2) = aa_complex_complex(uminus1482373934393186551omplex,B2) ) )
      & ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z2))),B2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B2),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_4406_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: real,A: real,B2: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z2))),B2) = aa_real_real(uminus_uminus_real,B2) ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z2))),B2) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(times_times_real(B2),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_4407_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: complex,A: complex,B2: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z2)),B2) = aa_complex_complex(uminus1482373934393186551omplex,B2) ) )
      & ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z2)),B2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B2),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_4408_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: real,A: real,B2: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),Z2)),B2) = aa_real_real(uminus_uminus_real,B2) ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),Z2)),B2) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(A),aa_real_real(times_times_real(B2),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_4409_minus__divide__diff__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z2))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z2))),Z2) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4410_minus__divide__diff__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(X),Z2))),Y) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,X)),aa_real_real(times_times_real(Y),Z2))),Z2) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4411_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_4412_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N) ).

% Suc3_eq_add_3
tff(fact_4413_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,bit0(K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% take_bit_Suc_bit0
tff(fact_4414_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),numeral_numeral_int(bit0(K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),numeral_numeral_int(K))),numeral_numeral_int(bit0(one))) ).

% take_bit_Suc_bit0
tff(fact_4415_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_4416_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% take_bit_nat_less_exp
tff(fact_4417_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_4418_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_4419_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_4420_le__minus__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_4421_minus__divide__le__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_4422_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4423_neg__minus__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B2))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_4424_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B2))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_4425_pos__minus__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_4426_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B2),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C)),B2)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_4427_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),aa_real_real(divide_divide_real(B2),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C)),B2)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_4428_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,bit0(Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,bit0(Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4429_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( aa_int_int(modulo_modulo_int(numeral_numeral_int(one)),numeral_numeral_int(bit0(Q3))) = aa_int_int(modulo_modulo_int(numeral_numeral_int(aa_num_num(bit1,N))),numeral_numeral_int(bit0(Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(numeral_numeral_int(N)),numeral_numeral_int(Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4430_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,bit0(Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,bit0(Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4431_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( aa_int_int(modulo_modulo_int(numeral_numeral_int(aa_num_num(bit1,M))),numeral_numeral_int(bit0(Q3))) = aa_int_int(modulo_modulo_int(numeral_numeral_int(one)),numeral_numeral_int(bit0(Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(numeral_numeral_int(M)),numeral_numeral_int(Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4432_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).

% Suc_div_eq_add3_div
tff(fact_4433_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4434_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4435_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4436_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).

% Suc_mod_eq_add3_mod
tff(fact_4437_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),A)) ) ).

% take_bit_eq_0_iff
tff(fact_4438_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = zero_zero_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)),A)) ) ).

% take_bit_eq_0_iff
tff(fact_4439_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_4440_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B2),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C)),B2)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_4441_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),aa_real_real(divide_divide_real(B2),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C)),B2)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_4442_square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)) ) ) ).

% square_le_1
tff(fact_4443_square__le__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_int)) ) ) ).

% square_le_1
tff(fact_4444_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_4445_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_4446_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_4447_take__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% take_bit_Suc
tff(fact_4448_take__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))))),numeral_numeral_int(bit0(one)))),aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))) ).

% take_bit_Suc
tff(fact_4449_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = A )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),one_one_nat) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4450_stable__imp__take__bit__eq,axiom,
    ! [A: int,N: nat] :
      ( ( aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))) = A )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
         => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = zero_zero_int ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
         => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)),one_one_int) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4451_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_num_nat(numeral_numeral_nat,bit0(one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_4452_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(ord_less_num(M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,numeral_numeral_int(aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(ord_less_num(M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit1,M),bit0(aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_4453_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(ord_less_num(M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(ord_less_num(M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit1,M),bit0(aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_4454_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(ord_less_eq_num(M),N))
       => ( unique5052692396658037445od_int(bit0(M),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,numeral_numeral_int(bit0(M))) ) )
      & ( ~ pp(aa_num_bool(ord_less_eq_num(M),N))
       => ( unique5052692396658037445od_int(bit0(M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(bit0(M),bit0(aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_4455_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(ord_less_eq_num(M),N))
       => ( unique5055182867167087721od_nat(bit0(M),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(M))) ) )
      & ( ~ pp(aa_num_bool(ord_less_eq_num(M),N))
       => ( unique5055182867167087721od_nat(bit0(M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(bit0(M),bit0(aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_4456_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_eo(L),Qr) ).

% divmod_step_def
tff(fact_4457_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ep(L)),Qr) ).

% divmod_step_def
tff(fact_4458_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_4459_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,L)),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)),numeral_numeral_int(K))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_4460_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(ord_less_num(M),N))
       => ( unique5052692396658037445od_int(M,N) = product_Pair_int_int(zero_zero_int,numeral_numeral_int(M)) ) )
      & ( ~ pp(aa_num_bool(ord_less_num(M),N))
       => ( unique5052692396658037445od_int(M,N) = unique5024387138958732305ep_int(N,unique5052692396658037445od_int(M,bit0(N))) ) ) ) ).

% divmod_divmod_step
tff(fact_4461_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(ord_less_num(M),N))
       => ( unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)) ) )
      & ( ~ pp(aa_num_bool(ord_less_num(M),N))
       => ( unique5055182867167087721od_nat(M,N) = unique5026877609467782581ep_nat(N,unique5055182867167087721od_nat(M,bit0(N))) ) ) ) ).

% divmod_divmod_step
tff(fact_4462_set__encode__insert,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(N),A2))
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_4463_insert__absorb2,axiom,
    ! [X: nat,A2: set_nat] : insert_nat(X,insert_nat(X,A2)) = insert_nat(X,A2) ).

% insert_absorb2
tff(fact_4464_insert__absorb2,axiom,
    ! [X: int,A2: set_int] : insert_int(X,insert_int(X,A2)) = insert_int(X,A2) ).

% insert_absorb2
tff(fact_4465_insert__absorb2,axiom,
    ! [X: real,A2: set_real] : insert_real(X,insert_real(X,A2)) = insert_real(X,A2) ).

% insert_absorb2
tff(fact_4466_insert__iff,axiom,
    ! [A: extended_enat,B2: extended_enat,A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),insert_Extended_enat(B2,A2)))
    <=> ( ( A = B2 )
        | pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2)) ) ) ).

% insert_iff
tff(fact_4467_insert__iff,axiom,
    ! [A: real,B2: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),insert_real(B2,A2)))
    <=> ( ( A = B2 )
        | pp(aa_set_real_bool(member_real(A),A2)) ) ) ).

% insert_iff
tff(fact_4468_insert__iff,axiom,
    ! [A: set_nat,B2: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(B2,A2)))
    <=> ( ( A = B2 )
        | pp(aa_set_set_nat_bool(member_set_nat(A),A2)) ) ) ).

% insert_iff
tff(fact_4469_insert__iff,axiom,
    ! [A: nat,B2: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),insert_nat(B2,A2)))
    <=> ( ( A = B2 )
        | pp(aa_set_nat_bool(member_nat(A),A2)) ) ) ).

% insert_iff
tff(fact_4470_insert__iff,axiom,
    ! [A: int,B2: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),insert_int(B2,A2)))
    <=> ( ( A = B2 )
        | pp(aa_set_int_bool(member_int(A),A2)) ) ) ).

% insert_iff
tff(fact_4471_insertCI,axiom,
    ! [A: extended_enat,B: set_Extended_enat,B2: extended_enat] :
      ( ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),B))
       => ( A = B2 ) )
     => pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),insert_Extended_enat(B2,B))) ) ).

% insertCI
tff(fact_4472_insertCI,axiom,
    ! [A: real,B: set_real,B2: real] :
      ( ( ~ pp(aa_set_real_bool(member_real(A),B))
       => ( A = B2 ) )
     => pp(aa_set_real_bool(member_real(A),insert_real(B2,B))) ) ).

% insertCI
tff(fact_4473_insertCI,axiom,
    ! [A: set_nat,B: set_set_nat,B2: set_nat] :
      ( ( ~ pp(aa_set_set_nat_bool(member_set_nat(A),B))
       => ( A = B2 ) )
     => pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(B2,B))) ) ).

% insertCI
tff(fact_4474_insertCI,axiom,
    ! [A: nat,B: set_nat,B2: nat] :
      ( ( ~ pp(aa_set_nat_bool(member_nat(A),B))
       => ( A = B2 ) )
     => pp(aa_set_nat_bool(member_nat(A),insert_nat(B2,B))) ) ).

% insertCI
tff(fact_4475_insertCI,axiom,
    ! [A: int,B: set_int,B2: int] :
      ( ( ~ pp(aa_set_int_bool(member_int(A),B))
       => ( A = B2 ) )
     => pp(aa_set_int_bool(member_int(A),insert_int(B2,B))) ) ).

% insertCI
tff(fact_4476_Diff__idemp,axiom,
    ! [A2: set_nat,B: set_nat] : minus_minus_set_nat(minus_minus_set_nat(A2,B),B) = minus_minus_set_nat(A2,B) ).

% Diff_idemp
tff(fact_4477_Diff__iff,axiom,
    ! [C: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),minus_925952699566721837d_enat(A2,B)))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),A2))
        & ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),B)) ) ) ).

% Diff_iff
tff(fact_4478_Diff__iff,axiom,
    ! [C: real,A2: set_real,B: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B)))
    <=> ( pp(aa_set_real_bool(member_real(C),A2))
        & ~ pp(aa_set_real_bool(member_real(C),B)) ) ) ).

% Diff_iff
tff(fact_4479_Diff__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B)))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(C),B)) ) ) ).

% Diff_iff
tff(fact_4480_Diff__iff,axiom,
    ! [C: int,A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B)))
    <=> ( pp(aa_set_int_bool(member_int(C),A2))
        & ~ pp(aa_set_int_bool(member_int(C),B)) ) ) ).

% Diff_iff
tff(fact_4481_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B)))
    <=> ( pp(aa_set_nat_bool(member_nat(C),A2))
        & ~ pp(aa_set_nat_bool(member_nat(C),B)) ) ) ).

% Diff_iff
tff(fact_4482_DiffI,axiom,
    ! [C: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),A2))
     => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),B))
       => pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),minus_925952699566721837d_enat(A2,B))) ) ) ).

% DiffI
tff(fact_4483_DiffI,axiom,
    ! [C: real,A2: set_real,B: set_real] :
      ( pp(aa_set_real_bool(member_real(C),A2))
     => ( ~ pp(aa_set_real_bool(member_real(C),B))
       => pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B))) ) ) ).

% DiffI
tff(fact_4484_DiffI,axiom,
    ! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(C),B))
       => pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B))) ) ) ).

% DiffI
tff(fact_4485_DiffI,axiom,
    ! [C: int,A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(member_int(C),A2))
     => ( ~ pp(aa_set_int_bool(member_int(C),B))
       => pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B))) ) ) ).

% DiffI
tff(fact_4486_DiffI,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(C),B))
       => pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B))) ) ) ).

% DiffI
tff(fact_4487_ComplI,axiom,
    ! [C: extended_enat,A2: set_Extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),A2))
     => pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),uminus417252749190364093d_enat(A2))) ) ).

% ComplI
tff(fact_4488_ComplI,axiom,
    ! [C: real,A2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(C),A2))
     => pp(aa_set_real_bool(member_real(C),uminus612125837232591019t_real(A2))) ) ).

% ComplI
tff(fact_4489_ComplI,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(C),A2))
     => pp(aa_set_set_nat_bool(member_set_nat(C),uminus613421341184616069et_nat(A2))) ) ).

% ComplI
tff(fact_4490_ComplI,axiom,
    ! [C: nat,A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(C),A2))
     => pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A2))) ) ).

% ComplI
tff(fact_4491_ComplI,axiom,
    ! [C: int,A2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(C),A2))
     => pp(aa_set_int_bool(member_int(C),uminus1532241313380277803et_int(A2))) ) ).

% ComplI
tff(fact_4492_Compl__iff,axiom,
    ! [C: extended_enat,A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),uminus417252749190364093d_enat(A2)))
    <=> ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),A2)) ) ).

% Compl_iff
tff(fact_4493_Compl__iff,axiom,
    ! [C: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),uminus612125837232591019t_real(A2)))
    <=> ~ pp(aa_set_real_bool(member_real(C),A2)) ) ).

% Compl_iff
tff(fact_4494_Compl__iff,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),uminus613421341184616069et_nat(A2)))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(C),A2)) ) ).

% Compl_iff
tff(fact_4495_Compl__iff,axiom,
    ! [C: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A2)))
    <=> ~ pp(aa_set_nat_bool(member_nat(C),A2)) ) ).

% Compl_iff
tff(fact_4496_Compl__iff,axiom,
    ! [C: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),uminus1532241313380277803et_int(A2)))
    <=> ~ pp(aa_set_int_bool(member_int(C),A2)) ) ).

% Compl_iff
tff(fact_4497_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or1266510415728281911st_int(L,U))) ).

% finite_atLeastAtMost_int
tff(fact_4498_singletonI,axiom,
    ! [A: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(A,bot_bot_set_set_nat))) ).

% singletonI
tff(fact_4499_singletonI,axiom,
    ! [A: extended_enat] : pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) ).

% singletonI
tff(fact_4500_singletonI,axiom,
    ! [A: real] : pp(aa_set_real_bool(member_real(A),insert_real(A,bot_bot_set_real))) ).

% singletonI
tff(fact_4501_singletonI,axiom,
    ! [A: nat] : pp(aa_set_nat_bool(member_nat(A),insert_nat(A,bot_bot_set_nat))) ).

% singletonI
tff(fact_4502_singletonI,axiom,
    ! [A: int] : pp(aa_set_int_bool(member_int(A),insert_int(A,bot_bot_set_int))) ).

% singletonI
tff(fact_4503_finite__insert,axiom,
    ! [A: real,A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,insert_real(A,A2)))
    <=> pp(aa_set_real_bool(finite_finite_real,A2)) ) ).

% finite_insert
tff(fact_4504_finite__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,insert_nat(A,A2)))
    <=> pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ).

% finite_insert
tff(fact_4505_finite__insert,axiom,
    ! [A: complex,A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,insert_complex(A,A2)))
    <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ).

% finite_insert
tff(fact_4506_finite__insert,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,insert_int(A,A2)))
    <=> pp(aa_set_int_bool(finite_finite_int,A2)) ) ).

% finite_insert
tff(fact_4507_finite__insert,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,insert_Extended_enat(A,A2)))
    <=> pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2)) ) ).

% finite_insert
tff(fact_4508_insert__subset,axiom,
    ! [X: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(insert_Extended_enat(X,A2)),B))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),B))
        & pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B)) ) ) ).

% insert_subset
tff(fact_4509_insert__subset,axiom,
    ! [X: real,A2: set_real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(insert_real(X,A2)),B))
    <=> ( pp(aa_set_real_bool(member_real(X),B))
        & pp(aa_set_real_bool(ord_less_eq_set_real(A2),B)) ) ) ).

% insert_subset
tff(fact_4510_insert__subset,axiom,
    ! [X: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(insert_set_nat(X,A2)),B))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(X),B))
        & pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B)) ) ) ).

% insert_subset
tff(fact_4511_insert__subset,axiom,
    ! [X: nat,A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,insert_nat(X,A2)),B))
    <=> ( pp(aa_set_nat_bool(member_nat(X),B))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B)) ) ) ).

% insert_subset
tff(fact_4512_insert__subset,axiom,
    ! [X: int,A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(X,A2)),B))
    <=> ( pp(aa_set_int_bool(member_int(X),B))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),B)) ) ) ).

% insert_subset
tff(fact_4513_insert__Diff1,axiom,
    ! [X: extended_enat,B: set_Extended_enat,A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),B))
     => ( minus_925952699566721837d_enat(insert_Extended_enat(X,A2),B) = minus_925952699566721837d_enat(A2,B) ) ) ).

% insert_Diff1
tff(fact_4514_insert__Diff1,axiom,
    ! [X: real,B: set_real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(X),B))
     => ( minus_minus_set_real(insert_real(X,A2),B) = minus_minus_set_real(A2,B) ) ) ).

% insert_Diff1
tff(fact_4515_insert__Diff1,axiom,
    ! [X: set_nat,B: set_set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),B))
     => ( minus_2163939370556025621et_nat(insert_set_nat(X,A2),B) = minus_2163939370556025621et_nat(A2,B) ) ) ).

% insert_Diff1
tff(fact_4516_insert__Diff1,axiom,
    ! [X: int,B: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(X),B))
     => ( minus_minus_set_int(insert_int(X,A2),B) = minus_minus_set_int(A2,B) ) ) ).

% insert_Diff1
tff(fact_4517_insert__Diff1,axiom,
    ! [X: nat,B: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),B))
     => ( minus_minus_set_nat(insert_nat(X,A2),B) = minus_minus_set_nat(A2,B) ) ) ).

% insert_Diff1
tff(fact_4518_Diff__insert0,axiom,
    ! [X: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
     => ( minus_925952699566721837d_enat(A2,insert_Extended_enat(X,B)) = minus_925952699566721837d_enat(A2,B) ) ) ).

% Diff_insert0
tff(fact_4519_Diff__insert0,axiom,
    ! [X: real,A2: set_real,B: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),A2))
     => ( minus_minus_set_real(A2,insert_real(X,B)) = minus_minus_set_real(A2,B) ) ) ).

% Diff_insert0
tff(fact_4520_Diff__insert0,axiom,
    ! [X: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2))
     => ( minus_2163939370556025621et_nat(A2,insert_set_nat(X,B)) = minus_2163939370556025621et_nat(A2,B) ) ) ).

% Diff_insert0
tff(fact_4521_Diff__insert0,axiom,
    ! [X: int,A2: set_int,B: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),A2))
     => ( minus_minus_set_int(A2,insert_int(X,B)) = minus_minus_set_int(A2,B) ) ) ).

% Diff_insert0
tff(fact_4522_Diff__insert0,axiom,
    ! [X: nat,A2: set_nat,B: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
     => ( minus_minus_set_nat(A2,insert_nat(X,B)) = minus_minus_set_nat(A2,B) ) ) ).

% Diff_insert0
tff(fact_4523_singleton__conv,axiom,
    ! [A: list_nat] : collect_list_nat(aTP_Lamm_eq(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv
tff(fact_4524_singleton__conv,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_er,A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv
tff(fact_4525_singleton__conv,axiom,
    ! [A: extended_enat] : collec4429806609662206161d_enat(aa_Ext291666158496884126t_bool(aTP_Lamm_es,A)) = insert_Extended_enat(A,bot_bo7653980558646680370d_enat) ).

% singleton_conv
tff(fact_4526_singleton__conv,axiom,
    ! [A: real] : collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_et,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv
tff(fact_4527_singleton__conv,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_eu,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv
tff(fact_4528_singleton__conv,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_bool(aTP_Lamm_ev,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv
tff(fact_4529_singleton__conv2,axiom,
    ! [A: list_nat] : collect_list_nat(fequal_list_nat(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv2
tff(fact_4530_singleton__conv2,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set5116768664143533222t_bool(fequal_set_nat,A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv2
tff(fact_4531_singleton__conv2,axiom,
    ! [A: extended_enat] : collec4429806609662206161d_enat(aa_Ext291666158496884126t_bool(fequal_Extended_enat,A)) = insert_Extended_enat(A,bot_bo7653980558646680370d_enat) ).

% singleton_conv2
tff(fact_4532_singleton__conv2,axiom,
    ! [A: real] : collect_real(aa_rea7809428808904475482l_bool(fequal_real,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv2
tff(fact_4533_singleton__conv2,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_bool(fequal_nat,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_4534_singleton__conv2,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_bool(fequal_int,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv2
tff(fact_4535_finite__interval__int1,axiom,
    ! [A: int,B2: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_ew(A),B2)))) ).

% finite_interval_int1
tff(fact_4536_singleton__insert__inj__eq_H,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,B2: extended_enat] :
      ( ( insert_Extended_enat(A,A2) = insert_Extended_enat(B2,bot_bo7653980558646680370d_enat) )
    <=> ( ( A = B2 )
        & pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),insert_Extended_enat(B2,bot_bo7653980558646680370d_enat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4537_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B2: real] :
      ( ( insert_real(A,A2) = insert_real(B2,bot_bot_set_real) )
    <=> ( ( A = B2 )
        & pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(B2,bot_bot_set_real))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4538_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B2: nat] :
      ( ( insert_nat(A,A2) = insert_nat(B2,bot_bot_set_nat) )
    <=> ( ( A = B2 )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(B2,bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4539_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B2: int] :
      ( ( insert_int(A,A2) = insert_int(B2,bot_bot_set_int) )
    <=> ( ( A = B2 )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(B2,bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4540_singleton__insert__inj__eq,axiom,
    ! [B2: extended_enat,A: extended_enat,A2: set_Extended_enat] :
      ( ( insert_Extended_enat(B2,bot_bo7653980558646680370d_enat) = insert_Extended_enat(A,A2) )
    <=> ( ( A = B2 )
        & pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),insert_Extended_enat(B2,bot_bo7653980558646680370d_enat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_4541_singleton__insert__inj__eq,axiom,
    ! [B2: real,A: real,A2: set_real] :
      ( ( insert_real(B2,bot_bot_set_real) = insert_real(A,A2) )
    <=> ( ( A = B2 )
        & pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(B2,bot_bot_set_real))) ) ) ).

% singleton_insert_inj_eq
tff(fact_4542_singleton__insert__inj__eq,axiom,
    ! [B2: nat,A: nat,A2: set_nat] :
      ( ( insert_nat(B2,bot_bot_set_nat) = insert_nat(A,A2) )
    <=> ( ( A = B2 )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(B2,bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_4543_singleton__insert__inj__eq,axiom,
    ! [B2: int,A: int,A2: set_int] :
      ( ( insert_int(B2,bot_bot_set_int) = insert_int(A,A2) )
    <=> ( ( A = B2 )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(B2,bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq
tff(fact_4544_atLeastAtMost__singleton__iff,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( set_or5403411693681687835d_enat(A,B2) = insert_Extended_enat(C,bot_bo7653980558646680370d_enat) )
    <=> ( ( A = B2 )
        & ( B2 = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4545_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2) = insert_nat(C,bot_bot_set_nat) )
    <=> ( ( A = B2 )
        & ( B2 = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4546_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( set_or1266510415728281911st_int(A,B2) = insert_int(C,bot_bot_set_int) )
    <=> ( ( A = B2 )
        & ( B2 = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4547_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( set_or1222579329274155063t_real(A,B2) = insert_real(C,bot_bot_set_real) )
    <=> ( ( A = B2 )
        & ( B2 = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4548_atLeastAtMost__singleton,axiom,
    ! [A: extended_enat] : set_or5403411693681687835d_enat(A,A) = insert_Extended_enat(A,bot_bo7653980558646680370d_enat) ).

% atLeastAtMost_singleton
tff(fact_4549_atLeastAtMost__singleton,axiom,
    ! [A: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(A),A) = insert_nat(A,bot_bot_set_nat) ).

% atLeastAtMost_singleton
tff(fact_4550_atLeastAtMost__singleton,axiom,
    ! [A: int] : set_or1266510415728281911st_int(A,A) = insert_int(A,bot_bot_set_int) ).

% atLeastAtMost_singleton
tff(fact_4551_atLeastAtMost__singleton,axiom,
    ! [A: real] : set_or1222579329274155063t_real(A,A) = insert_real(A,bot_bot_set_real) ).

% atLeastAtMost_singleton
tff(fact_4552_insert__Diff__single,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] : insert_Extended_enat(A,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = insert_Extended_enat(A,A2) ).

% insert_Diff_single
tff(fact_4553_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] : insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = insert_real(A,A2) ).

% insert_Diff_single
tff(fact_4554_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] : insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = insert_int(A,A2) ).

% insert_Diff_single
tff(fact_4555_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = insert_nat(A,A2) ).

% insert_Diff_single
tff(fact_4556_finite__Diff__insert,axiom,
    ! [A2: set_real,A: real,B: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A2,insert_real(A,B))))
    <=> pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A2,B))) ) ).

% finite_Diff_insert
tff(fact_4557_finite__Diff__insert,axiom,
    ! [A2: set_complex,A: complex,B: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,insert_complex(A,B))))
    <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B))) ) ).

% finite_Diff_insert
tff(fact_4558_finite__Diff__insert,axiom,
    ! [A2: set_int,A: int,B: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,insert_int(A,B))))
    <=> pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,B))) ) ).

% finite_Diff_insert
tff(fact_4559_finite__Diff__insert,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,B))))
    <=> pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A2,B))) ) ).

% finite_Diff_insert
tff(fact_4560_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,insert_nat(A,B))))
    <=> pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,B))) ) ).

% finite_Diff_insert
tff(fact_4561_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_4562_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,N) )
    <=> ( aa_num_nat(pred_numeral,K) = N ) ) ).

% eq_numeral_Suc
tff(fact_4563_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa_nat_nat(suc,N) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( N = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_4564_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B2: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),uminus613421341184616069et_nat(insert_set_nat(B2,bot_bot_set_set_nat))))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(B2),A2)) ) ).

% subset_Compl_singleton
tff(fact_4565_subset__Compl__singleton,axiom,
    ! [A2: set_Extended_enat,B2: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),uminus417252749190364093d_enat(insert_Extended_enat(B2,bot_bo7653980558646680370d_enat))))
    <=> ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(B2),A2)) ) ).

% subset_Compl_singleton
tff(fact_4566_subset__Compl__singleton,axiom,
    ! [A2: set_real,B2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),uminus612125837232591019t_real(insert_real(B2,bot_bot_set_real))))
    <=> ~ pp(aa_set_real_bool(member_real(B2),A2)) ) ).

% subset_Compl_singleton
tff(fact_4567_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),uminus5710092332889474511et_nat(insert_nat(B2,bot_bot_set_nat))))
    <=> ~ pp(aa_set_nat_bool(member_nat(B2),A2)) ) ).

% subset_Compl_singleton
tff(fact_4568_subset__Compl__singleton,axiom,
    ! [A2: set_int,B2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(insert_int(B2,bot_bot_set_int))))
    <=> ~ pp(aa_set_int_bool(member_int(B2),A2)) ) ).

% subset_Compl_singleton
tff(fact_4569_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% less_numeral_Suc
tff(fact_4570_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% less_Suc_numeral
tff(fact_4571_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% le_numeral_Suc
tff(fact_4572_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% le_Suc_numeral
tff(fact_4573_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_4574_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),N) ).

% diff_numeral_Suc
tff(fact_4575_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_4576_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),N)) ).

% max_numeral_Suc
tff(fact_4577_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5052692396658037445od_int(M,one) = product_Pair_int_int(numeral_numeral_int(M),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_4578_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5055182867167087721od_nat(M,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,M)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_4579_set__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_4580_set__replicate,axiom,
    ! [N: nat,X: extended_enat] :
      ( ( N != zero_zero_nat )
     => ( set_Extended_enat2(replic7216382294607269926d_enat(N,X)) = insert_Extended_enat(X,bot_bo7653980558646680370d_enat) ) ) ).

% set_replicate
tff(fact_4581_set__replicate,axiom,
    ! [N: nat,X: real] :
      ( ( N != zero_zero_nat )
     => ( set_real2(replicate_real(N,X)) = insert_real(X,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_4582_set__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( N != zero_zero_nat )
     => ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_4583_set__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( N != zero_zero_nat )
     => ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_4584_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,bit0(N)) = product_Pair_int_int(zero_zero_int,numeral_numeral_int(one)) ).

% divmod_algorithm_code(3)
tff(fact_4585_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,bit0(N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_4586_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,numeral_numeral_int(one)) ).

% divmod_algorithm_code(4)
tff(fact_4587_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_4588_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit1,M),bit0(N)) = produc4245557441103728435nt_int(aTP_Lamm_ex,unique5052692396658037445od_int(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_4589_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(aa_num_num(bit1,M),bit0(N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ey),unique5055182867167087721od_nat(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_4590_set__diff__eq,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] : minus_925952699566721837d_enat(A2,B) = collec4429806609662206161d_enat(aa_set3329637325543298558t_bool(aTP_Lamm_ez(A2),B)) ).

% set_diff_eq
tff(fact_4591_set__diff__eq,axiom,
    ! [A2: set_real,B: set_real] : minus_minus_set_real(A2,B) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_fa(A2),B)) ).

% set_diff_eq
tff(fact_4592_set__diff__eq,axiom,
    ! [A2: set_list_nat,B: set_list_nat] : minus_7954133019191499631st_nat(A2,B) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_fb(A2),B)) ).

% set_diff_eq
tff(fact_4593_set__diff__eq,axiom,
    ! [A2: set_set_nat,B: set_set_nat] : minus_2163939370556025621et_nat(A2,B) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_fc(A2),B)) ).

% set_diff_eq
tff(fact_4594_set__diff__eq,axiom,
    ! [A2: set_int,B: set_int] : minus_minus_set_int(A2,B) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_fd(A2),B)) ).

% set_diff_eq
tff(fact_4595_set__diff__eq,axiom,
    ! [A2: set_nat,B: set_nat] : minus_minus_set_nat(A2,B) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_fe(A2),B)) ).

% set_diff_eq
tff(fact_4596_minus__set__def,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] : minus_925952699566721837d_enat(A2,B) = collec4429806609662206161d_enat(minus_396011825176962637t_bool(aTP_Lamm_a(A2),aTP_Lamm_a(B))) ).

% minus_set_def
tff(fact_4597_minus__set__def,axiom,
    ! [A2: set_real,B: set_real] : minus_minus_set_real(A2,B) = collect_real(minus_6582619742941512235l_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B))) ).

% minus_set_def
tff(fact_4598_minus__set__def,axiom,
    ! [A2: set_list_nat,B: set_list_nat] : minus_7954133019191499631st_nat(A2,B) = collect_list_nat(minus_8064998606961450743t_bool(aTP_Lamm_ab(A2),aTP_Lamm_ab(B))) ).

% minus_set_def
tff(fact_4599_minus__set__def,axiom,
    ! [A2: set_set_nat,B: set_set_nat] : minus_2163939370556025621et_nat(A2,B) = collect_set_nat(minus_8322047804725131473t_bool(aTP_Lamm_ac(A2),aTP_Lamm_ac(B))) ).

% minus_set_def
tff(fact_4600_minus__set__def,axiom,
    ! [A2: set_int,B: set_int] : minus_minus_set_int(A2,B) = collect_int(minus_7727386867957376939t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B))) ).

% minus_set_def
tff(fact_4601_minus__set__def,axiom,
    ! [A2: set_nat,B: set_nat] : minus_minus_set_nat(A2,B) = collect_nat(minus_1628305797571225351t_bool(aTP_Lamm_ad(A2),aTP_Lamm_ad(B))) ).

% minus_set_def
tff(fact_4602_insert__Diff__if,axiom,
    ! [X: extended_enat,B: set_Extended_enat,A2: set_Extended_enat] :
      ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),B))
       => ( minus_925952699566721837d_enat(insert_Extended_enat(X,A2),B) = minus_925952699566721837d_enat(A2,B) ) )
      & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),B))
       => ( minus_925952699566721837d_enat(insert_Extended_enat(X,A2),B) = insert_Extended_enat(X,minus_925952699566721837d_enat(A2,B)) ) ) ) ).

% insert_Diff_if
tff(fact_4603_insert__Diff__if,axiom,
    ! [X: real,B: set_real,A2: set_real] :
      ( ( pp(aa_set_real_bool(member_real(X),B))
       => ( minus_minus_set_real(insert_real(X,A2),B) = minus_minus_set_real(A2,B) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X),B))
       => ( minus_minus_set_real(insert_real(X,A2),B) = insert_real(X,minus_minus_set_real(A2,B)) ) ) ) ).

% insert_Diff_if
tff(fact_4604_insert__Diff__if,axiom,
    ! [X: set_nat,B: set_set_nat,A2: set_set_nat] :
      ( ( pp(aa_set_set_nat_bool(member_set_nat(X),B))
       => ( minus_2163939370556025621et_nat(insert_set_nat(X,A2),B) = minus_2163939370556025621et_nat(A2,B) ) )
      & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),B))
       => ( minus_2163939370556025621et_nat(insert_set_nat(X,A2),B) = insert_set_nat(X,minus_2163939370556025621et_nat(A2,B)) ) ) ) ).

% insert_Diff_if
tff(fact_4605_insert__Diff__if,axiom,
    ! [X: int,B: set_int,A2: set_int] :
      ( ( pp(aa_set_int_bool(member_int(X),B))
       => ( minus_minus_set_int(insert_int(X,A2),B) = minus_minus_set_int(A2,B) ) )
      & ( ~ pp(aa_set_int_bool(member_int(X),B))
       => ( minus_minus_set_int(insert_int(X,A2),B) = insert_int(X,minus_minus_set_int(A2,B)) ) ) ) ).

% insert_Diff_if
tff(fact_4606_insert__Diff__if,axiom,
    ! [X: nat,B: set_nat,A2: set_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(X),B))
       => ( minus_minus_set_nat(insert_nat(X,A2),B) = minus_minus_set_nat(A2,B) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(X),B))
       => ( minus_minus_set_nat(insert_nat(X,A2),B) = insert_nat(X,minus_minus_set_nat(A2,B)) ) ) ) ).

% insert_Diff_if
tff(fact_4607_DiffD2,axiom,
    ! [C: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),minus_925952699566721837d_enat(A2,B)))
     => ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),B)) ) ).

% DiffD2
tff(fact_4608_DiffD2,axiom,
    ! [C: real,A2: set_real,B: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B)))
     => ~ pp(aa_set_real_bool(member_real(C),B)) ) ).

% DiffD2
tff(fact_4609_DiffD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B)))
     => ~ pp(aa_set_set_nat_bool(member_set_nat(C),B)) ) ).

% DiffD2
tff(fact_4610_DiffD2,axiom,
    ! [C: int,A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B)))
     => ~ pp(aa_set_int_bool(member_int(C),B)) ) ).

% DiffD2
tff(fact_4611_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B)))
     => ~ pp(aa_set_nat_bool(member_nat(C),B)) ) ).

% DiffD2
tff(fact_4612_DiffD1,axiom,
    ! [C: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),minus_925952699566721837d_enat(A2,B)))
     => pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),A2)) ) ).

% DiffD1
tff(fact_4613_DiffD1,axiom,
    ! [C: real,A2: set_real,B: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B)))
     => pp(aa_set_real_bool(member_real(C),A2)) ) ).

% DiffD1
tff(fact_4614_DiffD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B)))
     => pp(aa_set_set_nat_bool(member_set_nat(C),A2)) ) ).

% DiffD1
tff(fact_4615_DiffD1,axiom,
    ! [C: int,A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B)))
     => pp(aa_set_int_bool(member_int(C),A2)) ) ).

% DiffD1
tff(fact_4616_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B)))
     => pp(aa_set_nat_bool(member_nat(C),A2)) ) ).

% DiffD1
tff(fact_4617_DiffE,axiom,
    ! [C: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),minus_925952699566721837d_enat(A2,B)))
     => ~ ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),A2))
         => pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),B)) ) ) ).

% DiffE
tff(fact_4618_DiffE,axiom,
    ! [C: real,A2: set_real,B: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B)))
     => ~ ( pp(aa_set_real_bool(member_real(C),A2))
         => pp(aa_set_real_bool(member_real(C),B)) ) ) ).

% DiffE
tff(fact_4619_DiffE,axiom,
    ! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B)))
     => ~ ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(C),B)) ) ) ).

% DiffE
tff(fact_4620_DiffE,axiom,
    ! [C: int,A2: set_int,B: set_int] :
      ( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B)))
     => ~ ( pp(aa_set_int_bool(member_int(C),A2))
         => pp(aa_set_int_bool(member_int(C),B)) ) ) ).

% DiffE
tff(fact_4621_DiffE,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B)))
     => ~ ( pp(aa_set_nat_bool(member_nat(C),A2))
         => pp(aa_set_nat_bool(member_nat(C),B)) ) ) ).

% DiffE
tff(fact_4622_int__induct,axiom,
    ! [P: fun_int_bool,K: int,I: int] :
      ( pp(aa_int_bool(P,K))
     => ( ! [I4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I4))
           => ( pp(aa_int_bool(P,I4))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I4),one_one_int))) ) )
       => ( ! [I4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I4),K))
             => ( pp(aa_int_bool(P,I4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I4),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_induct
tff(fact_4623_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I4),K))
             => ( pp(aa_int_bool(P,I4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I4),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_le_induct
tff(fact_4624_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),M)) ) ) ).

% zdvd_zdiffD
tff(fact_4625_ComplD,axiom,
    ! [C: extended_enat,A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),uminus417252749190364093d_enat(A2)))
     => ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(C),A2)) ) ).

% ComplD
tff(fact_4626_ComplD,axiom,
    ! [C: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),uminus612125837232591019t_real(A2)))
     => ~ pp(aa_set_real_bool(member_real(C),A2)) ) ).

% ComplD
tff(fact_4627_ComplD,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),uminus613421341184616069et_nat(A2)))
     => ~ pp(aa_set_set_nat_bool(member_set_nat(C),A2)) ) ).

% ComplD
tff(fact_4628_ComplD,axiom,
    ! [C: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A2)))
     => ~ pp(aa_set_nat_bool(member_nat(C),A2)) ) ).

% ComplD
tff(fact_4629_ComplD,axiom,
    ! [C: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),uminus1532241313380277803et_int(A2)))
     => ~ pp(aa_set_int_bool(member_int(C),A2)) ) ).

% ComplD
tff(fact_4630_uminus__set__def,axiom,
    ! [A2: set_Extended_enat] : uminus417252749190364093d_enat(A2) = collec4429806609662206161d_enat(uminus5643383058349504957t_bool(aTP_Lamm_a(A2))) ).

% uminus_set_def
tff(fact_4631_uminus__set__def,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(uminus4638318004487240379l_bool(aTP_Lamm_aa(A2))) ).

% uminus_set_def
tff(fact_4632_uminus__set__def,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(uminus5119773305880778631t_bool(aTP_Lamm_ab(A2))) ).

% uminus_set_def
tff(fact_4633_uminus__set__def,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(uminus6226608070181861729t_bool(aTP_Lamm_ac(A2))) ).

% uminus_set_def
tff(fact_4634_uminus__set__def,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(uminus1995910650241132439t_bool(aTP_Lamm_ad(A2))) ).

% uminus_set_def
tff(fact_4635_uminus__set__def,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(uminus8094991720627284027t_bool(aTP_Lamm_ae(A2))) ).

% uminus_set_def
tff(fact_4636_Collect__neg__eq,axiom,
    ! [P: fun_real_bool] : collect_real(aTP_Lamm_ff(P)) = uminus612125837232591019t_real(collect_real(P)) ).

% Collect_neg_eq
tff(fact_4637_Collect__neg__eq,axiom,
    ! [P: fun_list_nat_bool] : collect_list_nat(aTP_Lamm_fg(P)) = uminus3195874150345416415st_nat(collect_list_nat(P)) ).

% Collect_neg_eq
tff(fact_4638_Collect__neg__eq,axiom,
    ! [P: fun_set_nat_bool] : collect_set_nat(aTP_Lamm_fh(P)) = uminus613421341184616069et_nat(collect_set_nat(P)) ).

% Collect_neg_eq
tff(fact_4639_Collect__neg__eq,axiom,
    ! [P: fun_nat_bool] : collect_nat(aTP_Lamm_fi(P)) = uminus5710092332889474511et_nat(collect_nat(P)) ).

% Collect_neg_eq
tff(fact_4640_Collect__neg__eq,axiom,
    ! [P: fun_int_bool] : collect_int(aTP_Lamm_fj(P)) = uminus1532241313380277803et_int(collect_int(P)) ).

% Collect_neg_eq
tff(fact_4641_Compl__eq,axiom,
    ! [A2: set_Extended_enat] : uminus417252749190364093d_enat(A2) = collec4429806609662206161d_enat(aTP_Lamm_fk(A2)) ).

% Compl_eq
tff(fact_4642_Compl__eq,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(aTP_Lamm_fl(A2)) ).

% Compl_eq
tff(fact_4643_Compl__eq,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(aTP_Lamm_fm(A2)) ).

% Compl_eq
tff(fact_4644_Compl__eq,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(aTP_Lamm_fn(A2)) ).

% Compl_eq
tff(fact_4645_Compl__eq,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(aTP_Lamm_fo(A2)) ).

% Compl_eq
tff(fact_4646_Compl__eq,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(aTP_Lamm_fp(A2)) ).

% Compl_eq
tff(fact_4647_insert__compr,axiom,
    ! [A: extended_enat,B: set_Extended_enat] : insert_Extended_enat(A,B) = collec4429806609662206161d_enat(aa_set3329637325543298558t_bool(aTP_Lamm_fq(A),B)) ).

% insert_compr
tff(fact_4648_insert__compr,axiom,
    ! [A: real,B: set_real] : insert_real(A,B) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_fr(A),B)) ).

% insert_compr
tff(fact_4649_insert__compr,axiom,
    ! [A: list_nat,B: set_list_nat] : insert_list_nat(A,B) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_fs(A),B)) ).

% insert_compr
tff(fact_4650_insert__compr,axiom,
    ! [A: set_nat,B: set_set_nat] : insert_set_nat(A,B) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_ft(A),B)) ).

% insert_compr
tff(fact_4651_insert__compr,axiom,
    ! [A: nat,B: set_nat] : insert_nat(A,B) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_fu(A),B)) ).

% insert_compr
tff(fact_4652_insert__compr,axiom,
    ! [A: int,B: set_int] : insert_int(A,B) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_fv(A),B)) ).

% insert_compr
tff(fact_4653_insert__Collect,axiom,
    ! [A: real,P: fun_real_bool] : insert_real(A,collect_real(P)) = collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_fw(A),P)) ).

% insert_Collect
tff(fact_4654_insert__Collect,axiom,
    ! [A: list_nat,P: fun_list_nat_bool] : insert_list_nat(A,collect_list_nat(P)) = collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_fx(A),P)) ).

% insert_Collect
tff(fact_4655_insert__Collect,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] : insert_set_nat(A,collect_set_nat(P)) = collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_fy(A),P)) ).

% insert_Collect
tff(fact_4656_insert__Collect,axiom,
    ! [A: nat,P: fun_nat_bool] : insert_nat(A,collect_nat(P)) = collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_fz(A),P)) ).

% insert_Collect
tff(fact_4657_insert__Collect,axiom,
    ! [A: int,P: fun_int_bool] : insert_int(A,collect_int(P)) = collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ga(A),P)) ).

% insert_Collect
tff(fact_4658_mk__disjoint__insert,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
     => ? [B7: set_Extended_enat] :
          ( ( A2 = insert_Extended_enat(A,B7) )
          & ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_4659_mk__disjoint__insert,axiom,
    ! [A: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ? [B7: set_real] :
          ( ( A2 = insert_real(A,B7) )
          & ~ pp(aa_set_real_bool(member_real(A),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_4660_mk__disjoint__insert,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ? [B7: set_set_nat] :
          ( ( A2 = insert_set_nat(A,B7) )
          & ~ pp(aa_set_set_nat_bool(member_set_nat(A),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_4661_mk__disjoint__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ? [B7: set_nat] :
          ( ( A2 = insert_nat(A,B7) )
          & ~ pp(aa_set_nat_bool(member_nat(A),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_4662_mk__disjoint__insert,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => ? [B7: set_int] :
          ( ( A2 = insert_int(A,B7) )
          & ~ pp(aa_set_int_bool(member_int(A),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_4663_insert__commute,axiom,
    ! [X: nat,Y: nat,A2: set_nat] : insert_nat(X,insert_nat(Y,A2)) = insert_nat(Y,insert_nat(X,A2)) ).

% insert_commute
tff(fact_4664_insert__commute,axiom,
    ! [X: int,Y: int,A2: set_int] : insert_int(X,insert_int(Y,A2)) = insert_int(Y,insert_int(X,A2)) ).

% insert_commute
tff(fact_4665_insert__commute,axiom,
    ! [X: real,Y: real,A2: set_real] : insert_real(X,insert_real(Y,A2)) = insert_real(Y,insert_real(X,A2)) ).

% insert_commute
tff(fact_4666_insert__eq__iff,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,B2: extended_enat,B: set_Extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
     => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(B2),B))
       => ( ( insert_Extended_enat(A,A2) = insert_Extended_enat(B2,B) )
        <=> ( ( ( A = B2 )
             => ( A2 = B ) )
            & ( ( A != B2 )
             => ? [C5: set_Extended_enat] :
                  ( ( A2 = insert_Extended_enat(B2,C5) )
                  & ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(B2),C5))
                  & ( B = insert_Extended_enat(A,C5) )
                  & ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),C5)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4667_insert__eq__iff,axiom,
    ! [A: real,A2: set_real,B2: real,B: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(A),A2))
     => ( ~ pp(aa_set_real_bool(member_real(B2),B))
       => ( ( insert_real(A,A2) = insert_real(B2,B) )
        <=> ( ( ( A = B2 )
             => ( A2 = B ) )
            & ( ( A != B2 )
             => ? [C5: set_real] :
                  ( ( A2 = insert_real(B2,C5) )
                  & ~ pp(aa_set_real_bool(member_real(B2),C5))
                  & ( B = insert_real(A,C5) )
                  & ~ pp(aa_set_real_bool(member_real(A),C5)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4668_insert__eq__iff,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_nat,B: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(B2),B))
       => ( ( insert_set_nat(A,A2) = insert_set_nat(B2,B) )
        <=> ( ( ( A = B2 )
             => ( A2 = B ) )
            & ( ( A != B2 )
             => ? [C5: set_set_nat] :
                  ( ( A2 = insert_set_nat(B2,C5) )
                  & ~ pp(aa_set_set_nat_bool(member_set_nat(B2),C5))
                  & ( B = insert_set_nat(A,C5) )
                  & ~ pp(aa_set_set_nat_bool(member_set_nat(A),C5)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4669_insert__eq__iff,axiom,
    ! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(B2),B))
       => ( ( insert_nat(A,A2) = insert_nat(B2,B) )
        <=> ( ( ( A = B2 )
             => ( A2 = B ) )
            & ( ( A != B2 )
             => ? [C5: set_nat] :
                  ( ( A2 = insert_nat(B2,C5) )
                  & ~ pp(aa_set_nat_bool(member_nat(B2),C5))
                  & ( B = insert_nat(A,C5) )
                  & ~ pp(aa_set_nat_bool(member_nat(A),C5)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4670_insert__eq__iff,axiom,
    ! [A: int,A2: set_int,B2: int,B: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(A),A2))
     => ( ~ pp(aa_set_int_bool(member_int(B2),B))
       => ( ( insert_int(A,A2) = insert_int(B2,B) )
        <=> ( ( ( A = B2 )
             => ( A2 = B ) )
            & ( ( A != B2 )
             => ? [C5: set_int] :
                  ( ( A2 = insert_int(B2,C5) )
                  & ~ pp(aa_set_int_bool(member_int(B2),C5))
                  & ( B = insert_int(A,C5) )
                  & ~ pp(aa_set_int_bool(member_int(A),C5)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4671_insert__absorb,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
     => ( insert_Extended_enat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4672_insert__absorb,axiom,
    ! [A: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ( insert_real(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4673_insert__absorb,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ( insert_set_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4674_insert__absorb,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ( insert_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4675_insert__absorb,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => ( insert_int(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4676_insert__ident,axiom,
    ! [X: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
     => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),B))
       => ( ( insert_Extended_enat(X,A2) = insert_Extended_enat(X,B) )
        <=> ( A2 = B ) ) ) ) ).

% insert_ident
tff(fact_4677_insert__ident,axiom,
    ! [X: real,A2: set_real,B: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),B))
       => ( ( insert_real(X,A2) = insert_real(X,B) )
        <=> ( A2 = B ) ) ) ) ).

% insert_ident
tff(fact_4678_insert__ident,axiom,
    ! [X: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),B))
       => ( ( insert_set_nat(X,A2) = insert_set_nat(X,B) )
        <=> ( A2 = B ) ) ) ) ).

% insert_ident
tff(fact_4679_insert__ident,axiom,
    ! [X: nat,A2: set_nat,B: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),B))
       => ( ( insert_nat(X,A2) = insert_nat(X,B) )
        <=> ( A2 = B ) ) ) ) ).

% insert_ident
tff(fact_4680_insert__ident,axiom,
    ! [X: int,A2: set_int,B: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),A2))
     => ( ~ pp(aa_set_int_bool(member_int(X),B))
       => ( ( insert_int(X,A2) = insert_int(X,B) )
        <=> ( A2 = B ) ) ) ) ).

% insert_ident
tff(fact_4681_Set_Oset__insert,axiom,
    ! [X: extended_enat,A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
     => ~ ! [B7: set_Extended_enat] :
            ( ( A2 = insert_Extended_enat(X,B7) )
           => pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),B7)) ) ) ).

% Set.set_insert
tff(fact_4682_Set_Oset__insert,axiom,
    ! [X: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(X),A2))
     => ~ ! [B7: set_real] :
            ( ( A2 = insert_real(X,B7) )
           => pp(aa_set_real_bool(member_real(X),B7)) ) ) ).

% Set.set_insert
tff(fact_4683_Set_Oset__insert,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
     => ~ ! [B7: set_set_nat] :
            ( ( A2 = insert_set_nat(X,B7) )
           => pp(aa_set_set_nat_bool(member_set_nat(X),B7)) ) ) ).

% Set.set_insert
tff(fact_4684_Set_Oset__insert,axiom,
    ! [X: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),A2))
     => ~ ! [B7: set_nat] :
            ( ( A2 = insert_nat(X,B7) )
           => pp(aa_set_nat_bool(member_nat(X),B7)) ) ) ).

% Set.set_insert
tff(fact_4685_Set_Oset__insert,axiom,
    ! [X: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(X),A2))
     => ~ ! [B7: set_int] :
            ( ( A2 = insert_int(X,B7) )
           => pp(aa_set_int_bool(member_int(X),B7)) ) ) ).

% Set.set_insert
tff(fact_4686_insertI2,axiom,
    ! [A: extended_enat,B: set_Extended_enat,B2: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),B))
     => pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),insert_Extended_enat(B2,B))) ) ).

% insertI2
tff(fact_4687_insertI2,axiom,
    ! [A: real,B: set_real,B2: real] :
      ( pp(aa_set_real_bool(member_real(A),B))
     => pp(aa_set_real_bool(member_real(A),insert_real(B2,B))) ) ).

% insertI2
tff(fact_4688_insertI2,axiom,
    ! [A: set_nat,B: set_set_nat,B2: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),B))
     => pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(B2,B))) ) ).

% insertI2
tff(fact_4689_insertI2,axiom,
    ! [A: nat,B: set_nat,B2: nat] :
      ( pp(aa_set_nat_bool(member_nat(A),B))
     => pp(aa_set_nat_bool(member_nat(A),insert_nat(B2,B))) ) ).

% insertI2
tff(fact_4690_insertI2,axiom,
    ! [A: int,B: set_int,B2: int] :
      ( pp(aa_set_int_bool(member_int(A),B))
     => pp(aa_set_int_bool(member_int(A),insert_int(B2,B))) ) ).

% insertI2
tff(fact_4691_insertI1,axiom,
    ! [A: extended_enat,B: set_Extended_enat] : pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),insert_Extended_enat(A,B))) ).

% insertI1
tff(fact_4692_insertI1,axiom,
    ! [A: real,B: set_real] : pp(aa_set_real_bool(member_real(A),insert_real(A,B))) ).

% insertI1
tff(fact_4693_insertI1,axiom,
    ! [A: set_nat,B: set_set_nat] : pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(A,B))) ).

% insertI1
tff(fact_4694_insertI1,axiom,
    ! [A: nat,B: set_nat] : pp(aa_set_nat_bool(member_nat(A),insert_nat(A,B))) ).

% insertI1
tff(fact_4695_insertI1,axiom,
    ! [A: int,B: set_int] : pp(aa_set_int_bool(member_int(A),insert_int(A,B))) ).

% insertI1
tff(fact_4696_insertE,axiom,
    ! [A: extended_enat,B2: extended_enat,A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),insert_Extended_enat(B2,A2)))
     => ( ( A != B2 )
       => pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2)) ) ) ).

% insertE
tff(fact_4697_insertE,axiom,
    ! [A: real,B2: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),insert_real(B2,A2)))
     => ( ( A != B2 )
       => pp(aa_set_real_bool(member_real(A),A2)) ) ) ).

% insertE
tff(fact_4698_insertE,axiom,
    ! [A: set_nat,B2: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(B2,A2)))
     => ( ( A != B2 )
       => pp(aa_set_set_nat_bool(member_set_nat(A),A2)) ) ) ).

% insertE
tff(fact_4699_insertE,axiom,
    ! [A: nat,B2: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),insert_nat(B2,A2)))
     => ( ( A != B2 )
       => pp(aa_set_nat_bool(member_nat(A),A2)) ) ) ).

% insertE
tff(fact_4700_insertE,axiom,
    ! [A: int,B2: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),insert_int(B2,A2)))
     => ( ( A != B2 )
       => pp(aa_set_int_bool(member_int(A),A2)) ) ) ).

% insertE
tff(fact_4701_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I4))
             => ( pp(aa_int_bool(P,I4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I4),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_ge_induct
tff(fact_4702_singletonD,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(B2),insert_set_nat(A,bot_bot_set_set_nat)))
     => ( B2 = A ) ) ).

% singletonD
tff(fact_4703_singletonD,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B2),insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))
     => ( B2 = A ) ) ).

% singletonD
tff(fact_4704_singletonD,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_set_real_bool(member_real(B2),insert_real(A,bot_bot_set_real)))
     => ( B2 = A ) ) ).

% singletonD
tff(fact_4705_singletonD,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_set_nat_bool(member_nat(B2),insert_nat(A,bot_bot_set_nat)))
     => ( B2 = A ) ) ).

% singletonD
tff(fact_4706_singletonD,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_set_int_bool(member_int(B2),insert_int(A,bot_bot_set_int)))
     => ( B2 = A ) ) ).

% singletonD
tff(fact_4707_singleton__iff,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(B2),insert_set_nat(A,bot_bot_set_set_nat)))
    <=> ( B2 = A ) ) ).

% singleton_iff
tff(fact_4708_singleton__iff,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B2),insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))
    <=> ( B2 = A ) ) ).

% singleton_iff
tff(fact_4709_singleton__iff,axiom,
    ! [B2: real,A: real] :
      ( pp(aa_set_real_bool(member_real(B2),insert_real(A,bot_bot_set_real)))
    <=> ( B2 = A ) ) ).

% singleton_iff
tff(fact_4710_singleton__iff,axiom,
    ! [B2: nat,A: nat] :
      ( pp(aa_set_nat_bool(member_nat(B2),insert_nat(A,bot_bot_set_nat)))
    <=> ( B2 = A ) ) ).

% singleton_iff
tff(fact_4711_singleton__iff,axiom,
    ! [B2: int,A: int] :
      ( pp(aa_set_int_bool(member_int(B2),insert_int(A,bot_bot_set_int)))
    <=> ( B2 = A ) ) ).

% singleton_iff
tff(fact_4712_doubleton__eq__iff,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( insert_Extended_enat(A,insert_Extended_enat(B2,bot_bo7653980558646680370d_enat)) = insert_Extended_enat(C,insert_Extended_enat(D,bot_bo7653980558646680370d_enat)) )
    <=> ( ( ( A = C )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_4713_doubleton__eq__iff,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( insert_real(A,insert_real(B2,bot_bot_set_real)) = insert_real(C,insert_real(D,bot_bot_set_real)) )
    <=> ( ( ( A = C )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_4714_doubleton__eq__iff,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( insert_nat(A,insert_nat(B2,bot_bot_set_nat)) = insert_nat(C,insert_nat(D,bot_bot_set_nat)) )
    <=> ( ( ( A = C )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_4715_doubleton__eq__iff,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( insert_int(A,insert_int(B2,bot_bot_set_int)) = insert_int(C,insert_int(D,bot_bot_set_int)) )
    <=> ( ( ( A = C )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_4716_insert__not__empty,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] : insert_Extended_enat(A,A2) != bot_bo7653980558646680370d_enat ).

% insert_not_empty
tff(fact_4717_insert__not__empty,axiom,
    ! [A: real,A2: set_real] : insert_real(A,A2) != bot_bot_set_real ).

% insert_not_empty
tff(fact_4718_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,A2) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_4719_insert__not__empty,axiom,
    ! [A: int,A2: set_int] : insert_int(A,A2) != bot_bot_set_int ).

% insert_not_empty
tff(fact_4720_singleton__inject,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( insert_Extended_enat(A,bot_bo7653980558646680370d_enat) = insert_Extended_enat(B2,bot_bo7653980558646680370d_enat) )
     => ( A = B2 ) ) ).

% singleton_inject
tff(fact_4721_singleton__inject,axiom,
    ! [A: real,B2: real] :
      ( ( insert_real(A,bot_bot_set_real) = insert_real(B2,bot_bot_set_real) )
     => ( A = B2 ) ) ).

% singleton_inject
tff(fact_4722_singleton__inject,axiom,
    ! [A: nat,B2: nat] :
      ( ( insert_nat(A,bot_bot_set_nat) = insert_nat(B2,bot_bot_set_nat) )
     => ( A = B2 ) ) ).

% singleton_inject
tff(fact_4723_singleton__inject,axiom,
    ! [A: int,B2: int] :
      ( ( insert_int(A,bot_bot_set_int) = insert_int(B2,bot_bot_set_int) )
     => ( A = B2 ) ) ).

% singleton_inject
tff(fact_4724_Compl__insert,axiom,
    ! [X: extended_enat,A2: set_Extended_enat] : uminus417252749190364093d_enat(insert_Extended_enat(X,A2)) = minus_925952699566721837d_enat(uminus417252749190364093d_enat(A2),insert_Extended_enat(X,bot_bo7653980558646680370d_enat)) ).

% Compl_insert
tff(fact_4725_Compl__insert,axiom,
    ! [X: real,A2: set_real] : uminus612125837232591019t_real(insert_real(X,A2)) = minus_minus_set_real(uminus612125837232591019t_real(A2),insert_real(X,bot_bot_set_real)) ).

% Compl_insert
tff(fact_4726_Compl__insert,axiom,
    ! [X: int,A2: set_int] : uminus1532241313380277803et_int(insert_int(X,A2)) = minus_minus_set_int(uminus1532241313380277803et_int(A2),insert_int(X,bot_bot_set_int)) ).

% Compl_insert
tff(fact_4727_Compl__insert,axiom,
    ! [X: nat,A2: set_nat] : uminus5710092332889474511et_nat(insert_nat(X,A2)) = minus_minus_set_nat(uminus5710092332889474511et_nat(A2),insert_nat(X,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_4728_Diff__insert,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B: set_Extended_enat] : minus_925952699566721837d_enat(A2,insert_Extended_enat(A,B)) = minus_925952699566721837d_enat(minus_925952699566721837d_enat(A2,B),insert_Extended_enat(A,bot_bo7653980558646680370d_enat)) ).

% Diff_insert
tff(fact_4729_Diff__insert,axiom,
    ! [A2: set_real,A: real,B: set_real] : minus_minus_set_real(A2,insert_real(A,B)) = minus_minus_set_real(minus_minus_set_real(A2,B),insert_real(A,bot_bot_set_real)) ).

% Diff_insert
tff(fact_4730_Diff__insert,axiom,
    ! [A2: set_int,A: int,B: set_int] : minus_minus_set_int(A2,insert_int(A,B)) = minus_minus_set_int(minus_minus_set_int(A2,B),insert_int(A,bot_bot_set_int)) ).

% Diff_insert
tff(fact_4731_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B)) = minus_minus_set_nat(minus_minus_set_nat(A2,B),insert_nat(A,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_4732_insert__Diff,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ( insert_set_nat(A,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_4733_insert__Diff,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
     => ( insert_Extended_enat(A,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = A2 ) ) ).

% insert_Diff
tff(fact_4734_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ( insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = A2 ) ) ).

% insert_Diff
tff(fact_4735_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => ( insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = A2 ) ) ).

% insert_Diff
tff(fact_4736_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ( insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_4737_Diff__insert2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B: set_Extended_enat] : minus_925952699566721837d_enat(A2,insert_Extended_enat(A,B)) = minus_925952699566721837d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)),B) ).

% Diff_insert2
tff(fact_4738_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B: set_real] : minus_minus_set_real(A2,insert_real(A,B)) = minus_minus_set_real(minus_minus_set_real(A2,insert_real(A,bot_bot_set_real)),B) ).

% Diff_insert2
tff(fact_4739_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B: set_int] : minus_minus_set_int(A2,insert_int(A,B)) = minus_minus_set_int(minus_minus_set_int(A2,insert_int(A,bot_bot_set_int)),B) ).

% Diff_insert2
tff(fact_4740_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B)) = minus_minus_set_nat(minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat)),B) ).

% Diff_insert2
tff(fact_4741_Diff__insert__absorb,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2))
     => ( minus_2163939370556025621et_nat(insert_set_nat(X,A2),insert_set_nat(X,bot_bot_set_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_4742_Diff__insert__absorb,axiom,
    ! [X: extended_enat,A2: set_Extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
     => ( minus_925952699566721837d_enat(insert_Extended_enat(X,A2),insert_Extended_enat(X,bot_bo7653980558646680370d_enat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_4743_Diff__insert__absorb,axiom,
    ! [X: real,A2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),A2))
     => ( minus_minus_set_real(insert_real(X,A2),insert_real(X,bot_bot_set_real)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_4744_Diff__insert__absorb,axiom,
    ! [X: int,A2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),A2))
     => ( minus_minus_set_int(insert_int(X,A2),insert_int(X,bot_bot_set_int)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_4745_Diff__insert__absorb,axiom,
    ! [X: nat,A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
     => ( minus_minus_set_nat(insert_nat(X,A2),insert_nat(X,bot_bot_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_4746_finite_OinsertI,axiom,
    ! [A2: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => pp(aa_set_real_bool(finite_finite_real,insert_real(A,A2))) ) ).

% finite.insertI
tff(fact_4747_finite_OinsertI,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_set_nat_bool(finite_finite_nat,insert_nat(A,A2))) ) ).

% finite.insertI
tff(fact_4748_finite_OinsertI,axiom,
    ! [A2: set_complex,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,insert_complex(A,A2))) ) ).

% finite.insertI
tff(fact_4749_finite_OinsertI,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => pp(aa_set_int_bool(finite_finite_int,insert_int(A,A2))) ) ).

% finite.insertI
tff(fact_4750_finite_OinsertI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,insert_Extended_enat(A,A2))) ) ).

% finite.insertI
tff(fact_4751_insert__mono,axiom,
    ! [C4: set_real,D5: set_real,A: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(C4),D5))
     => pp(aa_set_real_bool(ord_less_eq_set_real(insert_real(A,C4)),insert_real(A,D5))) ) ).

% insert_mono
tff(fact_4752_insert__mono,axiom,
    ! [C4: set_nat,D5: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C4),D5))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,insert_nat(A,C4)),insert_nat(A,D5))) ) ).

% insert_mono
tff(fact_4753_insert__mono,axiom,
    ! [C4: set_int,D5: set_int,A: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C4),D5))
     => pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(A,C4)),insert_int(A,D5))) ) ).

% insert_mono
tff(fact_4754_subset__insert,axiom,
    ! [X: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),insert_Extended_enat(X,B)))
      <=> pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B)) ) ) ).

% subset_insert
tff(fact_4755_subset__insert,axiom,
    ! [X: real,A2: set_real,B: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),A2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X,B)))
      <=> pp(aa_set_real_bool(ord_less_eq_set_real(A2),B)) ) ) ).

% subset_insert
tff(fact_4756_subset__insert,axiom,
    ! [X: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),insert_set_nat(X,B)))
      <=> pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B)) ) ) ).

% subset_insert
tff(fact_4757_subset__insert,axiom,
    ! [X: nat,A2: set_nat,B: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(X,B)))
      <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B)) ) ) ).

% subset_insert
tff(fact_4758_subset__insert,axiom,
    ! [X: int,A2: set_int,B: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X,B)))
      <=> pp(aa_set_int_bool(ord_less_eq_set_int(A2),B)) ) ) ).

% subset_insert
tff(fact_4759_subset__insertI,axiom,
    ! [B: set_real,A: real] : pp(aa_set_real_bool(ord_less_eq_set_real(B),insert_real(A,B))) ).

% subset_insertI
tff(fact_4760_subset__insertI,axiom,
    ! [B: set_nat,A: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),insert_nat(A,B))) ).

% subset_insertI
tff(fact_4761_subset__insertI,axiom,
    ! [B: set_int,A: int] : pp(aa_set_int_bool(ord_less_eq_set_int(B),insert_int(A,B))) ).

% subset_insertI
tff(fact_4762_subset__insertI2,axiom,
    ! [A2: set_real,B: set_real,B2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(B2,B))) ) ).

% subset_insertI2
tff(fact_4763_subset__insertI2,axiom,
    ! [A2: set_nat,B: set_nat,B2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(B2,B))) ) ).

% subset_insertI2
tff(fact_4764_subset__insertI2,axiom,
    ! [A2: set_int,B: set_int,B2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(B2,B))) ) ).

% subset_insertI2
tff(fact_4765_insert__subsetI,axiom,
    ! [X: extended_enat,A2: set_Extended_enat,X6: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(X6),A2))
       => pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(insert_Extended_enat(X,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_4766_insert__subsetI,axiom,
    ! [X: real,A2: set_real,X6: set_real] :
      ( pp(aa_set_real_bool(member_real(X),A2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(X6),A2))
       => pp(aa_set_real_bool(ord_less_eq_set_real(insert_real(X,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_4767_insert__subsetI,axiom,
    ! [X: set_nat,A2: set_set_nat,X6: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(X6),A2))
       => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(insert_set_nat(X,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_4768_insert__subsetI,axiom,
    ! [X: nat,A2: set_nat,X6: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X6),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,insert_nat(X,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_4769_insert__subsetI,axiom,
    ! [X: int,A2: set_int,X6: set_int] :
      ( pp(aa_set_int_bool(member_int(X),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X6),A2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(X,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_4770_subset__Diff__insert,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,X: extended_enat,C4: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),minus_925952699566721837d_enat(B,insert_Extended_enat(X,C4))))
    <=> ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),minus_925952699566721837d_enat(B,C4)))
        & ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2)) ) ) ).

% subset_Diff_insert
tff(fact_4771_subset__Diff__insert,axiom,
    ! [A2: set_real,B: set_real,X: real,C4: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),minus_minus_set_real(B,insert_real(X,C4))))
    <=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),minus_minus_set_real(B,C4)))
        & ~ pp(aa_set_real_bool(member_real(X),A2)) ) ) ).

% subset_Diff_insert
tff(fact_4772_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B: set_set_nat,X: set_nat,C4: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B,insert_set_nat(X,C4))))
    <=> ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B,C4)))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2)) ) ) ).

% subset_Diff_insert
tff(fact_4773_subset__Diff__insert,axiom,
    ! [A2: set_nat,B: set_nat,X: nat,C4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),minus_minus_set_nat(B,insert_nat(X,C4))))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),minus_minus_set_nat(B,C4)))
        & ~ pp(aa_set_nat_bool(member_nat(X),A2)) ) ) ).

% subset_Diff_insert
tff(fact_4774_subset__Diff__insert,axiom,
    ! [A2: set_int,B: set_int,X: int,C4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),minus_minus_set_int(B,insert_int(X,C4))))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),minus_minus_set_int(B,C4)))
        & ~ pp(aa_set_int_bool(member_int(X),A2)) ) ) ).

% subset_Diff_insert
tff(fact_4775_Collect__conv__if,axiom,
    ! [P: fun_list_nat_bool,A: list_nat] :
      ( ( pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_gb(P),A)) = insert_list_nat(A,bot_bot_set_list_nat) ) )
      & ( ~ pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_gb(P),A)) = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
tff(fact_4776_Collect__conv__if,axiom,
    ! [P: fun_set_nat_bool,A: set_nat] :
      ( ( pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_gc(P),A)) = insert_set_nat(A,bot_bot_set_set_nat) ) )
      & ( ~ pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_gc(P),A)) = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
tff(fact_4777_Collect__conv__if,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,A: extended_enat] :
      ( ( pp(aa_Ext2772979874139974381t_bool(P,A))
       => ( collec4429806609662206161d_enat(aa_Ext291666158496884126t_bool(aTP_Lamm_gd(P),A)) = insert_Extended_enat(A,bot_bo7653980558646680370d_enat) ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(P,A))
       => ( collec4429806609662206161d_enat(aa_Ext291666158496884126t_bool(aTP_Lamm_gd(P),A)) = bot_bo7653980558646680370d_enat ) ) ) ).

% Collect_conv_if
tff(fact_4778_Collect__conv__if,axiom,
    ! [P: fun_real_bool,A: real] :
      ( ( pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ge(P),A)) = insert_real(A,bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ge(P),A)) = bot_bot_set_real ) ) ) ).

% Collect_conv_if
tff(fact_4779_Collect__conv__if,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ( pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gf(P),A)) = insert_nat(A,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gf(P),A)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
tff(fact_4780_Collect__conv__if,axiom,
    ! [P: fun_int_bool,A: int] :
      ( ( pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_gg(P),A)) = insert_int(A,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_gg(P),A)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if
tff(fact_4781_Collect__conv__if2,axiom,
    ! [P: fun_list_nat_bool,A: list_nat] :
      ( ( pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_gh(P),A)) = insert_list_nat(A,bot_bot_set_list_nat) ) )
      & ( ~ pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_gh(P),A)) = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
tff(fact_4782_Collect__conv__if2,axiom,
    ! [P: fun_set_nat_bool,A: set_nat] :
      ( ( pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_gi(P),A)) = insert_set_nat(A,bot_bot_set_set_nat) ) )
      & ( ~ pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_gi(P),A)) = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
tff(fact_4783_Collect__conv__if2,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,A: extended_enat] :
      ( ( pp(aa_Ext2772979874139974381t_bool(P,A))
       => ( collec4429806609662206161d_enat(aa_Ext291666158496884126t_bool(aTP_Lamm_gj(P),A)) = insert_Extended_enat(A,bot_bo7653980558646680370d_enat) ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(P,A))
       => ( collec4429806609662206161d_enat(aa_Ext291666158496884126t_bool(aTP_Lamm_gj(P),A)) = bot_bo7653980558646680370d_enat ) ) ) ).

% Collect_conv_if2
tff(fact_4784_Collect__conv__if2,axiom,
    ! [P: fun_real_bool,A: real] :
      ( ( pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_gk(P),A)) = insert_real(A,bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_gk(P),A)) = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
tff(fact_4785_Collect__conv__if2,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ( pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gl(P),A)) = insert_nat(A,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gl(P),A)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
tff(fact_4786_Collect__conv__if2,axiom,
    ! [P: fun_int_bool,A: int] :
      ( ( pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_gm(P),A)) = insert_int(A,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_gm(P),A)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
tff(fact_4787_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A4: set_complex] :
              ( ? [A3: complex] : A = insert_complex(A3,A4)
             => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4)) ) ) ) ).

% finite.cases
tff(fact_4788_finite_Ocases,axiom,
    ! [A: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A))
     => ( ( A != bot_bo7653980558646680370d_enat )
       => ~ ! [A4: set_Extended_enat] :
              ( ? [A3: extended_enat] : A = insert_Extended_enat(A3,A4)
             => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4)) ) ) ) ).

% finite.cases
tff(fact_4789_finite_Ocases,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( A != bot_bot_set_real )
       => ~ ! [A4: set_real] :
              ( ? [A3: real] : A = insert_real(A3,A4)
             => ~ pp(aa_set_real_bool(finite_finite_real,A4)) ) ) ) ).

% finite.cases
tff(fact_4790_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A4: set_nat] :
              ( ? [A3: nat] : A = insert_nat(A3,A4)
             => ~ pp(aa_set_nat_bool(finite_finite_nat,A4)) ) ) ) ).

% finite.cases
tff(fact_4791_finite_Ocases,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( A != bot_bot_set_int )
       => ~ ! [A4: set_int] :
              ( ? [A3: int] : A = insert_int(A3,A4)
             => ~ pp(aa_set_int_bool(finite_finite_int,A4)) ) ) ) ).

% finite.cases
tff(fact_4792_finite_Osimps,axiom,
    ! [A: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
    <=> ( ( A = bot_bot_set_complex )
        | ? [A7: set_complex,A6: complex] :
            ( ( A = insert_complex(A6,A7) )
            & pp(aa_set_complex_bool(finite3207457112153483333omplex,A7)) ) ) ) ).

% finite.simps
tff(fact_4793_finite_Osimps,axiom,
    ! [A: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A))
    <=> ( ( A = bot_bo7653980558646680370d_enat )
        | ? [A7: set_Extended_enat,A6: extended_enat] :
            ( ( A = insert_Extended_enat(A6,A7) )
            & pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A7)) ) ) ) ).

% finite.simps
tff(fact_4794_finite_Osimps,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
    <=> ( ( A = bot_bot_set_real )
        | ? [A7: set_real,A6: real] :
            ( ( A = insert_real(A6,A7) )
            & pp(aa_set_real_bool(finite_finite_real,A7)) ) ) ) ).

% finite.simps
tff(fact_4795_finite_Osimps,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
    <=> ( ( A = bot_bot_set_nat )
        | ? [A7: set_nat,A6: nat] :
            ( ( A = insert_nat(A6,A7) )
            & pp(aa_set_nat_bool(finite_finite_nat,A7)) ) ) ) ).

% finite.simps
tff(fact_4796_finite_Osimps,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
    <=> ( ( A = bot_bot_set_int )
        | ? [A7: set_int,A6: int] :
            ( ( A = insert_int(A6,A7) )
            & pp(aa_set_int_bool(finite_finite_int,A7)) ) ) ) ).

% finite.simps
tff(fact_4797_finite__induct,axiom,
    ! [F3: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
       => ( ! [X4: set_nat,F4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
             => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X4),F4))
               => ( pp(aa_set_set_nat_bool(P,F4))
                 => pp(aa_set_set_nat_bool(P,insert_set_nat(X4,F4))) ) ) )
         => pp(aa_set_set_nat_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_4798_finite__induct,axiom,
    ! [F3: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,F4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
             => ( ~ pp(aa_set_complex_bool(member_complex(X4),F4))
               => ( pp(aa_set_complex_bool(P,F4))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,F4))) ) ) )
         => pp(aa_set_complex_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_4799_finite__induct,axiom,
    ! [F3: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F3))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [X4: extended_enat,F4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F4))
             => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),F4))
               => ( pp(aa_set3347977610044113549t_bool(P,F4))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X4,F4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_4800_finite__induct,axiom,
    ! [F3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,F4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,F4))
             => ( ~ pp(aa_set_real_bool(member_real(X4),F4))
               => ( pp(aa_set_real_bool(P,F4))
                 => pp(aa_set_real_bool(P,insert_real(X4,F4))) ) ) )
         => pp(aa_set_real_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_4801_finite__induct,axiom,
    ! [F3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X4: nat,F4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,F4))
             => ( ~ pp(aa_set_nat_bool(member_nat(X4),F4))
               => ( pp(aa_set_nat_bool(P,F4))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,F4))) ) ) )
         => pp(aa_set_nat_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_4802_finite__induct,axiom,
    ! [F3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X4: int,F4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,F4))
             => ( ~ pp(aa_set_int_bool(member_int(X4),F4))
               => ( pp(aa_set_int_bool(P,F4))
                 => pp(aa_set_int_bool(P,insert_int(X4,F4))) ) ) )
         => pp(aa_set_int_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_4803_finite__ne__induct,axiom,
    ! [F3: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( ( F3 != bot_bot_set_set_nat )
       => ( ! [X4: set_nat] : pp(aa_set_set_nat_bool(P,insert_set_nat(X4,bot_bot_set_set_nat)))
         => ( ! [X4: set_nat,F4: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
               => ( ( F4 != bot_bot_set_set_nat )
                 => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X4),F4))
                   => ( pp(aa_set_set_nat_bool(P,F4))
                     => pp(aa_set_set_nat_bool(P,insert_set_nat(X4,F4))) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_4804_finite__ne__induct,axiom,
    ! [F3: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( ( F3 != bot_bot_set_complex )
       => ( ! [X4: complex] : pp(aa_set_complex_bool(P,insert_complex(X4,bot_bot_set_complex)))
         => ( ! [X4: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( ( F4 != bot_bot_set_complex )
                 => ( ~ pp(aa_set_complex_bool(member_complex(X4),F4))
                   => ( pp(aa_set_complex_bool(P,F4))
                     => pp(aa_set_complex_bool(P,insert_complex(X4,F4))) ) ) ) )
           => pp(aa_set_complex_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_4805_finite__ne__induct,axiom,
    ! [F3: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F3))
     => ( ( F3 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] : pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X4,bot_bo7653980558646680370d_enat)))
         => ( ! [X4: extended_enat,F4: set_Extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F4))
               => ( ( F4 != bot_bo7653980558646680370d_enat )
                 => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),F4))
                   => ( pp(aa_set3347977610044113549t_bool(P,F4))
                     => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X4,F4))) ) ) ) )
           => pp(aa_set3347977610044113549t_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_4806_finite__ne__induct,axiom,
    ! [F3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( ( F3 != bot_bot_set_real )
       => ( ! [X4: real] : pp(aa_set_real_bool(P,insert_real(X4,bot_bot_set_real)))
         => ( ! [X4: real,F4: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F4))
               => ( ( F4 != bot_bot_set_real )
                 => ( ~ pp(aa_set_real_bool(member_real(X4),F4))
                   => ( pp(aa_set_real_bool(P,F4))
                     => pp(aa_set_real_bool(P,insert_real(X4,F4))) ) ) ) )
           => pp(aa_set_real_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_4807_finite__ne__induct,axiom,
    ! [F3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( ( F3 != bot_bot_set_nat )
       => ( ! [X4: nat] : pp(aa_set_nat_bool(P,insert_nat(X4,bot_bot_set_nat)))
         => ( ! [X4: nat,F4: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F4))
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ pp(aa_set_nat_bool(member_nat(X4),F4))
                   => ( pp(aa_set_nat_bool(P,F4))
                     => pp(aa_set_nat_bool(P,insert_nat(X4,F4))) ) ) ) )
           => pp(aa_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_4808_finite__ne__induct,axiom,
    ! [F3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( ( F3 != bot_bot_set_int )
       => ( ! [X4: int] : pp(aa_set_int_bool(P,insert_int(X4,bot_bot_set_int)))
         => ( ! [X4: int,F4: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F4))
               => ( ( F4 != bot_bot_set_int )
                 => ( ~ pp(aa_set_int_bool(member_int(X4),F4))
                   => ( pp(aa_set_int_bool(P,F4))
                     => pp(aa_set_int_bool(P,insert_int(X4,F4))) ) ) ) )
           => pp(aa_set_int_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_4809_infinite__finite__induct,axiom,
    ! [P: fun_set_set_nat_bool,A2: set_set_nat] :
      ( ! [A4: set_set_nat] :
          ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
         => pp(aa_set_set_nat_bool(P,A4)) )
     => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
       => ( ! [X4: set_nat,F4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
             => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X4),F4))
               => ( pp(aa_set_set_nat_bool(P,F4))
                 => pp(aa_set_set_nat_bool(P,insert_set_nat(X4,F4))) ) ) )
         => pp(aa_set_set_nat_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_4810_infinite__finite__induct,axiom,
    ! [P: fun_set_complex_bool,A2: set_complex] :
      ( ! [A4: set_complex] :
          ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
         => pp(aa_set_complex_bool(P,A4)) )
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,F4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
             => ( ~ pp(aa_set_complex_bool(member_complex(X4),F4))
               => ( pp(aa_set_complex_bool(P,F4))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,F4))) ) ) )
         => pp(aa_set_complex_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_4811_infinite__finite__induct,axiom,
    ! [P: fun_se2015444840911768724t_bool,A2: set_Extended_enat] :
      ( ! [A4: set_Extended_enat] :
          ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
         => pp(aa_set3347977610044113549t_bool(P,A4)) )
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [X4: extended_enat,F4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F4))
             => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),F4))
               => ( pp(aa_set3347977610044113549t_bool(P,F4))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X4,F4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_4812_infinite__finite__induct,axiom,
    ! [P: fun_set_real_bool,A2: set_real] :
      ( ! [A4: set_real] :
          ( ~ pp(aa_set_real_bool(finite_finite_real,A4))
         => pp(aa_set_real_bool(P,A4)) )
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,F4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,F4))
             => ( ~ pp(aa_set_real_bool(member_real(X4),F4))
               => ( pp(aa_set_real_bool(P,F4))
                 => pp(aa_set_real_bool(P,insert_real(X4,F4))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_4813_infinite__finite__induct,axiom,
    ! [P: fun_set_nat_bool,A2: set_nat] :
      ( ! [A4: set_nat] :
          ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
         => pp(aa_set_nat_bool(P,A4)) )
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X4: nat,F4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,F4))
             => ( ~ pp(aa_set_nat_bool(member_nat(X4),F4))
               => ( pp(aa_set_nat_bool(P,F4))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,F4))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_4814_infinite__finite__induct,axiom,
    ! [P: fun_set_int_bool,A2: set_int] :
      ( ! [A4: set_int] :
          ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
         => pp(aa_set_int_bool(P,A4)) )
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X4: int,F4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,F4))
             => ( ~ pp(aa_set_int_bool(member_int(X4),F4))
               => ( pp(aa_set_int_bool(P,F4))
                 => pp(aa_set_int_bool(P,insert_int(X4,F4))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_4815_infinite__remove,axiom,
    ! [S2: set_complex,A: complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ).

% infinite_remove
tff(fact_4816_infinite__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) ).

% infinite_remove
tff(fact_4817_infinite__remove,axiom,
    ! [S2: set_real,A: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,S2))
     => ~ pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ).

% infinite_remove
tff(fact_4818_infinite__remove,axiom,
    ! [S2: set_int,A: int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
     => ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) ).

% infinite_remove
tff(fact_4819_infinite__remove,axiom,
    ! [S2: set_nat,A: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(S2,insert_nat(A,bot_bot_set_nat)))) ) ).

% infinite_remove
tff(fact_4820_infinite__coinduct,axiom,
    ! [X6: fun_set_complex_bool,A2: set_complex] :
      ( pp(aa_set_complex_bool(X6,A2))
     => ( ! [A4: set_complex] :
            ( pp(aa_set_complex_bool(X6,A4))
           => ? [X2: complex] :
                ( pp(aa_set_complex_bool(member_complex(X2),A4))
                & ( pp(aa_set_complex_bool(X6,minus_811609699411566653omplex(A4,insert_complex(X2,bot_bot_set_complex))))
                  | ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,insert_complex(X2,bot_bot_set_complex)))) ) ) )
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% infinite_coinduct
tff(fact_4821_infinite__coinduct,axiom,
    ! [X6: fun_se2015444840911768724t_bool,A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(X6,A2))
     => ( ! [A4: set_Extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(X6,A4))
           => ? [X2: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A4))
                & ( pp(aa_set3347977610044113549t_bool(X6,minus_925952699566721837d_enat(A4,insert_Extended_enat(X2,bot_bo7653980558646680370d_enat))))
                  | ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A4,insert_Extended_enat(X2,bot_bo7653980558646680370d_enat)))) ) ) )
       => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2)) ) ) ).

% infinite_coinduct
tff(fact_4822_infinite__coinduct,axiom,
    ! [X6: fun_set_real_bool,A2: set_real] :
      ( pp(aa_set_real_bool(X6,A2))
     => ( ! [A4: set_real] :
            ( pp(aa_set_real_bool(X6,A4))
           => ? [X2: real] :
                ( pp(aa_set_real_bool(member_real(X2),A4))
                & ( pp(aa_set_real_bool(X6,minus_minus_set_real(A4,insert_real(X2,bot_bot_set_real))))
                  | ~ pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A4,insert_real(X2,bot_bot_set_real)))) ) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,A2)) ) ) ).

% infinite_coinduct
tff(fact_4823_infinite__coinduct,axiom,
    ! [X6: fun_set_int_bool,A2: set_int] :
      ( pp(aa_set_int_bool(X6,A2))
     => ( ! [A4: set_int] :
            ( pp(aa_set_int_bool(X6,A4))
           => ? [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),A4))
                & ( pp(aa_set_int_bool(X6,minus_minus_set_int(A4,insert_int(X2,bot_bot_set_int))))
                  | ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A4,insert_int(X2,bot_bot_set_int)))) ) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% infinite_coinduct
tff(fact_4824_infinite__coinduct,axiom,
    ! [X6: fun_set_nat_bool,A2: set_nat] :
      ( pp(aa_set_nat_bool(X6,A2))
     => ( ! [A4: set_nat] :
            ( pp(aa_set_nat_bool(X6,A4))
           => ? [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),A4))
                & ( pp(aa_set_nat_bool(X6,minus_minus_set_nat(A4,insert_nat(X2,bot_bot_set_nat))))
                  | ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A4,insert_nat(X2,bot_bot_set_nat)))) ) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% infinite_coinduct
tff(fact_4825_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( pp(aa_set_set_nat_bool(P,A2))
       => ( ! [A3: set_nat,A4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
             => ( pp(aa_set_set_nat_bool(member_set_nat(A3),A4))
               => ( pp(aa_set_set_nat_bool(P,A4))
                 => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A4,insert_set_nat(A3,bot_bot_set_set_nat)))) ) ) )
         => pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_4826_finite__empty__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(P,A2))
       => ( ! [A3: complex,A4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
             => ( pp(aa_set_complex_bool(member_complex(A3),A4))
               => ( pp(aa_set_complex_bool(P,A4))
                 => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A4,insert_complex(A3,bot_bot_set_complex)))) ) ) )
         => pp(aa_set_complex_bool(P,bot_bot_set_complex)) ) ) ) ).

% finite_empty_induct
tff(fact_4827_finite__empty__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(P,A2))
       => ( ! [A3: extended_enat,A4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
             => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),A4))
               => ( pp(aa_set3347977610044113549t_bool(P,A4))
                 => pp(aa_set3347977610044113549t_bool(P,minus_925952699566721837d_enat(A4,insert_Extended_enat(A3,bot_bo7653980558646680370d_enat)))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat)) ) ) ) ).

% finite_empty_induct
tff(fact_4828_finite__empty__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,A2))
       => ( ! [A3: real,A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( pp(aa_set_real_bool(member_real(A3),A4))
               => ( pp(aa_set_real_bool(P,A4))
                 => pp(aa_set_real_bool(P,minus_minus_set_real(A4,insert_real(A3,bot_bot_set_real)))) ) ) )
         => pp(aa_set_real_bool(P,bot_bot_set_real)) ) ) ) ).

% finite_empty_induct
tff(fact_4829_finite__empty__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,A2))
       => ( ! [A3: int,A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( pp(aa_set_int_bool(member_int(A3),A4))
               => ( pp(aa_set_int_bool(P,A4))
                 => pp(aa_set_int_bool(P,minus_minus_set_int(A4,insert_int(A3,bot_bot_set_int)))) ) ) )
         => pp(aa_set_int_bool(P,bot_bot_set_int)) ) ) ) ).

% finite_empty_induct
tff(fact_4830_finite__empty__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,A2))
       => ( ! [A3: nat,A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( pp(aa_set_nat_bool(member_nat(A3),A4))
               => ( pp(aa_set_nat_bool(P,A4))
                 => pp(aa_set_nat_bool(P,minus_minus_set_nat(A4,insert_nat(A3,bot_bot_set_nat)))) ) ) )
         => pp(aa_set_nat_bool(P,bot_bot_set_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_4831_subset__singleton__iff,axiom,
    ! [X6: set_Extended_enat,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(X6),insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))
    <=> ( ( X6 = bot_bo7653980558646680370d_enat )
        | ( X6 = insert_Extended_enat(A,bot_bo7653980558646680370d_enat) ) ) ) ).

% subset_singleton_iff
tff(fact_4832_subset__singleton__iff,axiom,
    ! [X6: set_real,A: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(X6),insert_real(A,bot_bot_set_real)))
    <=> ( ( X6 = bot_bot_set_real )
        | ( X6 = insert_real(A,bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_4833_subset__singleton__iff,axiom,
    ! [X6: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X6),insert_nat(A,bot_bot_set_nat)))
    <=> ( ( X6 = bot_bot_set_nat )
        | ( X6 = insert_nat(A,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_4834_subset__singleton__iff,axiom,
    ! [X6: set_int,A: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X6),insert_int(A,bot_bot_set_int)))
    <=> ( ( X6 = bot_bot_set_int )
        | ( X6 = insert_int(A,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_4835_subset__singletonD,axiom,
    ! [A2: set_Extended_enat,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))
     => ( ( A2 = bot_bo7653980558646680370d_enat )
        | ( A2 = insert_Extended_enat(X,bot_bo7653980558646680370d_enat) ) ) ) ).

% subset_singletonD
tff(fact_4836_subset__singletonD,axiom,
    ! [A2: set_real,X: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X,bot_bot_set_real)))
     => ( ( A2 = bot_bot_set_real )
        | ( A2 = insert_real(X,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_4837_subset__singletonD,axiom,
    ! [A2: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(X,bot_bot_set_nat)))
     => ( ( A2 = bot_bot_set_nat )
        | ( A2 = insert_nat(X,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_4838_subset__singletonD,axiom,
    ! [A2: set_int,X: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X,bot_bot_set_int)))
     => ( ( A2 = bot_bot_set_int )
        | ( A2 = insert_int(X,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_4839_Diff__single__insert,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat))),B))
     => pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),insert_Extended_enat(X,B))) ) ).

% Diff_single_insert
tff(fact_4840_Diff__single__insert,axiom,
    ! [A2: set_real,X: real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))),B))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X,B))) ) ).

% Diff_single_insert
tff(fact_4841_Diff__single__insert,axiom,
    ! [A2: set_nat,X: nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))),B))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(X,B))) ) ).

% Diff_single_insert
tff(fact_4842_Diff__single__insert,axiom,
    ! [A2: set_int,X: int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X,B))) ) ).

% Diff_single_insert
tff(fact_4843_subset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),insert_set_nat(X,B)))
    <=> ( ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
         => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))),B)) )
        & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2))
         => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B)) ) ) ) ).

% subset_insert_iff
tff(fact_4844_subset__insert__iff,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),insert_Extended_enat(X,B)))
    <=> ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
         => pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat))),B)) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
         => pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B)) ) ) ) ).

% subset_insert_iff
tff(fact_4845_subset__insert__iff,axiom,
    ! [A2: set_real,X: real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X,B)))
    <=> ( ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))),B)) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_set_real_bool(ord_less_eq_set_real(A2),B)) ) ) ) ).

% subset_insert_iff
tff(fact_4846_subset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(X,B)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))),B)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B)) ) ) ) ).

% subset_insert_iff
tff(fact_4847_subset__insert__iff,axiom,
    ! [A2: set_int,X: int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X,B)))
    <=> ( ( pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B)) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B)) ) ) ) ).

% subset_insert_iff
tff(fact_4848_atLeastAtMost__singleton_H,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( A = B2 )
     => ( set_or5403411693681687835d_enat(A,B2) = insert_Extended_enat(A,bot_bo7653980558646680370d_enat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_4849_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2) = insert_nat(A,bot_bot_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_4850_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
     => ( set_or1266510415728281911st_int(A,B2) = insert_int(A,bot_bot_set_int) ) ) ).

% atLeastAtMost_singleton'
tff(fact_4851_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B2: real] :
      ( ( A = B2 )
     => ( set_or1222579329274155063t_real(A,B2) = insert_real(A,bot_bot_set_real) ) ) ).

% atLeastAtMost_singleton'
tff(fact_4852_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_4853_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_bool,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,S4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S4))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,Y4)),aa_complex_real(F,X4))) )
               => ( pp(aa_set_complex_bool(P,S4))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,S4))) ) ) )
         => pp(aa_set_complex_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_4854_finite__ranking__induct,axiom,
    ! [S2: set_Extended_enat,P: fun_se2015444840911768724t_bool,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [X4: extended_enat,S4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S4))
             => ( ! [Y4: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y4),S4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,Y4)),aa_Ext2541572767299617110t_real(F,X4))) )
               => ( pp(aa_set3347977610044113549t_bool(P,S4))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X4,S4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_4855_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_bool,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,S4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S4))
             => ( ! [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),S4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,Y4)),aa_real_real(F,X4))) )
               => ( pp(aa_set_real_bool(P,S4))
                 => pp(aa_set_real_bool(P,insert_real(X4,S4))) ) ) )
         => pp(aa_set_real_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_4856_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X4: nat,S4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S4))
             => ( ! [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),S4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,Y4)),aa_nat_real(F,X4))) )
               => ( pp(aa_set_nat_bool(P,S4))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,S4))) ) ) )
         => pp(aa_set_nat_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_4857_finite__ranking__induct,axiom,
    ! [S2: set_int,P: fun_set_int_bool,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X4: int,S4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S4))
             => ( ! [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),S4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,Y4)),aa_int_real(F,X4))) )
               => ( pp(aa_set_int_bool(P,S4))
                 => pp(aa_set_int_bool(P,insert_int(X4,S4))) ) ) )
         => pp(aa_set_int_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_4858_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_bool,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,S4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S4))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,Y4)),aa_complex_nat(F,X4))) )
               => ( pp(aa_set_complex_bool(P,S4))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,S4))) ) ) )
         => pp(aa_set_complex_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_4859_finite__ranking__induct,axiom,
    ! [S2: set_Extended_enat,P: fun_se2015444840911768724t_bool,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [X4: extended_enat,S4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S4))
             => ( ! [Y4: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y4),S4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,Y4)),aa_Extended_enat_nat(F,X4))) )
               => ( pp(aa_set3347977610044113549t_bool(P,S4))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X4,S4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_4860_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_bool,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,S4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S4))
             => ( ! [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),S4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,Y4)),aa_real_nat(F,X4))) )
               => ( pp(aa_set_real_bool(P,S4))
                 => pp(aa_set_real_bool(P,insert_real(X4,S4))) ) ) )
         => pp(aa_set_real_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_4861_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X4: nat,S4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S4))
             => ( ! [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),S4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,Y4)),aa_nat_nat(F,X4))) )
               => ( pp(aa_set_nat_bool(P,S4))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,S4))) ) ) )
         => pp(aa_set_nat_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_4862_finite__ranking__induct,axiom,
    ! [S2: set_int,P: fun_set_int_bool,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X4: int,S4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S4))
             => ( ! [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),S4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,Y4)),aa_int_nat(F,X4))) )
               => ( pp(aa_set_int_bool(P,S4))
                 => pp(aa_set_int_bool(P,insert_int(X4,S4))) ) ) )
         => pp(aa_set_int_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_4863_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B3: nat,A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( ! [X2: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X2),A4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),B3)) )
               => ( pp(aa_set_nat_bool(P,A4))
                 => pp(aa_set_nat_bool(P,insert_nat(B3,A4))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4864_finite__linorder__max__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [B3: extended_enat,A4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
             => ( ! [X2: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A4))
                   => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X2),B3)) )
               => ( pp(aa_set3347977610044113549t_bool(P,A4))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(B3,A4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4865_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B3: real,A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( ! [X2: real] :
                    ( pp(aa_set_real_bool(member_real(X2),A4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B3)) )
               => ( pp(aa_set_real_bool(P,A4))
                 => pp(aa_set_real_bool(P,insert_real(B3,A4))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4866_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B3: int,A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( ! [X2: int] :
                    ( pp(aa_set_int_bool(member_int(X2),A4))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),B3)) )
               => ( pp(aa_set_int_bool(P,A4))
                 => pp(aa_set_int_bool(P,insert_int(B3,A4))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4867_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B3: nat,A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( ! [X2: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X2),A4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),X2)) )
               => ( pp(aa_set_nat_bool(P,A4))
                 => pp(aa_set_nat_bool(P,insert_nat(B3,A4))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4868_finite__linorder__min__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [B3: extended_enat,A4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
             => ( ! [X2: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A4))
                   => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B3),X2)) )
               => ( pp(aa_set3347977610044113549t_bool(P,A4))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(B3,A4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4869_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B3: real,A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( ! [X2: real] :
                    ( pp(aa_set_real_bool(member_real(X2),A4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),X2)) )
               => ( pp(aa_set_real_bool(P,A4))
                 => pp(aa_set_real_bool(P,insert_real(B3,A4))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4870_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B3: int,A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( ! [X2: int] :
                    ( pp(aa_set_int_bool(member_int(X2),A4))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),X2)) )
               => ( pp(aa_set_int_bool(P,A4))
                 => pp(aa_set_int_bool(P,insert_int(B3,A4))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4871_finite__subset__induct_H,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F3),A2))
       => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
         => ( ! [A3: set_nat,F4: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
               => ( pp(aa_set_set_nat_bool(member_set_nat(A3),A2))
                 => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F4),A2))
                   => ( ~ pp(aa_set_set_nat_bool(member_set_nat(A3),F4))
                     => ( pp(aa_set_set_nat_bool(P,F4))
                       => pp(aa_set_set_nat_bool(P,insert_set_nat(A3,F4))) ) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4872_finite__subset__induct_H,axiom,
    ! [F3: set_complex,A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F3),A2))
       => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
         => ( ! [A3: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( pp(aa_set_complex_bool(member_complex(A3),A2))
                 => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F4),A2))
                   => ( ~ pp(aa_set_complex_bool(member_complex(A3),F4))
                     => ( pp(aa_set_complex_bool(P,F4))
                       => pp(aa_set_complex_bool(P,insert_complex(A3,F4))) ) ) ) ) )
           => pp(aa_set_complex_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4873_finite__subset__induct_H,axiom,
    ! [F3: set_Extended_enat,A2: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(F3),A2))
       => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
         => ( ! [A3: extended_enat,F4: set_Extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F4))
               => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),A2))
                 => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(F4),A2))
                   => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),F4))
                     => ( pp(aa_set3347977610044113549t_bool(P,F4))
                       => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(A3,F4))) ) ) ) ) )
           => pp(aa_set3347977610044113549t_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4874_finite__subset__induct_H,axiom,
    ! [F3: set_real,A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(F3),A2))
       => ( pp(aa_set_real_bool(P,bot_bot_set_real))
         => ( ! [A3: real,F4: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F4))
               => ( pp(aa_set_real_bool(member_real(A3),A2))
                 => ( pp(aa_set_real_bool(ord_less_eq_set_real(F4),A2))
                   => ( ~ pp(aa_set_real_bool(member_real(A3),F4))
                     => ( pp(aa_set_real_bool(P,F4))
                       => pp(aa_set_real_bool(P,insert_real(A3,F4))) ) ) ) ) )
           => pp(aa_set_real_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4875_finite__subset__induct_H,axiom,
    ! [F3: set_nat,A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F3),A2))
       => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
         => ( ! [A3: nat,F4: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F4))
               => ( pp(aa_set_nat_bool(member_nat(A3),A2))
                 => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F4),A2))
                   => ( ~ pp(aa_set_nat_bool(member_nat(A3),F4))
                     => ( pp(aa_set_nat_bool(P,F4))
                       => pp(aa_set_nat_bool(P,insert_nat(A3,F4))) ) ) ) ) )
           => pp(aa_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4876_finite__subset__induct_H,axiom,
    ! [F3: set_int,A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(F3),A2))
       => ( pp(aa_set_int_bool(P,bot_bot_set_int))
         => ( ! [A3: int,F4: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F4))
               => ( pp(aa_set_int_bool(member_int(A3),A2))
                 => ( pp(aa_set_int_bool(ord_less_eq_set_int(F4),A2))
                   => ( ~ pp(aa_set_int_bool(member_int(A3),F4))
                     => ( pp(aa_set_int_bool(P,F4))
                       => pp(aa_set_int_bool(P,insert_int(A3,F4))) ) ) ) ) )
           => pp(aa_set_int_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4877_finite__subset__induct,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F3),A2))
       => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
         => ( ! [A3: set_nat,F4: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
               => ( pp(aa_set_set_nat_bool(member_set_nat(A3),A2))
                 => ( ~ pp(aa_set_set_nat_bool(member_set_nat(A3),F4))
                   => ( pp(aa_set_set_nat_bool(P,F4))
                     => pp(aa_set_set_nat_bool(P,insert_set_nat(A3,F4))) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_4878_finite__subset__induct,axiom,
    ! [F3: set_complex,A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F3),A2))
       => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
         => ( ! [A3: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( pp(aa_set_complex_bool(member_complex(A3),A2))
                 => ( ~ pp(aa_set_complex_bool(member_complex(A3),F4))
                   => ( pp(aa_set_complex_bool(P,F4))
                     => pp(aa_set_complex_bool(P,insert_complex(A3,F4))) ) ) ) )
           => pp(aa_set_complex_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_4879_finite__subset__induct,axiom,
    ! [F3: set_Extended_enat,A2: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(F3),A2))
       => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
         => ( ! [A3: extended_enat,F4: set_Extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F4))
               => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),A2))
                 => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),F4))
                   => ( pp(aa_set3347977610044113549t_bool(P,F4))
                     => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(A3,F4))) ) ) ) )
           => pp(aa_set3347977610044113549t_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_4880_finite__subset__induct,axiom,
    ! [F3: set_real,A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(F3),A2))
       => ( pp(aa_set_real_bool(P,bot_bot_set_real))
         => ( ! [A3: real,F4: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F4))
               => ( pp(aa_set_real_bool(member_real(A3),A2))
                 => ( ~ pp(aa_set_real_bool(member_real(A3),F4))
                   => ( pp(aa_set_real_bool(P,F4))
                     => pp(aa_set_real_bool(P,insert_real(A3,F4))) ) ) ) )
           => pp(aa_set_real_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_4881_finite__subset__induct,axiom,
    ! [F3: set_nat,A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F3),A2))
       => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
         => ( ! [A3: nat,F4: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F4))
               => ( pp(aa_set_nat_bool(member_nat(A3),A2))
                 => ( ~ pp(aa_set_nat_bool(member_nat(A3),F4))
                   => ( pp(aa_set_nat_bool(P,F4))
                     => pp(aa_set_nat_bool(P,insert_nat(A3,F4))) ) ) ) )
           => pp(aa_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_4882_finite__subset__induct,axiom,
    ! [F3: set_int,A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(F3),A2))
       => ( pp(aa_set_int_bool(P,bot_bot_set_int))
         => ( ! [A3: int,F4: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F4))
               => ( pp(aa_set_int_bool(member_int(A3),A2))
                 => ( ~ pp(aa_set_int_bool(member_int(A3),F4))
                   => ( pp(aa_set_int_bool(P,F4))
                     => pp(aa_set_int_bool(P,insert_int(A3,F4))) ) ) ) )
           => pp(aa_set_int_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_4883_finite__remove__induct,axiom,
    ! [B: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B))
     => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
       => ( ! [A4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
             => ( ( A4 != bot_bot_set_set_nat )
               => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A4),B))
                 => ( ! [X2: set_nat] :
                        ( pp(aa_set_set_nat_bool(member_set_nat(X2),A4))
                       => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A4,insert_set_nat(X2,bot_bot_set_set_nat)))) )
                   => pp(aa_set_set_nat_bool(P,A4)) ) ) ) )
         => pp(aa_set_set_nat_bool(P,B)) ) ) ) ).

% finite_remove_induct
tff(fact_4884_finite__remove__induct,axiom,
    ! [B: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [A4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
             => ( ( A4 != bot_bot_set_complex )
               => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B))
                 => ( ! [X2: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X2),A4))
                       => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A4,insert_complex(X2,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P,A4)) ) ) ) )
         => pp(aa_set_complex_bool(P,B)) ) ) ) ).

% finite_remove_induct
tff(fact_4885_finite__remove__induct,axiom,
    ! [B: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [A4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
             => ( ( A4 != bot_bo7653980558646680370d_enat )
               => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A4),B))
                 => ( ! [X2: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A4))
                       => pp(aa_set3347977610044113549t_bool(P,minus_925952699566721837d_enat(A4,insert_Extended_enat(X2,bot_bo7653980558646680370d_enat)))) )
                   => pp(aa_set3347977610044113549t_bool(P,A4)) ) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,B)) ) ) ) ).

% finite_remove_induct
tff(fact_4886_finite__remove__induct,axiom,
    ! [B: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( ( A4 != bot_bot_set_real )
               => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B))
                 => ( ! [X2: real] :
                        ( pp(aa_set_real_bool(member_real(X2),A4))
                       => pp(aa_set_real_bool(P,minus_minus_set_real(A4,insert_real(X2,bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P,A4)) ) ) ) )
         => pp(aa_set_real_bool(P,B)) ) ) ) ).

% finite_remove_induct
tff(fact_4887_finite__remove__induct,axiom,
    ! [B: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( ( A4 != bot_bot_set_nat )
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B))
                 => ( ! [X2: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X2),A4))
                       => pp(aa_set_nat_bool(P,minus_minus_set_nat(A4,insert_nat(X2,bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P,A4)) ) ) ) )
         => pp(aa_set_nat_bool(P,B)) ) ) ) ).

% finite_remove_induct
tff(fact_4888_finite__remove__induct,axiom,
    ! [B: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( ( A4 != bot_bot_set_int )
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B))
                 => ( ! [X2: int] :
                        ( pp(aa_set_int_bool(member_int(X2),A4))
                       => pp(aa_set_int_bool(P,minus_minus_set_int(A4,insert_int(X2,bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P,A4)) ) ) ) )
         => pp(aa_set_int_bool(P,B)) ) ) ) ).

% finite_remove_induct
tff(fact_4889_remove__induct,axiom,
    ! [P: fun_set_set_nat_bool,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
     => ( ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B))
         => pp(aa_set_set_nat_bool(P,B)) )
       => ( ! [A4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
             => ( ( A4 != bot_bot_set_set_nat )
               => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A4),B))
                 => ( ! [X2: set_nat] :
                        ( pp(aa_set_set_nat_bool(member_set_nat(X2),A4))
                       => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A4,insert_set_nat(X2,bot_bot_set_set_nat)))) )
                   => pp(aa_set_set_nat_bool(P,A4)) ) ) ) )
         => pp(aa_set_set_nat_bool(P,B)) ) ) ) ).

% remove_induct
tff(fact_4890_remove__induct,axiom,
    ! [P: fun_set_complex_bool,B: set_complex] :
      ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
     => ( ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
         => pp(aa_set_complex_bool(P,B)) )
       => ( ! [A4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
             => ( ( A4 != bot_bot_set_complex )
               => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B))
                 => ( ! [X2: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X2),A4))
                       => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A4,insert_complex(X2,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P,A4)) ) ) ) )
         => pp(aa_set_complex_bool(P,B)) ) ) ) ).

% remove_induct
tff(fact_4891_remove__induct,axiom,
    ! [P: fun_se2015444840911768724t_bool,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
     => ( ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
         => pp(aa_set3347977610044113549t_bool(P,B)) )
       => ( ! [A4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
             => ( ( A4 != bot_bo7653980558646680370d_enat )
               => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A4),B))
                 => ( ! [X2: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A4))
                       => pp(aa_set3347977610044113549t_bool(P,minus_925952699566721837d_enat(A4,insert_Extended_enat(X2,bot_bo7653980558646680370d_enat)))) )
                   => pp(aa_set3347977610044113549t_bool(P,A4)) ) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,B)) ) ) ) ).

% remove_induct
tff(fact_4892_remove__induct,axiom,
    ! [P: fun_set_real_bool,B: set_real] :
      ( pp(aa_set_real_bool(P,bot_bot_set_real))
     => ( ( ~ pp(aa_set_real_bool(finite_finite_real,B))
         => pp(aa_set_real_bool(P,B)) )
       => ( ! [A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( ( A4 != bot_bot_set_real )
               => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B))
                 => ( ! [X2: real] :
                        ( pp(aa_set_real_bool(member_real(X2),A4))
                       => pp(aa_set_real_bool(P,minus_minus_set_real(A4,insert_real(X2,bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P,A4)) ) ) ) )
         => pp(aa_set_real_bool(P,B)) ) ) ) ).

% remove_induct
tff(fact_4893_remove__induct,axiom,
    ! [P: fun_set_nat_bool,B: set_nat] :
      ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
     => ( ( ~ pp(aa_set_nat_bool(finite_finite_nat,B))
         => pp(aa_set_nat_bool(P,B)) )
       => ( ! [A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( ( A4 != bot_bot_set_nat )
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B))
                 => ( ! [X2: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X2),A4))
                       => pp(aa_set_nat_bool(P,minus_minus_set_nat(A4,insert_nat(X2,bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P,A4)) ) ) ) )
         => pp(aa_set_nat_bool(P,B)) ) ) ) ).

% remove_induct
tff(fact_4894_remove__induct,axiom,
    ! [P: fun_set_int_bool,B: set_int] :
      ( pp(aa_set_int_bool(P,bot_bot_set_int))
     => ( ( ~ pp(aa_set_int_bool(finite_finite_int,B))
         => pp(aa_set_int_bool(P,B)) )
       => ( ! [A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( ( A4 != bot_bot_set_int )
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B))
                 => ( ! [X2: int] :
                        ( pp(aa_set_int_bool(member_int(X2),A4))
                       => pp(aa_set_int_bool(P,minus_minus_set_int(A4,insert_int(X2,bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P,A4)) ) ) ) )
         => pp(aa_set_int_bool(P,B)) ) ) ) ).

% remove_induct
tff(fact_4895_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_4896_finite__induct__select,axiom,
    ! [S2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [T4: set_complex] :
              ( pp(aa_set_complex_bool(ord_less_set_complex(T4),S2))
             => ( pp(aa_set_complex_bool(P,T4))
               => ? [X2: complex] :
                    ( pp(aa_set_complex_bool(member_complex(X2),minus_811609699411566653omplex(S2,T4)))
                    & pp(aa_set_complex_bool(P,insert_complex(X2,T4))) ) ) )
         => pp(aa_set_complex_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_4897_finite__induct__select,axiom,
    ! [S2: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [T4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(T4),S2))
             => ( pp(aa_set3347977610044113549t_bool(P,T4))
               => ? [X2: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),minus_925952699566721837d_enat(S2,T4)))
                    & pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X2,T4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_4898_finite__induct__select,axiom,
    ! [S2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [T4: set_real] :
              ( pp(aa_set_real_bool(ord_less_set_real(T4),S2))
             => ( pp(aa_set_real_bool(P,T4))
               => ? [X2: real] :
                    ( pp(aa_set_real_bool(member_real(X2),minus_minus_set_real(S2,T4)))
                    & pp(aa_set_real_bool(P,insert_real(X2,T4))) ) ) )
         => pp(aa_set_real_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_4899_finite__induct__select,axiom,
    ! [S2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [T4: set_int] :
              ( pp(aa_set_int_bool(ord_less_set_int(T4),S2))
             => ( pp(aa_set_int_bool(P,T4))
               => ? [X2: int] :
                    ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(S2,T4)))
                    & pp(aa_set_int_bool(P,insert_int(X2,T4))) ) ) )
         => pp(aa_set_int_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_4900_finite__induct__select,axiom,
    ! [S2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [T4: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,T4),S2))
             => ( pp(aa_set_nat_bool(P,T4))
               => ? [X2: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X2),minus_minus_set_nat(S2,T4)))
                    & pp(aa_set_nat_bool(P,insert_nat(X2,T4))) ) ) )
         => pp(aa_set_nat_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_4901_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),insert_set_nat(X,B)))
    <=> ( ( pp(aa_set_set_nat_bool(member_set_nat(X),B))
         => pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),B)) )
        & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),B))
         => ( ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
             => pp(aa_set_set_nat_bool(ord_less_set_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))),B)) )
            & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2))
             => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_4902_psubset__insert__iff,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,B: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(A2),insert_Extended_enat(X,B)))
    <=> ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),B))
         => pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(A2),B)) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),B))
         => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
             => pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat))),B)) )
            & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
             => pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_4903_psubset__insert__iff,axiom,
    ! [A2: set_real,X: real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A2),insert_real(X,B)))
    <=> ( ( pp(aa_set_real_bool(member_real(X),B))
         => pp(aa_set_real_bool(ord_less_set_real(A2),B)) )
        & ( ~ pp(aa_set_real_bool(member_real(X),B))
         => ( ( pp(aa_set_real_bool(member_real(X),A2))
             => pp(aa_set_real_bool(ord_less_set_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))),B)) )
            & ( ~ pp(aa_set_real_bool(member_real(X),A2))
             => pp(aa_set_real_bool(ord_less_eq_set_real(A2),B)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_4904_psubset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),insert_nat(X,B)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X),B))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),B))
         => ( ( pp(aa_set_nat_bool(member_nat(X),A2))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))),B)) )
            & ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_4905_psubset__insert__iff,axiom,
    ! [A2: set_int,X: int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),insert_int(X,B)))
    <=> ( ( pp(aa_set_int_bool(member_int(X),B))
         => pp(aa_set_int_bool(ord_less_set_int(A2),B)) )
        & ( ~ pp(aa_set_int_bool(member_int(X),B))
         => ( ( pp(aa_set_int_bool(member_int(X),A2))
             => pp(aa_set_int_bool(ord_less_set_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B)) )
            & ( ~ pp(aa_set_int_bool(member_int(X),A2))
             => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_4906_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(M),N) ) ) ).

% atLeastAtMost_insertL
tff(fact_4907_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_4908_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),N) = insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_4909_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X: real] : pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X))),insert_real(X,set_real2(Xs)))) ).

% set_update_subset_insert
tff(fact_4910_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X)),insert_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_4911_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,X))),insert_nat(X,set_nat2(Xs)))) ).

% set_update_subset_insert
tff(fact_4912_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X: int] : pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X))),insert_int(X,set_int2(Xs)))) ).

% set_update_subset_insert
tff(fact_4913_set__replicate__Suc,axiom,
    ! [N: nat,X: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,N),X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_4914_set__replicate__Suc,axiom,
    ! [N: nat,X: extended_enat] : set_Extended_enat2(replic7216382294607269926d_enat(aa_nat_nat(suc,N),X)) = insert_Extended_enat(X,bot_bo7653980558646680370d_enat) ).

% set_replicate_Suc
tff(fact_4915_set__replicate__Suc,axiom,
    ! [N: nat,X: real] : set_real2(replicate_real(aa_nat_nat(suc,N),X)) = insert_real(X,bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_4916_set__replicate__Suc,axiom,
    ! [N: nat,X: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,N),X)) = insert_nat(X,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_4917_set__replicate__Suc,axiom,
    ! [N: nat,X: int] : set_int2(replicate_int(aa_nat_nat(suc,N),X)) = insert_int(X,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_4918_set__replicate__conv__if,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% set_replicate_conv_if
tff(fact_4919_set__replicate__conv__if,axiom,
    ! [N: nat,X: extended_enat] :
      ( ( ( N = zero_zero_nat )
       => ( set_Extended_enat2(replic7216382294607269926d_enat(N,X)) = bot_bo7653980558646680370d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( set_Extended_enat2(replic7216382294607269926d_enat(N,X)) = insert_Extended_enat(X,bot_bo7653980558646680370d_enat) ) ) ) ).

% set_replicate_conv_if
tff(fact_4920_set__replicate__conv__if,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( set_real2(replicate_real(N,X)) = bot_bot_set_real ) )
      & ( ( N != zero_zero_nat )
       => ( set_real2(replicate_real(N,X)) = insert_real(X,bot_bot_set_real) ) ) ) ).

% set_replicate_conv_if
tff(fact_4921_set__replicate__conv__if,axiom,
    ! [N: nat,X: nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X)) = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_4922_set__replicate__conv__if,axiom,
    ! [N: nat,X: int] :
      ( ( ( N = zero_zero_nat )
       => ( set_int2(replicate_int(N,X)) = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ) ).

% set_replicate_conv_if
tff(fact_4923_set__decode__plus__power__2,axiom,
    ! [N: nat,Z2: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(N),aa_nat_set_nat(nat_set_decode,Z2)))
     => ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),Z2)) = insert_nat(N,aa_nat_set_nat(nat_set_decode,Z2)) ) ) ).

% set_decode_plus_power_2
tff(fact_4924_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ep(L)),Qr) ).

% divmod_step_nat_def
tff(fact_4925_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,numeral_numeral_int(aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(K))))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_4926_divmod__nat__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( ( N = zero_zero_nat )
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M) ) )
      & ( ~ ( ( N = zero_zero_nat )
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gn),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_4927_mask__numeral,axiom,
    ! [N: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(pred_numeral,N)))) ).

% mask_numeral
tff(fact_4928_mask__numeral,axiom,
    ! [N: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(pred_numeral,N)))) ).

% mask_numeral
tff(fact_4929_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_complex(ring_17405671764205052669omplex,K) = zero_zero_complex ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),numeral_numeral_int(bit0(one)))),zero_zero_int),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one)))))),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one)))))),one_one_complex)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_4930_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_int(ring_1_of_int_int,K) = zero_zero_int ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),numeral_numeral_int(bit0(one)))),zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(ring_1_of_int_int,aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(ring_1_of_int_int,aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one)))))),one_one_int)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_4931_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_real(ring_1_of_int_real,K) = zero_zero_real ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),numeral_numeral_int(bit0(one)))),zero_zero_int),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one)))))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one)))))),one_one_real)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_4932_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,N),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(K),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_concat_bit(N,aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one)))),L))) ).

% concat_bit_Suc
tff(fact_4933_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4934_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4935_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4936_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4937_of__int__eq__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( aa_int_real(ring_1_of_int_real,W2) = aa_int_real(ring_1_of_int_real,Z2) )
    <=> ( W2 = Z2 ) ) ).

% of_int_eq_iff
tff(fact_4938_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% mask_nat_positive_iff
tff(fact_4939_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_real(ring_1_of_int_real,zero_n2684676970156552555ol_int(P)) = zero_n3304061248610475627l_real(P) ).

% of_int_of_bool
tff(fact_4940_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_int(ring_1_of_int_int,zero_n2684676970156552555ol_int(P)) = zero_n2684676970156552555ol_int(P) ).

% of_int_of_bool
tff(fact_4941_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(bit_concat_bit(zero_zero_nat,K),L) = L ).

% concat_bit_0
tff(fact_4942_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z2) = zero_zero_int )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4943_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z2) = zero_zero_complex )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4944_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z2) = zero_zero_real )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4945_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_int = aa_int_int(ring_1_of_int_int,Z2) )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4946_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_complex = aa_int_complex(ring_17405671764205052669omplex,Z2) )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4947_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_real = aa_int_real(ring_1_of_int_real,Z2) )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4948_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_4949_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_4950_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_4951_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z2)) ) ).

% of_int_le_iff
tff(fact_4952_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z2)) ) ).

% of_int_le_iff
tff(fact_4953_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,numeral_numeral_int(K)) = numeral_numeral_int(K) ).

% of_int_numeral
tff(fact_4954_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,numeral_numeral_int(K)) = numeral_numeral_real(K) ).

% of_int_numeral
tff(fact_4955_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z2) = numeral_numeral_int(N) )
    <=> ( Z2 = numeral_numeral_int(N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4956_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z2) = numeral_numeral_real(N) )
    <=> ( Z2 = numeral_numeral_int(N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4957_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ).

% of_int_less_iff
tff(fact_4958_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ).

% of_int_less_iff
tff(fact_4959_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_4960_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_4961_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_4962_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z2) = one_one_int )
    <=> ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4963_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z2) = one_one_complex )
    <=> ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4964_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z2) = one_one_real )
    <=> ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4965_of__int__mult,axiom,
    ! [W2: int,Z2: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z2)) ).

% of_int_mult
tff(fact_4966_of__int__mult,axiom,
    ! [W2: int,Z2: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z2)) = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z2)) ).

% of_int_mult
tff(fact_4967_of__int__mult,axiom,
    ! [W2: int,Z2: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z2)) = aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,W2)),aa_int_complex(ring_17405671764205052669omplex,Z2)) ).

% of_int_mult
tff(fact_4968_of__int__add,axiom,
    ! [W2: int,Z2: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z2)) ).

% of_int_add
tff(fact_4969_of__int__add,axiom,
    ! [W2: int,Z2: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z2)) = aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z2)) ).

% of_int_add
tff(fact_4970_of__int__minus,axiom,
    ! [Z2: int] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,Z2)) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,Z2)) ).

% of_int_minus
tff(fact_4971_of__int__minus,axiom,
    ! [Z2: int] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,Z2)) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,Z2)) ).

% of_int_minus
tff(fact_4972_mask__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,zero_zero_nat) = zero_zero_int ).

% mask_0
tff(fact_4973_mask__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,zero_zero_nat) = zero_zero_nat ).

% mask_0
tff(fact_4974_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(bit_se2000444600071755411sk_int,N) = zero_zero_int )
    <=> ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_4975_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(bit_se2002935070580805687sk_nat,N) = zero_zero_nat )
    <=> ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_4976_of__int__diff,axiom,
    ! [W2: int,Z2: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z2)) ).

% of_int_diff
tff(fact_4977_of__int__diff,axiom,
    ! [W2: int,Z2: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z2)) = aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z2)) ).

% of_int_diff
tff(fact_4978_of__int__power,axiom,
    ! [Z2: int,N: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z2),N)) = aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,Z2)),N) ).

% of_int_power
tff(fact_4979_of__int__power,axiom,
    ! [Z2: int,N: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z2),N)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z2)),N) ).

% of_int_power
tff(fact_4980_of__int__power,axiom,
    ! [Z2: int,N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z2),N)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z2)),N) ).

% of_int_power
tff(fact_4981_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( ( aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B2)),W2) = aa_int_real(ring_1_of_int_real,X) )
    <=> ( aa_nat_int(power_power_int(B2),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4982_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B2)),W2) = aa_int_complex(ring_17405671764205052669omplex,X) )
    <=> ( aa_nat_int(power_power_int(B2),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4983_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W2) = aa_int_int(ring_1_of_int_int,X) )
    <=> ( aa_nat_int(power_power_int(B2),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4984_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( ( aa_int_real(ring_1_of_int_real,X) = aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B2)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B2),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4985_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,X) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B2)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B2),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4986_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( ( aa_int_int(ring_1_of_int_int,X) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B2),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4987_mask__Suc__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% mask_Suc_0
tff(fact_4988_mask__Suc__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% mask_Suc_0
tff(fact_4989_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2)) ) ).

% of_int_0_le_iff
tff(fact_4990_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2)) ) ).

% of_int_0_le_iff
tff(fact_4991_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_4992_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_4993_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2)) ) ).

% of_int_0_less_iff
tff(fact_4994_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2)) ) ).

% of_int_0_less_iff
tff(fact_4995_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_4996_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_4997_of__int__numeral__le__iff,axiom,
    ! [N: num,Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(N)),Z2)) ) ).

% of_int_numeral_le_iff
tff(fact_4998_of__int__numeral__le__iff,axiom,
    ! [N: num,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(N)),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(N)),Z2)) ) ).

% of_int_numeral_le_iff
tff(fact_4999_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),numeral_numeral_real(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),numeral_numeral_int(N))) ) ).

% of_int_le_numeral_iff
tff(fact_5000_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z2)),numeral_numeral_int(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),numeral_numeral_int(N))) ) ).

% of_int_le_numeral_iff
tff(fact_5001_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z2)),numeral_numeral_int(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),numeral_numeral_int(N))) ) ).

% of_int_less_numeral_iff
tff(fact_5002_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),numeral_numeral_real(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),numeral_numeral_int(N))) ) ).

% of_int_less_numeral_iff
tff(fact_5003_of__int__numeral__less__iff,axiom,
    ! [N: num,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,numeral_numeral_int(N)),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,numeral_numeral_int(N)),Z2)) ) ).

% of_int_numeral_less_iff
tff(fact_5004_of__int__numeral__less__iff,axiom,
    ! [N: num,Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(N)),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,numeral_numeral_int(N)),Z2)) ) ).

% of_int_numeral_less_iff
tff(fact_5005_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z2)) ) ).

% of_int_1_le_iff
tff(fact_5006_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z2)) ) ).

% of_int_1_le_iff
tff(fact_5007_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_5008_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z2)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_5009_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_5010_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z2)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_5011_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z2)) ) ).

% of_int_1_less_iff
tff(fact_5012_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z2)) ) ).

% of_int_1_less_iff
tff(fact_5013_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B2)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B2),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5014_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B2),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5015_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B2)),W2)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B2),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5016_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W2)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B2),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5017_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(numeral_numeral_int(X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5018_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(numeral_numeral_int(X)),N) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(numeral_numeral_int(X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5019_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(numeral_numeral_real(X)),N) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(numeral_numeral_int(X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5020_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(numeral_numeral_int(X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5021_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(numeral_numeral_int(X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(numeral_numeral_int(X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5022_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(numeral_numeral_real(X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(numeral_numeral_int(X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5023_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(inc(M))) ).

% add_neg_numeral_special(6)
tff(fact_5024_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(M))) ).

% add_neg_numeral_special(6)
tff(fact_5025_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(inc(M))) ).

% add_neg_numeral_special(6)
tff(fact_5026_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(inc(N))) ).

% add_neg_numeral_special(5)
tff(fact_5027_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(N))) ).

% add_neg_numeral_special(5)
tff(fact_5028_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(inc(N))) ).

% add_neg_numeral_special(5)
tff(fact_5029_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B2)),W2)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B2),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5030_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W2)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B2),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5031_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B2)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B2),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5032_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B2),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5033_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(numeral_numeral_real(X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(numeral_numeral_int(X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5034_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(numeral_numeral_int(X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(numeral_numeral_int(X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5035_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(numeral_numeral_real(X)),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5036_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(X)),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5037_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(numeral_numeral_int(X)),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(numeral_numeral_int(X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5038_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(numeral_numeral_real(X)),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(numeral_numeral_int(X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5039_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(numeral_numeral_int(X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(numeral_numeral_int(X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5040_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(numeral_numeral_real(X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(numeral_numeral_int(X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5041_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5042_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5043_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5044_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5045_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5046_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5047_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5048_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5049_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5050_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5051_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z))) ).

% ex_le_of_int
tff(fact_5052_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),X)) ).

% ex_of_int_less
tff(fact_5053_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z))) ).

% ex_less_of_int
tff(fact_5054_mult__of__int__commute,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,X)) ).

% mult_of_int_commute
tff(fact_5055_mult__of__int__commute,axiom,
    ! [X: int,Y: real] : aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,X)),Y) = aa_real_real(times_times_real(Y),aa_int_real(ring_1_of_int_real,X)) ).

% mult_of_int_commute
tff(fact_5056_mult__of__int__commute,axiom,
    ! [X: int,Y: complex] : aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_int_complex(ring_17405671764205052669omplex,X)) ).

% mult_of_int_commute
tff(fact_5057_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)) = aa_real_real(ord_max_real(aa_int_real(ring_1_of_int_real,X)),aa_int_real(ring_1_of_int_real,Y)) ).

% of_int_max
tff(fact_5058_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(ring_1_of_int_int,X)),aa_int_int(ring_1_of_int_int,Y)) ).

% of_int_max
tff(fact_5059_less__eq__mask,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ).

% less_eq_mask
tff(fact_5060_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L: int,R2: int] : aa_int_int(bit_concat_bit(N,K),aa_int_int(bit_concat_bit(M,L),R2)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N),aa_int_int(bit_concat_bit(N,K),L)),R2) ).

% concat_bit_assoc
tff(fact_5061_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N),set_or1266510415728281911st_int(M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_5062_simp__from__to,axiom,
    ! [J: int,I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ) ).

% simp_from_to
tff(fact_5063_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ) ).

% less_mask
tff(fact_5064_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z2))) ) ).

% of_int_nonneg
tff(fact_5065_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z2))) ) ).

% of_int_nonneg
tff(fact_5066_of__int__pos,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z2))) ) ).

% of_int_pos
tff(fact_5067_of__int__pos,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z2))) ) ).

% of_int_pos
tff(fact_5068_floor__exists1,axiom,
    ! [X: real] :
    ? [X4: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X4)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int))))
      & ! [Y4: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y4)),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int)))) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
tff(fact_5069_floor__exists,axiom,
    ! [X: real] :
    ? [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))) ) ).

% floor_exists
tff(fact_5070_numeral__inc,axiom,
    ! [X: num] : numera6690914467698888265omplex(inc(X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% numeral_inc
tff(fact_5071_numeral__inc,axiom,
    ! [X: num] : aa_num_nat(numeral_numeral_nat,inc(X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% numeral_inc
tff(fact_5072_numeral__inc,axiom,
    ! [X: num] : numera1916890842035813515d_enat(inc(X)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(X)),one_on7984719198319812577d_enat) ).

% numeral_inc
tff(fact_5073_numeral__inc,axiom,
    ! [X: num] : numeral_numeral_int(inc(X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(X)),one_one_int) ).

% numeral_inc
tff(fact_5074_numeral__inc,axiom,
    ! [X: num] : numeral_numeral_real(inc(X)) = aa_real_real(plus_plus_real(numeral_numeral_real(X)),one_one_real) ).

% numeral_inc
tff(fact_5075_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(K))) = aa_int_int(uminus_uminus_int,numeral_numeral_int(K)) ).

% of_int_neg_numeral
tff(fact_5076_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,numeral_numeral_int(K))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(K)) ).

% of_int_neg_numeral
tff(fact_5077_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) ).

% Suc_mask_eq_exp
tff(fact_5078_mask__nat__less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% mask_nat_less_exp
tff(fact_5079_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_nat_int(bit_se2000444600071755411sk_int,N)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_5080_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_5081_round__unique,axiom,
    ! [X: real,Y: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(X),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))),aa_int_real(ring_1_of_int_real,Y)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))))
       => ( aa_real_int(archim8280529875227126926d_real,X) = Y ) ) ) ).

% round_unique
tff(fact_5082_of__int__round__gt,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(X),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X)))) ).

% of_int_round_gt
tff(fact_5083_of__int__round__ge,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(X),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X)))) ).

% of_int_round_ge
tff(fact_5084_of__int__round__le,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))))) ).

% of_int_round_le
tff(fact_5085_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N))
     => ( groups4538972089207619220nt_int(aTP_Lamm_ce,set_or1266510415728281911st_int(M,N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),numeral_numeral_int(bit0(one))) ) ) ).

% Sum_Icc_int
tff(fact_5086_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa2) = Y )
     => ( ( ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y = aa_int_int(uminus_uminus_int,zero_n2684676970156552555ol_int(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),X)),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xa2))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_n2684676970156552555ol_int(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),X)),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xa2))))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(Xa2),numeral_numeral_int(bit0(one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_5087_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( ( ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
          & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(uminus_uminus_int,zero_n2684676970156552555ol_int(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),K)),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),L))))) ) )
      & ( ~ ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_n2684676970156552555ol_int(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),K)),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L),numeral_numeral_int(bit0(one)))))) ) ) ) ).

% and_int.simps
tff(fact_5088_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_5089_and__zero__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),zero_zero_nat) = zero_zero_nat ).

% and_zero_eq
tff(fact_5090_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_5091_zero__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_and_eq
tff(fact_5092_bit_Oconj__zero__left,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),X) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_5093_bit_Oconj__zero__right,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(X),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_5094_sum_Oneutral__const,axiom,
    ! [A2: set_int] : groups4538972089207619220nt_int(aTP_Lamm_bz,A2) = zero_zero_int ).

% sum.neutral_const
tff(fact_5095_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_by),A2) = zero_zero_nat ).

% sum.neutral_const
tff(fact_5096_sum_Oneutral__const,axiom,
    ! [A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_cb,A2) = zero_zero_complex ).

% sum.neutral_const
tff(fact_5097_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_go),A2) = zero_zero_real ).

% sum.neutral_const
tff(fact_5098_sum_Oempty,axiom,
    ! [G: fun_Ex7660298398204468809at_nat] : groups2027974829824023292at_nat(G,bot_bo7653980558646680370d_enat) = zero_zero_nat ).

% sum.empty
tff(fact_5099_sum_Oempty,axiom,
    ! [G: fun_Ex1451620480498224165t_real] : groups4148127829035722712t_real(G,bot_bo7653980558646680370d_enat) = zero_zero_real ).

% sum.empty
tff(fact_5100_sum_Oempty,axiom,
    ! [G: fun_Ex3482447378695272101at_int] : groups2025484359314973016at_int(G,bot_bo7653980558646680370d_enat) = zero_zero_int ).

% sum.empty
tff(fact_5101_sum_Oempty,axiom,
    ! [G: fun_Ex3741802288643809319omplex] : groups6818542070133387226omplex(G,bot_bo7653980558646680370d_enat) = zero_zero_complex ).

% sum.empty
tff(fact_5102_sum_Oempty,axiom,
    ! [G: fun_Ex5852502975177633731d_enat] : groups2433450451889696826d_enat(G,bot_bo7653980558646680370d_enat) = zero_z5237406670263579293d_enat ).

% sum.empty
tff(fact_5103_sum_Oempty,axiom,
    ! [G: fun_real_nat] : groups1935376822645274424al_nat(G,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_5104_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_5105_sum_Oempty,axiom,
    ! [G: fun_real_int] : groups1932886352136224148al_int(G,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_5106_sum_Oempty,axiom,
    ! [G: fun_real_complex] : groups5754745047067104278omplex(G,bot_bot_set_real) = zero_zero_complex ).

% sum.empty
tff(fact_5107_sum_Oempty,axiom,
    ! [G: fun_re6065469302365115155d_enat] : groups2800946370649118462d_enat(G,bot_bot_set_real) = zero_z5237406670263579293d_enat ).

% sum.empty
tff(fact_5108_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5693394587270226106ex_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_5109_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups4541462559716669496nt_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_5110_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2027974829824023292at_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_5111_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_5112_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_5113_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups4148127829035722712t_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_5114_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),A2) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_5115_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_int] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5690904116761175830ex_int(G,A2) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_5116_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2025484359314973016at_int(G,A2) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_5117_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_complex] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2073611262835488442omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_5118_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( ( groups5693394587270226106ex_nat(F,F3) = zero_zero_nat )
      <=> ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),F3))
           => ( aa_complex_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5119_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( ( groups4541462559716669496nt_nat(F,F3) = zero_zero_nat )
      <=> ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),F3))
           => ( aa_int_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5120_sum__eq__0__iff,axiom,
    ! [F3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F3))
     => ( ( groups2027974829824023292at_nat(F,F3) = zero_zero_nat )
      <=> ! [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),F3))
           => ( aa_Extended_enat_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5121_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: fun_na8797796813480665199d_enat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( ( groups7108830773950497114d_enat(F,F3) = zero_z5237406670263579293d_enat )
      <=> ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),F3))
           => ( aa_nat_Extended_enat(F,X5) = zero_z5237406670263579293d_enat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5122_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( ( groups1752964319039525884d_enat(F,F3) = zero_z5237406670263579293d_enat )
      <=> ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),F3))
           => ( aa_com3683005661447327866d_enat(F,X5) = zero_z5237406670263579293d_enat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5123_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: fun_in5914218760682845459d_enat] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( ( groups4225252721152677374d_enat(F,F3) = zero_z5237406670263579293d_enat )
      <=> ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),F3))
           => ( aa_int_Extended_enat(F,X5) = zero_z5237406670263579293d_enat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5124_sum__eq__0__iff,axiom,
    ! [F3: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F3))
     => ( ( groups2433450451889696826d_enat(F,F3) = zero_z5237406670263579293d_enat )
      <=> ! [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),F3))
           => ( aa_Ext7969882492699152700d_enat(F,X5) = zero_z5237406670263579293d_enat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5125_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F3) = zero_zero_nat )
      <=> ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),F3))
           => ( aa_nat_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5126_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_5127_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gp(A),B2),S2) = aa_real_nat(B2,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gp(A),B2),S2) = zero_zero_nat ) ) ) ) ).

% sum.delta
tff(fact_5128_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_gq(A),B2),S2) = aa_complex_nat(B2,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_gq(A),B2),S2) = zero_zero_nat ) ) ) ) ).

% sum.delta
tff(fact_5129_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_gr(A),B2),S2) = aa_int_nat(B2,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_gr(A),B2),S2) = zero_zero_nat ) ) ) ) ).

% sum.delta
tff(fact_5130_sum_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2027974829824023292at_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_gs(A),B2),S2) = aa_Extended_enat_nat(B2,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2027974829824023292at_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_gs(A),B2),S2) = zero_zero_nat ) ) ) ) ).

% sum.delta
tff(fact_5131_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_gt(A),B2),S2) = aa_real_real(B2,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_gt(A),B2),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5132_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_gu(A),B2),S2) = aa_complex_real(B2,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_gu(A),B2),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5133_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_gv(A),B2),S2) = aa_int_real(B2,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_gv(A),B2),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5134_sum_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_gw(A),B2),S2) = aa_Ext2541572767299617110t_real(B2,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_gw(A),B2),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5135_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B2: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aTP_Lamm_gx(A),B2),S2) = aa_real_int(B2,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aTP_Lamm_gx(A),B2),S2) = zero_zero_int ) ) ) ) ).

% sum.delta
tff(fact_5136_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B2: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_gy(A),B2)),S2) = aa_nat_int(B2,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_gy(A),B2)),S2) = zero_zero_int ) ) ) ) ).

% sum.delta
tff(fact_5137_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gz(A),B2),S2) = aa_real_nat(B2,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gz(A),B2),S2) = zero_zero_nat ) ) ) ) ).

% sum.delta'
tff(fact_5138_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ha(A),B2),S2) = aa_complex_nat(B2,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ha(A),B2),S2) = zero_zero_nat ) ) ) ) ).

% sum.delta'
tff(fact_5139_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_hb(A),B2),S2) = aa_int_nat(B2,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_hb(A),B2),S2) = zero_zero_nat ) ) ) ) ).

% sum.delta'
tff(fact_5140_sum_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2027974829824023292at_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_hc(A),B2),S2) = aa_Extended_enat_nat(B2,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2027974829824023292at_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_hc(A),B2),S2) = zero_zero_nat ) ) ) ) ).

% sum.delta'
tff(fact_5141_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_hd(A),B2),S2) = aa_real_real(B2,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_hd(A),B2),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5142_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_he(A),B2),S2) = aa_complex_real(B2,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_he(A),B2),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5143_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_hf(A),B2),S2) = aa_int_real(B2,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_hf(A),B2),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5144_sum_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_hg(A),B2),S2) = aa_Ext2541572767299617110t_real(B2,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_hg(A),B2),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5145_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B2: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aTP_Lamm_hh(A),B2),S2) = aa_real_int(B2,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aTP_Lamm_hh(A),B2),S2) = zero_zero_int ) ) ) ) ).

% sum.delta'
tff(fact_5146_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B2: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_hi(A),B2)),S2) = aa_nat_int(B2,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_hi(A),B2)),S2) = zero_zero_int ) ) ) ) ).

% sum.delta'
tff(fact_5147_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5148_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups5693394587270226106ex_nat(G,insert_complex(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5149_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X),A2))
       => ( groups4541462559716669496nt_nat(G,insert_int(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,X)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5150_sum_Oinsert,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
       => ( groups2027974829824023292at_nat(G,insert_Extended_enat(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(G,X)),groups2027974829824023292at_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5151_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1932886352136224148al_int(G,insert_real(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X)),groups1932886352136224148al_int(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5152_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),insert_nat(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,X)),aa_set_nat_int(groups3539618377306564664at_int(G),A2)) ) ) ) ).

% sum.insert
tff(fact_5153_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups5690904116761175830ex_int(G,insert_complex(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5154_sum_Oinsert,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
       => ( groups2025484359314973016at_int(G,insert_Extended_enat(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Extended_enat_int(G,X)),groups2025484359314973016at_int(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5155_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups8097168146408367636l_real(G,insert_real(X,A2)) = aa_real_real(plus_plus_real(aa_real_real(G,X)),groups8097168146408367636l_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5156_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(plus_plus_real(aa_complex_real(G,X)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5157_of__int__sum,axiom,
    ! [F: fun_complex_int,A2: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_hj(F),A2) ).

% of_int_sum
tff(fact_5158_of__int__sum,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,aa_set_nat_int(groups3539618377306564664at_int(F),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hk(F)),A2) ).

% of_int_sum
tff(fact_5159_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A2)) = groups8778361861064173332t_real(aTP_Lamm_hl(F),A2) ).

% of_int_sum
tff(fact_5160_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_hm(F),A2) ).

% of_int_sum
tff(fact_5161_and__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(bit0(X))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_5162_and__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,bit0(X))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_5163_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),numeral_numeral_int(bit0(Y))) = zero_zero_int ).

% and_numerals(1)
tff(fact_5164_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,bit0(Y))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_5165_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(aa_num_num(bit1,X))),numeral_numeral_int(aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(X)),numeral_numeral_int(Y)))) ).

% and_numerals(7)
tff(fact_5166_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% and_numerals(7)
tff(fact_5167_sum_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( aa_int_int(G,X4) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A2) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_5168_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_5169_sum_Oneutral,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => ( aa_complex_complex(G,X4) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A2) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_5170_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_real(G,X4) = zero_zero_real ) )
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),A2) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_5171_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_Ex7660298398204468809at_nat,A2: set_Extended_enat] :
      ( ( groups2027974829824023292at_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),A2))
           => ( aa_Extended_enat_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5172_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_nat,A2: set_real] :
      ( ( groups1935376822645274424al_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => ( aa_real_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5173_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_nat,A2: set_int] :
      ( ( groups4541462559716669496nt_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5174_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_Ex1451620480498224165t_real,A2: set_Extended_enat] :
      ( ( groups4148127829035722712t_real(G,A2) != zero_zero_real )
     => ~ ! [A3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),A2))
           => ( aa_Ext2541572767299617110t_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5175_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups8097168146408367636l_real(G,A2) != zero_zero_real )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => ( aa_real_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5176_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups8778361861064173332t_real(G,A2) != zero_zero_real )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5177_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_Ex3482447378695272101at_int,A2: set_Extended_enat] :
      ( ( groups2025484359314973016at_int(G,A2) != zero_zero_int )
     => ~ ! [A3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),A2))
           => ( aa_Extended_enat_int(G,A3) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5178_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_int,A2: set_real] :
      ( ( groups1932886352136224148al_int(G,A2) != zero_zero_int )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => ( aa_real_int(G,A3) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5179_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_int,A2: set_nat] :
      ( ( aa_set_nat_int(groups3539618377306564664at_int(G),A2) != zero_zero_int )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => ( aa_nat_int(G,A3) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5180_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_Ex3741802288643809319omplex,A2: set_Extended_enat] :
      ( ( groups6818542070133387226omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),A2))
           => ( aa_Ext8748583412541189208omplex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5181_sum__mono,axiom,
    ! [K5: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( ! [I4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),K5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,I4)),aa_Ext2541572767299617110t_real(G,I4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,K5)),groups4148127829035722712t_real(G,K5))) ) ).

% sum_mono
tff(fact_5182_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I4: real] :
          ( pp(aa_set_real_bool(member_real(I4),K5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I4)),aa_real_real(G,I4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,K5)),groups8097168146408367636l_real(G,K5))) ) ).

% sum_mono
tff(fact_5183_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I4: int] :
          ( pp(aa_set_int_bool(member_int(I4),K5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I4)),aa_int_real(G,I4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,K5)),groups8778361861064173332t_real(G,K5))) ) ).

% sum_mono
tff(fact_5184_sum__mono,axiom,
    ! [K5: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( ! [I4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,I4)),aa_Extended_enat_nat(G,I4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups2027974829824023292at_nat(F,K5)),groups2027974829824023292at_nat(G,K5))) ) ).

% sum_mono
tff(fact_5185_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I4: real] :
          ( pp(aa_set_real_bool(member_real(I4),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I4)),aa_real_nat(G,I4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5))) ) ).

% sum_mono
tff(fact_5186_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I4: int] :
          ( pp(aa_set_int_bool(member_int(I4),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I4)),aa_int_nat(G,I4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5))) ) ).

% sum_mono
tff(fact_5187_sum__mono,axiom,
    ! [K5: set_Extended_enat,F: fun_Ex3482447378695272101at_int,G: fun_Ex3482447378695272101at_int] :
      ( ! [I4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Extended_enat_int(F,I4)),aa_Extended_enat_int(G,I4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups2025484359314973016at_int(F,K5)),groups2025484359314973016at_int(G,K5))) ) ).

% sum_mono
tff(fact_5188_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I4: real] :
          ( pp(aa_set_real_bool(member_real(I4),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I4)),aa_real_int(G,I4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5))) ) ).

% sum_mono
tff(fact_5189_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I4: nat] :
          ( pp(aa_set_nat_bool(member_nat(I4),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I4)),aa_nat_int(G,I4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),K5)),aa_set_nat_int(groups3539618377306564664at_int(G),K5))) ) ).

% sum_mono
tff(fact_5190_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_int,G: fun_int_int] :
      ( ! [I4: int] :
          ( pp(aa_set_int_bool(member_int(I4),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,I4)),aa_int_int(G,I4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4538972089207619220nt_int(F,K5)),groups4538972089207619220nt_int(G,K5))) ) ).

% sum_mono
tff(fact_5191_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_hn(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_5192_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_ho(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_5193_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_hp(G),H),A2) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).

% sum.distrib
tff(fact_5194_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_hq(G),H)),A2) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),A2)),aa_set_nat_real(groups6591440286371151544t_real(H),A2)) ).

% sum.distrib
tff(fact_5195_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B: set_int,G: fun_real_fun_int_int,R: fun_re8661203372455305201t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( groups1932886352136224148al_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_hs(B),G),R),A2) = groups4538972089207619220nt_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_hu(A2),G),R),B) ) ) ) ).

% sum.swap_restrict
tff(fact_5196_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B: set_int,G: fun_nat_fun_int_int,R: fun_nat_fun_int_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun3504929462025808522at_int(aa_fun3325927984641084153at_int(aTP_Lamm_hw(B),G),R)),A2) = groups4538972089207619220nt_int(aa_fun4504283173926863078nt_int(aa_fun5512849559334105941nt_int(aTP_Lamm_hy(A2),G),R),B) ) ) ) ).

% sum.swap_restrict
tff(fact_5197_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B: set_int,G: fun_co333780937270390912nt_int,R: fun_co2381122952212249071t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( groups5690904116761175830ex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_ia(B),G),R),A2) = groups4538972089207619220nt_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_id(A2),G),R),B) ) ) ) ).

% sum.swap_restrict
tff(fact_5198_sum_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B: set_int,G: fun_Ex6820543082444009806nt_int,R: fun_Ex7222163901356174497t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( groups2025484359314973016at_int(aa_fun1926933649907246598at_int(aa_fun1321938580149485067at_int(aTP_Lamm_if(B),G),R),A2) = groups4538972089207619220nt_int(aa_fun1757119979448242868nt_int(aa_fun547143500490336389nt_int(aTP_Lamm_ii(A2),G),R),B) ) ) ) ).

% sum.swap_restrict
tff(fact_5199_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B: set_nat,G: fun_real_fun_nat_nat,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups1935376822645274424al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_ik(B),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_im(A2),G),R)),B) ) ) ) ).

% sum.swap_restrict
tff(fact_5200_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B: set_nat,G: fun_co636942507509095880at_nat,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups5693394587270226106ex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_io(B),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_iq(A2),G),R)),B) ) ) ) ).

% sum.swap_restrict
tff(fact_5201_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups4541462559716669496nt_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_is(B),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_iv(A2),G),R)),B) ) ) ) ).

% sum.swap_restrict
tff(fact_5202_sum_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B: set_nat,G: fun_Ex7123704652682714774at_nat,R: fun_Ex7682446853002821117t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups2027974829824023292at_nat(aa_fun6947057244751970126at_nat(aa_fun635179117725900299at_nat(aTP_Lamm_ix(B),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun1897775202592251224at_nat(aa_fun594951741515554929at_nat(aTP_Lamm_ja(A2),G),R)),B) ) ) ) ).

% sum.swap_restrict
tff(fact_5203_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B: set_complex,G: fun_re7208826350674946562omplex,R: fun_re3010880487650781423x_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => ( groups5754745047067104278omplex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_jc(B),G),R),A2) = groups7754918857620584856omplex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_je(A2),G),R),B) ) ) ) ).

% sum.swap_restrict
tff(fact_5204_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B: set_complex,G: fun_na1591161823130732966omplex,R: fun_na5941084329111710283x_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => ( groups2073611262835488442omplex(aa_fun2700696402887535882omplex(aa_fun8019143150332516725omplex(aTP_Lamm_jg(B),G),R),A2) = groups7754918857620584856omplex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_ji(A2),G),R),B) ) ) ) ).

% sum.swap_restrict
tff(fact_5205_sum__nonpos,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(F,X4)),zero_z5237406670263579293d_enat)) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups2433450451889696826d_enat(F,A2)),zero_z5237406670263579293d_enat)) ) ).

% sum_nonpos
tff(fact_5206_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_re6065469302365115155d_enat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,X4)),zero_z5237406670263579293d_enat)) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups2800946370649118462d_enat(F,A2)),zero_z5237406670263579293d_enat)) ) ).

% sum_nonpos
tff(fact_5207_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_na8797796813480665199d_enat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),zero_z5237406670263579293d_enat)) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups7108830773950497114d_enat(F,A2)),zero_z5237406670263579293d_enat)) ) ).

% sum_nonpos
tff(fact_5208_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_in5914218760682845459d_enat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,X4)),zero_z5237406670263579293d_enat)) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups4225252721152677374d_enat(F,A2)),zero_z5237406670263579293d_enat)) ) ).

% sum_nonpos
tff(fact_5209_sum__nonpos,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_5210_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_5211_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_5212_sum__nonpos,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups2027974829824023292at_nat(F,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_5213_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_5214_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_5215_sum__nonneg,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,X4))) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),groups2433450451889696826d_enat(F,A2))) ) ).

% sum_nonneg
tff(fact_5216_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_re6065469302365115155d_enat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,X4))) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),groups2800946370649118462d_enat(F,A2))) ) ).

% sum_nonneg
tff(fact_5217_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_na8797796813480665199d_enat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,X4))) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),groups7108830773950497114d_enat(F,A2))) ) ).

% sum_nonneg
tff(fact_5218_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_in5914218760682845459d_enat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,X4))) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),groups4225252721152677374d_enat(F,A2))) ) ).

% sum_nonneg
tff(fact_5219_sum__nonneg,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups4148127829035722712t_real(F,A2))) ) ).

% sum_nonneg
tff(fact_5220_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A2))) ) ).

% sum_nonneg
tff(fact_5221_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A2))) ) ).

% sum_nonneg
tff(fact_5222_sum__nonneg,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_Extended_enat_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups2027974829824023292at_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_5223_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_5224_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_5225_sum__mono__inv,axiom,
    ! [F: fun_real_real,I6: set_real,G: fun_real_real,I: real] :
      ( ( groups8097168146408367636l_real(F,I6) = groups8097168146408367636l_real(G,I6) )
     => ( ! [I4: real] :
            ( pp(aa_set_real_bool(member_real(I4),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I4)),aa_real_real(G,I4))) )
       => ( pp(aa_set_real_bool(member_real(I),I6))
         => ( pp(aa_set_real_bool(finite_finite_real,I6))
           => ( aa_real_real(F,I) = aa_real_real(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5226_sum__mono__inv,axiom,
    ! [F: fun_complex_real,I6: set_complex,G: fun_complex_real,I: complex] :
      ( ( groups5808333547571424918x_real(F,I6) = groups5808333547571424918x_real(G,I6) )
     => ( ! [I4: complex] :
            ( pp(aa_set_complex_bool(member_complex(I4),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I4)),aa_complex_real(G,I4))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I6))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
           => ( aa_complex_real(F,I) = aa_complex_real(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5227_sum__mono__inv,axiom,
    ! [F: fun_int_real,I6: set_int,G: fun_int_real,I: int] :
      ( ( groups8778361861064173332t_real(F,I6) = groups8778361861064173332t_real(G,I6) )
     => ( ! [I4: int] :
            ( pp(aa_set_int_bool(member_int(I4),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I4)),aa_int_real(G,I4))) )
       => ( pp(aa_set_int_bool(member_int(I),I6))
         => ( pp(aa_set_int_bool(finite_finite_int,I6))
           => ( aa_int_real(F,I) = aa_int_real(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5228_sum__mono__inv,axiom,
    ! [F: fun_Ex1451620480498224165t_real,I6: set_Extended_enat,G: fun_Ex1451620480498224165t_real,I: extended_enat] :
      ( ( groups4148127829035722712t_real(F,I6) = groups4148127829035722712t_real(G,I6) )
     => ( ! [I4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,I4)),aa_Ext2541572767299617110t_real(G,I4))) )
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I6))
         => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I6))
           => ( aa_Ext2541572767299617110t_real(F,I) = aa_Ext2541572767299617110t_real(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5229_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I6: set_real,G: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(F,I6) = groups1935376822645274424al_nat(G,I6) )
     => ( ! [I4: real] :
            ( pp(aa_set_real_bool(member_real(I4),I6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I4)),aa_real_nat(G,I4))) )
       => ( pp(aa_set_real_bool(member_real(I),I6))
         => ( pp(aa_set_real_bool(finite_finite_real,I6))
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5230_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I6: set_complex,G: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(F,I6) = groups5693394587270226106ex_nat(G,I6) )
     => ( ! [I4: complex] :
            ( pp(aa_set_complex_bool(member_complex(I4),I6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I4)),aa_complex_nat(G,I4))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I6))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5231_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I6: set_int,G: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(F,I6) = groups4541462559716669496nt_nat(G,I6) )
     => ( ! [I4: int] :
            ( pp(aa_set_int_bool(member_int(I4),I6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I4)),aa_int_nat(G,I4))) )
       => ( pp(aa_set_int_bool(member_int(I),I6))
         => ( pp(aa_set_int_bool(finite_finite_int,I6))
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5232_sum__mono__inv,axiom,
    ! [F: fun_Ex7660298398204468809at_nat,I6: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,I: extended_enat] :
      ( ( groups2027974829824023292at_nat(F,I6) = groups2027974829824023292at_nat(G,I6) )
     => ( ! [I4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,I4)),aa_Extended_enat_nat(G,I4))) )
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I6))
         => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I6))
           => ( aa_Extended_enat_nat(F,I) = aa_Extended_enat_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5233_sum__mono__inv,axiom,
    ! [F: fun_real_int,I6: set_real,G: fun_real_int,I: real] :
      ( ( groups1932886352136224148al_int(F,I6) = groups1932886352136224148al_int(G,I6) )
     => ( ! [I4: real] :
            ( pp(aa_set_real_bool(member_real(I4),I6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I4)),aa_real_int(G,I4))) )
       => ( pp(aa_set_real_bool(member_real(I),I6))
         => ( pp(aa_set_real_bool(finite_finite_real,I6))
           => ( aa_real_int(F,I) = aa_real_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5234_sum__mono__inv,axiom,
    ! [F: fun_nat_int,I6: set_nat,G: fun_nat_int,I: nat] :
      ( ( aa_set_nat_int(groups3539618377306564664at_int(F),I6) = aa_set_nat_int(groups3539618377306564664at_int(G),I6) )
     => ( ! [I4: nat] :
            ( pp(aa_set_nat_bool(member_nat(I4),I6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I4)),aa_nat_int(G,I4))) )
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
           => ( aa_nat_int(F,I) = aa_nat_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5235_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_nat,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1935376822645274424al_nat(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bt(A2),P))) = groups1935376822645274424al_nat(aa_fun1504556115071737857al_nat(aTP_Lamm_jj(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5236_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_nat,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5693394587270226106ex_nat(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_jk(A2),P))) = groups5693394587270226106ex_nat(aa_fun9038227978352879617ex_nat(aTP_Lamm_jl(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5237_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_nat,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups4541462559716669496nt_nat(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bx(A2),P))) = groups4541462559716669496nt_nat(aa_fun7908831221058585985nt_nat(aTP_Lamm_jm(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5238_sum_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2027974829824023292at_nat(G,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_bs(A2),P))) = groups2027974829824023292at_nat(aa_fun943526657271735063at_nat(aTP_Lamm_jn(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5239_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bt(A2),P))) = groups8097168146408367636l_real(aa_fun6348708079608492509l_real(aTP_Lamm_jo(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5240_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_jk(A2),P))) = groups5808333547571424918x_real(aa_fun2902006202505214429x_real(aTP_Lamm_jp(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5241_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bx(A2),P))) = groups8778361861064173332t_real(aa_fun6974987029372589917t_real(aTP_Lamm_jq(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5242_sum_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups4148127829035722712t_real(G,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_bs(A2),P))) = groups4148127829035722712t_real(aa_fun1090353380788292083t_real(aTP_Lamm_jr(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5243_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_int,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1932886352136224148al_int(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bt(A2),P))) = groups1932886352136224148al_int(aa_fun6550077132417316957al_int(aTP_Lamm_js(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5244_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_int,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bw(A2),P))) = aa_set_nat_int(groups3539618377306564664at_int(aa_fun4986539647837908701at_int(aTP_Lamm_jt(G),P)),A2) ) ) ).

% sum.inter_filter
tff(fact_5245_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_re6065469302365115155d_enat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,X4))) )
       => ( ( groups2800946370649118462d_enat(F,A2) = zero_z5237406670263579293d_enat )
        <=> ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A2))
             => ( aa_rea1194391308913012860d_enat(F,X5) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5246_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: fun_na8797796813480665199d_enat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,X4))) )
       => ( ( groups7108830773950497114d_enat(F,A2) = zero_z5237406670263579293d_enat )
        <=> ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),A2))
             => ( aa_nat_Extended_enat(F,X5) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5247_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,X4))) )
       => ( ( groups1752964319039525884d_enat(F,A2) = zero_z5237406670263579293d_enat )
        <=> ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A2))
             => ( aa_com3683005661447327866d_enat(F,X5) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5248_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_in5914218760682845459d_enat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,X4))) )
       => ( ( groups4225252721152677374d_enat(F,A2) = zero_z5237406670263579293d_enat )
        <=> ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A2))
             => ( aa_int_Extended_enat(F,X5) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5249_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,X4))) )
       => ( ( groups2433450451889696826d_enat(F,A2) = zero_z5237406670263579293d_enat )
        <=> ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
             => ( aa_Ext7969882492699152700d_enat(F,X5) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5250_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( ( groups8097168146408367636l_real(F,A2) = zero_zero_real )
        <=> ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A2))
             => ( aa_real_real(F,X5) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5251_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
       => ( ( groups5808333547571424918x_real(F,A2) = zero_zero_real )
        <=> ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A2))
             => ( aa_complex_real(F,X5) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5252_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
       => ( ( groups8778361861064173332t_real(F,A2) = zero_zero_real )
        <=> ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A2))
             => ( aa_int_real(F,X5) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5253_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X4))) )
       => ( ( groups4148127829035722712t_real(F,A2) = zero_zero_real )
        <=> ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
             => ( aa_Ext2541572767299617110t_real(F,X5) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5254_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
       => ( ( groups1935376822645274424al_nat(F,A2) = zero_zero_nat )
        <=> ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A2))
             => ( aa_real_nat(F,X5) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5255_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: fun_na8797796813480665199d_enat,I: fun_nat_nat,F: fun_na8797796813480665199d_enat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),T))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(G,X4))) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_nat(I,Xa) = X4 )
                    & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(G,Xa))) ) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups7108830773950497114d_enat(F,S)),groups7108830773950497114d_enat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5256_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: fun_co6496365337113880337d_enat,I: fun_complex_nat,F: fun_na8797796813480665199d_enat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(G,X4))) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_nat(I,Xa) = X4 )
                    & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),aa_com3683005661447327866d_enat(G,Xa))) ) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups7108830773950497114d_enat(F,S)),groups1752964319039525884d_enat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5257_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: fun_in5914218760682845459d_enat,I: fun_int_nat,F: fun_na8797796813480665199d_enat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(G,X4))) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_nat(I,Xa) = X4 )
                    & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),aa_int_Extended_enat(G,Xa))) ) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups7108830773950497114d_enat(F,S)),groups4225252721152677374d_enat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5258_sum__le__included,axiom,
    ! [S: set_nat,T: set_Extended_enat,G: fun_Ex5852502975177633731d_enat,I: fun_Ex7660298398204468809at_nat,F: fun_na8797796813480665199d_enat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),T))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(G,X4))) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => ? [Xa: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),T))
                    & ( aa_Extended_enat_nat(I,Xa) = X4 )
                    & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),aa_Ext7969882492699152700d_enat(G,Xa))) ) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups7108830773950497114d_enat(F,S)),groups2433450451889696826d_enat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5259_sum__le__included,axiom,
    ! [S: set_complex,T: set_nat,G: fun_na8797796813480665199d_enat,I: fun_nat_complex,F: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),T))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_complex(I,Xa) = X4 )
                    & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_com3683005661447327866d_enat(F,X4)),aa_nat_Extended_enat(G,Xa))) ) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups1752964319039525884d_enat(F,S)),groups7108830773950497114d_enat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5260_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: fun_co6496365337113880337d_enat,I: fun_complex_complex,F: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_complex(I,Xa) = X4 )
                    & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_com3683005661447327866d_enat(F,X4)),aa_com3683005661447327866d_enat(G,Xa))) ) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups1752964319039525884d_enat(F,S)),groups1752964319039525884d_enat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5261_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: fun_in5914218760682845459d_enat,I: fun_int_complex,F: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_complex(I,Xa) = X4 )
                    & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_com3683005661447327866d_enat(F,X4)),aa_int_Extended_enat(G,Xa))) ) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups1752964319039525884d_enat(F,S)),groups4225252721152677374d_enat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5262_sum__le__included,axiom,
    ! [S: set_complex,T: set_Extended_enat,G: fun_Ex5852502975177633731d_enat,I: fun_Ex3741802288643809319omplex,F: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),T))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ? [Xa: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),T))
                    & ( aa_Ext8748583412541189208omplex(I,Xa) = X4 )
                    & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_com3683005661447327866d_enat(F,X4)),aa_Ext7969882492699152700d_enat(G,Xa))) ) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups1752964319039525884d_enat(F,S)),groups2433450451889696826d_enat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5263_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: fun_na8797796813480665199d_enat,I: fun_nat_int,F: fun_in5914218760682845459d_enat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),T))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_int(I,Xa) = X4 )
                    & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,X4)),aa_nat_Extended_enat(G,Xa))) ) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups4225252721152677374d_enat(F,S)),groups7108830773950497114d_enat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5264_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: fun_co6496365337113880337d_enat,I: fun_complex_int,F: fun_in5914218760682845459d_enat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_int(I,Xa) = X4 )
                    & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,X4)),aa_com3683005661447327866d_enat(G,Xa))) ) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups4225252721152677374d_enat(F,S)),groups1752964319039525884d_enat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5265_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,X4))) )
       => ( ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),A2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X2)),aa_complex_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5266_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,X4))) )
       => ( ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),A2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X2)),aa_int_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5267_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(G,X4))) )
       => ( ? [X2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X2)),aa_Ext2541572767299617110t_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups4148127829035722712t_real(F,A2)),groups4148127829035722712t_real(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5268_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4))) )
       => ( ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X2)),aa_complex_nat(G,X2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5269_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4))) )
       => ( ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(G,X2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5270_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(G,X4))) )
       => ( ? [X2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,X2)),aa_Extended_enat_nat(G,X2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups2027974829824023292at_nat(F,A2)),groups2027974829824023292at_nat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5271_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(G,X4))) )
       => ( ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,X2)),aa_nat_int(G,X2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),A2)),aa_set_nat_int(groups3539618377306564664at_int(G),A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5272_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F,X4)),aa_complex_int(G,X4))) )
       => ( ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_complex_int(F,X2)),aa_complex_int(G,X2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5273_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Extended_enat_int(F,X4)),aa_Extended_enat_int(G,X4))) )
       => ( ? [X2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Extended_enat_int(F,X2)),aa_Extended_enat_int(G,X2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups2025484359314973016at_int(F,A2)),groups2025484359314973016at_int(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5274_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(G,X4))) )
       => ( ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(F,X2)),aa_int_int(G,X2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5275_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,zero_zero_nat),zero_zero_nat))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y22)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y22))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_complex_nat(H,X4)),aa_complex_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups5693394587270226106ex_nat(H,S2)),groups5693394587270226106ex_nat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5276_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,zero_zero_nat),zero_zero_nat))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y22)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y22))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_int_nat(H,X4)),aa_int_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups4541462559716669496nt_nat(H,S2)),groups4541462559716669496nt_nat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5277_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S2: set_Extended_enat,H: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,zero_zero_nat),zero_zero_nat))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y22)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y22))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_Extended_enat_nat(H,X4)),aa_Extended_enat_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups2027974829824023292at_nat(H,S2)),groups2027974829824023292at_nat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5278_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(plus_plus_real(X1),Y1)),aa_real_real(plus_plus_real(X23),Y22))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_complex_real(H,X4)),aa_complex_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups5808333547571424918x_real(H,S2)),groups5808333547571424918x_real(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5279_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(plus_plus_real(X1),Y1)),aa_real_real(plus_plus_real(X23),Y22))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_int_real(H,X4)),aa_int_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups8778361861064173332t_real(H,S2)),groups8778361861064173332t_real(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5280_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(plus_plus_real(X1),Y1)),aa_real_real(plus_plus_real(X23),Y22))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_Ext2541572767299617110t_real(H,X4)),aa_Ext2541572767299617110t_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups4148127829035722712t_real(H,S2)),groups4148127829035722712t_real(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5281_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_bool,S2: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(R,zero_zero_int),zero_zero_int))
     => ( ! [X1: int,Y1: int,X23: int,Y22: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(R,X1),X23))
              & pp(aa_int_bool(aa_int_fun_int_bool(R,Y1),Y22)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X1),Y1)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X23),Y22))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_nat_int(H,X4)),aa_nat_int(G,X4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_set_nat_int(groups3539618377306564664at_int(H),S2)),aa_set_nat_int(groups3539618377306564664at_int(G),S2))) ) ) ) ) ).

% sum.related
tff(fact_5282_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_bool,S2: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(R,zero_zero_int),zero_zero_int))
     => ( ! [X1: int,Y1: int,X23: int,Y22: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(R,X1),X23))
              & pp(aa_int_bool(aa_int_fun_int_bool(R,Y1),Y22)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X1),Y1)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X23),Y22))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_complex_int(H,X4)),aa_complex_int(G,X4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,groups5690904116761175830ex_int(H,S2)),groups5690904116761175830ex_int(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5283_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_bool,S2: set_Extended_enat,H: fun_Ex3482447378695272101at_int,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(R,zero_zero_int),zero_zero_int))
     => ( ! [X1: int,Y1: int,X23: int,Y22: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(R,X1),X23))
              & pp(aa_int_bool(aa_int_fun_int_bool(R,Y1),Y22)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X1),Y1)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X23),Y22))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_Extended_enat_int(H,X4)),aa_Extended_enat_int(G,X4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,groups2025484359314973016at_int(H,S2)),groups2025484359314973016at_int(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5284_sum_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,zero_zero_complex),zero_zero_complex))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y22)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(plus_plus_complex(X1),Y1)),aa_complex_complex(plus_plus_complex(X23),Y22))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_nat_complex(H,X4)),aa_nat_complex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups2073611262835488442omplex(H,S2)),groups2073611262835488442omplex(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5285_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_5286_sum__strict__mono,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(G,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups2027974829824023292at_nat(F,A2)),groups2027974829824023292at_nat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_5287_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(G,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_5288_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_5289_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_co6496365337113880337d_enat,G: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_com3683005661447327866d_enat(F,X4)),aa_com3683005661447327866d_enat(G,X4))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,groups1752964319039525884d_enat(F,A2)),groups1752964319039525884d_enat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_5290_sum__strict__mono,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex5852502975177633731d_enat,G: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(G,X4))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,groups2433450451889696826d_enat(F,A2)),groups2433450451889696826d_enat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_5291_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_re6065469302365115155d_enat,G: fun_re6065469302365115155d_enat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(G,X4))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,groups2800946370649118462d_enat(F,A2)),groups2800946370649118462d_enat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_5292_sum__strict__mono,axiom,
    ! [A2: set_nat,F: fun_na8797796813480665199d_enat,G: fun_na8797796813480665199d_enat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(G,X4))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,groups7108830773950497114d_enat(F,A2)),groups7108830773950497114d_enat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_5293_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_in5914218760682845459d_enat,G: fun_in5914218760682845459d_enat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,X4)),aa_int_Extended_enat(G,X4))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,groups4225252721152677374d_enat(F,A2)),groups4225252721152677374d_enat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_5294_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X4)),aa_complex_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_5295_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = groups1935376822645274424al_nat(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5296_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups5693394587270226106ex_nat(G,insert_complex(X,A2)) = groups5693394587270226106ex_nat(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups5693394587270226106ex_nat(G,insert_complex(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5297_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(X),A2))
         => ( groups4541462559716669496nt_nat(G,insert_int(X,A2)) = groups4541462559716669496nt_nat(G,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A2))
         => ( groups4541462559716669496nt_nat(G,insert_int(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,X)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5298_sum_Oinsert__if,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
         => ( groups2027974829824023292at_nat(G,insert_Extended_enat(X,A2)) = groups2027974829824023292at_nat(G,A2) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
         => ( groups2027974829824023292at_nat(G,insert_Extended_enat(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(G,X)),groups2027974829824023292at_nat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5299_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1932886352136224148al_int(G,insert_real(X,A2)) = groups1932886352136224148al_int(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1932886352136224148al_int(G,insert_real(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X)),groups1932886352136224148al_int(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5300_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A2))
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),insert_nat(X,A2)) = aa_set_nat_int(groups3539618377306564664at_int(G),A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),insert_nat(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,X)),aa_set_nat_int(groups3539618377306564664at_int(G),A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5301_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups5690904116761175830ex_int(G,insert_complex(X,A2)) = groups5690904116761175830ex_int(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups5690904116761175830ex_int(G,insert_complex(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5302_sum_Oinsert__if,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
         => ( groups2025484359314973016at_int(G,insert_Extended_enat(X,A2)) = groups2025484359314973016at_int(G,A2) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
         => ( groups2025484359314973016at_int(G,insert_Extended_enat(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Extended_enat_int(G,X)),groups2025484359314973016at_int(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5303_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups8097168146408367636l_real(G,insert_real(X,A2)) = groups8097168146408367636l_real(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups8097168146408367636l_real(G,insert_real(X,A2)) = aa_real_real(plus_plus_real(aa_real_real(G,X)),groups8097168146408367636l_real(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5304_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = groups5808333547571424918x_real(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(plus_plus_real(aa_complex_real(G,X)),groups5808333547571424918x_real(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5305_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T3: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T5))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A3)),minus_minus_set_real(T3,T5))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T5)))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T5)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T5))
                       => ( aa_real_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_real_nat(H,aa_real_real(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups1935376822645274424al_nat(G,S2) = groups1935376822645274424al_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5306_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T3: set_complex,G: fun_real_nat,H: fun_complex_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T3,T5))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T5)))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T5)))
                   => pp(aa_set_real_bool(member_real(aa_complex_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T5))
                       => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_complex_nat(H,aa_real_complex(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups1935376822645274424al_nat(G,S2) = groups5693394587270226106ex_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5307_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T3: set_int,G: fun_real_nat,H: fun_int_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T5))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T3,T5))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T5)))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T5)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T5))
                       => ( aa_int_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_int_nat(H,aa_real_int(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups1935376822645274424al_nat(G,S2) = groups4541462559716669496nt_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5308_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_Extended_enat,S2: set_real,I: fun_Ex1451620480498224165t_real,J: fun_re6065469302365115155d_enat,T3: set_Extended_enat,G: fun_real_nat,H: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T5))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_Ext2541572767299617110t_real(I,aa_rea1194391308913012860d_enat(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set3347977610044113549t_bool(member_Extended_enat(aa_rea1194391308913012860d_enat(J,A3)),minus_925952699566721837d_enat(T3,T5))) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T3,T5)))
                 => ( aa_rea1194391308913012860d_enat(J,aa_Ext2541572767299617110t_real(I,B3)) = B3 ) )
             => ( ! [B3: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T3,T5)))
                   => pp(aa_set_real_bool(member_real(aa_Ext2541572767299617110t_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),T5))
                       => ( aa_Extended_enat_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_Extended_enat_nat(H,aa_rea1194391308913012860d_enat(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups1935376822645274424al_nat(G,S2) = groups2027974829824023292at_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5309_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_real,S2: set_complex,I: fun_real_complex,J: fun_complex_real,T3: set_real,G: fun_complex_nat,H: fun_real_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T5))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_real_complex(I,aa_complex_real(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_complex_real(J,A3)),minus_minus_set_real(T3,T5))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T5)))
                 => ( aa_complex_real(J,aa_real_complex(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T5)))
                   => pp(aa_set_complex_bool(member_complex(aa_real_complex(I,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T5))
                       => ( aa_real_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_real_nat(H,aa_complex_real(J,A3)) = aa_complex_nat(G,A3) ) )
                     => ( groups5693394587270226106ex_nat(G,S2) = groups1935376822645274424al_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5310_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_complex,S2: set_complex,I: fun_complex_complex,J: fun_complex_complex,T3: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_complex_complex(I,aa_complex_complex(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_complex_complex(J,A3)),minus_811609699411566653omplex(T3,T5))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T5)))
                 => ( aa_complex_complex(J,aa_complex_complex(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T5)))
                   => pp(aa_set_complex_bool(member_complex(aa_complex_complex(I,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T5))
                       => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_complex_nat(H,aa_complex_complex(J,A3)) = aa_complex_nat(G,A3) ) )
                     => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5311_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_int,S2: set_complex,I: fun_int_complex,J: fun_complex_int,T3: set_int,G: fun_complex_nat,H: fun_int_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T5))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_int_complex(I,aa_complex_int(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_complex_int(J,A3)),minus_minus_set_int(T3,T5))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T5)))
                 => ( aa_complex_int(J,aa_int_complex(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T5)))
                   => pp(aa_set_complex_bool(member_complex(aa_int_complex(I,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T5))
                       => ( aa_int_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_int_nat(H,aa_complex_int(J,A3)) = aa_complex_nat(G,A3) ) )
                     => ( groups5693394587270226106ex_nat(G,S2) = groups4541462559716669496nt_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5312_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_Extended_enat,S2: set_complex,I: fun_Ex3741802288643809319omplex,J: fun_co6496365337113880337d_enat,T3: set_Extended_enat,G: fun_complex_nat,H: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T5))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_Ext8748583412541189208omplex(I,aa_com3683005661447327866d_enat(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set3347977610044113549t_bool(member_Extended_enat(aa_com3683005661447327866d_enat(J,A3)),minus_925952699566721837d_enat(T3,T5))) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T3,T5)))
                 => ( aa_com3683005661447327866d_enat(J,aa_Ext8748583412541189208omplex(I,B3)) = B3 ) )
             => ( ! [B3: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T3,T5)))
                   => pp(aa_set_complex_bool(member_complex(aa_Ext8748583412541189208omplex(I,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),T5))
                       => ( aa_Extended_enat_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_Extended_enat_nat(H,aa_com3683005661447327866d_enat(J,A3)) = aa_complex_nat(G,A3) ) )
                     => ( groups5693394587270226106ex_nat(G,S2) = groups2027974829824023292at_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5313_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T3: set_real,G: fun_int_nat,H: fun_real_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T5))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_int_real(J,A3)),minus_minus_set_real(T3,T5))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T5)))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T5)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T5))
                       => ( aa_real_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_real_nat(H,aa_int_real(J,A3)) = aa_int_nat(G,A3) ) )
                     => ( groups4541462559716669496nt_nat(G,S2) = groups1935376822645274424al_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5314_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_complex,S2: set_int,I: fun_complex_int,J: fun_int_complex,T3: set_complex,G: fun_int_nat,H: fun_complex_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_complex_int(I,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_int_complex(J,A3)),minus_811609699411566653omplex(T3,T5))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T5)))
                 => ( aa_int_complex(J,aa_complex_int(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T5)))
                   => pp(aa_set_int_bool(member_int(aa_complex_int(I,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T5))
                       => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_complex_nat(H,aa_int_complex(J,A3)) = aa_int_nat(G,A3) ) )
                     => ( groups4541462559716669496nt_nat(G,S2) = groups5693394587270226106ex_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5315_round__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,X)),aa_real_int(archim8280529875227126926d_real,Y))) ) ).

% round_mono
tff(fact_5316_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_re6065469302365115155d_enat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I4: real] :
            ( pp(aa_set_real_bool(member_real(I4),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,I4))) )
       => ( ( groups2800946370649118462d_enat(F,S) = zero_z5237406670263579293d_enat )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => ( aa_rea1194391308913012860d_enat(F,I) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5317_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_na8797796813480665199d_enat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ! [I4: nat] :
            ( pp(aa_set_nat_bool(member_nat(I4),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,I4))) )
       => ( ( groups7108830773950497114d_enat(F,S) = zero_z5237406670263579293d_enat )
         => ( pp(aa_set_nat_bool(member_nat(I),S))
           => ( aa_nat_Extended_enat(F,I) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5318_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_co6496365337113880337d_enat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I4: complex] :
            ( pp(aa_set_complex_bool(member_complex(I4),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,I4))) )
       => ( ( groups1752964319039525884d_enat(F,S) = zero_z5237406670263579293d_enat )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => ( aa_com3683005661447327866d_enat(F,I) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5319_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_in5914218760682845459d_enat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I4: int] :
            ( pp(aa_set_int_bool(member_int(I4),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,I4))) )
       => ( ( groups4225252721152677374d_enat(F,S) = zero_z5237406670263579293d_enat )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => ( aa_int_Extended_enat(F,I) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5320_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: fun_Ex5852502975177633731d_enat,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ! [I4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,I4))) )
       => ( ( groups2433450451889696826d_enat(F,S) = zero_z5237406670263579293d_enat )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S))
           => ( aa_Ext7969882492699152700d_enat(F,I) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5321_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I4: real] :
            ( pp(aa_set_real_bool(member_real(I4),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I4))) )
       => ( ( groups8097168146408367636l_real(F,S) = zero_zero_real )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5322_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I4: complex] :
            ( pp(aa_set_complex_bool(member_complex(I4),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I4))) )
       => ( ( groups5808333547571424918x_real(F,S) = zero_zero_real )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5323_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I4: int] :
            ( pp(aa_set_int_bool(member_int(I4),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I4))) )
       => ( ( groups8778361861064173332t_real(F,S) = zero_zero_real )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5324_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: fun_Ex1451620480498224165t_real,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ! [I4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I4))) )
       => ( ( groups4148127829035722712t_real(F,S) = zero_zero_real )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S))
           => ( aa_Ext2541572767299617110t_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5325_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_nat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I4: real] :
            ( pp(aa_set_real_bool(member_real(I4),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I4))) )
       => ( ( groups1935376822645274424al_nat(F,S) = zero_zero_nat )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => ( aa_real_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5326_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_re6065469302365115155d_enat,B: extended_enat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I4: real] :
            ( pp(aa_set_real_bool(member_real(I4),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,I4))) )
       => ( ( groups2800946370649118462d_enat(F,S) = B )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5327_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_na8797796813480665199d_enat,B: extended_enat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ! [I4: nat] :
            ( pp(aa_set_nat_bool(member_nat(I4),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,I4))) )
       => ( ( groups7108830773950497114d_enat(F,S) = B )
         => ( pp(aa_set_nat_bool(member_nat(I),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5328_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_co6496365337113880337d_enat,B: extended_enat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I4: complex] :
            ( pp(aa_set_complex_bool(member_complex(I4),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,I4))) )
       => ( ( groups1752964319039525884d_enat(F,S) = B )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_com3683005661447327866d_enat(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5329_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_in5914218760682845459d_enat,B: extended_enat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I4: int] :
            ( pp(aa_set_int_bool(member_int(I4),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,I4))) )
       => ( ( groups4225252721152677374d_enat(F,S) = B )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5330_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: fun_Ex5852502975177633731d_enat,B: extended_enat,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ! [I4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,I4))) )
       => ( ( groups2433450451889696826d_enat(F,S) = B )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5331_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_real,B: real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I4: real] :
            ( pp(aa_set_real_bool(member_real(I4),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I4))) )
       => ( ( groups8097168146408367636l_real(F,S) = B )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5332_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_real,B: real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I4: complex] :
            ( pp(aa_set_complex_bool(member_complex(I4),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I4))) )
       => ( ( groups5808333547571424918x_real(F,S) = B )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5333_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_real,B: real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I4: int] :
            ( pp(aa_set_int_bool(member_int(I4),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I4))) )
       => ( ( groups8778361861064173332t_real(F,S) = B )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5334_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: fun_Ex1451620480498224165t_real,B: real,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ! [I4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I4))) )
       => ( ( groups4148127829035722712t_real(F,S) = B )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5335_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_nat,B: nat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I4: real] :
            ( pp(aa_set_real_bool(member_real(I4),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I4))) )
       => ( ( groups1935376822645274424al_nat(F,S) = B )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5336_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1935376822645274424al_nat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_ju(G)))) = groups1935376822645274424al_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5337_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_jv(G)))) = groups5693394587270226106ex_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5338_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_jw(G)))) = groups4541462559716669496nt_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5339_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2027974829824023292at_nat(G,minus_925952699566721837d_enat(A2,collec4429806609662206161d_enat(aTP_Lamm_jx(G)))) = groups2027974829824023292at_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5340_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups8097168146408367636l_real(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_jy(G)))) = groups8097168146408367636l_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5341_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_jz(G)))) = groups5808333547571424918x_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5342_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ka(G)))) = groups8778361861064173332t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5343_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups4148127829035722712t_real(G,minus_925952699566721837d_enat(A2,collec4429806609662206161d_enat(aTP_Lamm_kb(G)))) = groups4148127829035722712t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5344_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1932886352136224148al_int(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_kc(G)))) = groups1932886352136224148al_int(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5345_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_kd(G)))) = groups5690904116761175830ex_int(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5346_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_re6065469302365115155d_enat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,I)))
         => ( ! [I4: real] :
                ( pp(aa_set_real_bool(member_real(I4),I6))
               => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,I4))) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups2800946370649118462d_enat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5347_sum__pos2,axiom,
    ! [I6: set_nat,I: nat,F: fun_na8797796813480665199d_enat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( pp(aa_set_nat_bool(member_nat(I),I6))
       => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,I)))
         => ( ! [I4: nat] :
                ( pp(aa_set_nat_bool(member_nat(I4),I6))
               => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,I4))) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups7108830773950497114d_enat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5348_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I),I6))
       => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,I)))
         => ( ! [I4: complex] :
                ( pp(aa_set_complex_bool(member_complex(I4),I6))
               => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,I4))) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups1752964319039525884d_enat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5349_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_in5914218760682845459d_enat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I),I6))
       => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,I)))
         => ( ! [I4: int] :
                ( pp(aa_set_int_bool(member_int(I4),I6))
               => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,I4))) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups4225252721152677374d_enat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5350_sum__pos2,axiom,
    ! [I6: set_Extended_enat,I: extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I6))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I6))
       => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,I)))
         => ( ! [I4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
               => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,I4))) )
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups2433450451889696826d_enat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5351_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I)))
         => ( ! [I4: real] :
                ( pp(aa_set_real_bool(member_real(I4),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5352_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I)))
         => ( ! [I4: complex] :
                ( pp(aa_set_complex_bool(member_complex(I4),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5353_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I)))
         => ( ! [I4: int] :
                ( pp(aa_set_int_bool(member_int(I4),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5354_sum__pos2,axiom,
    ! [I6: set_Extended_enat,I: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I6))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I)))
         => ( ! [I4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups4148127829035722712t_real(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5355_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,I)))
         => ( ! [I4: real] :
                ( pp(aa_set_real_bool(member_real(I4),I6))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5356_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( pp(aa_set_complex_bool(member_complex(I4),I6))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5357_sum__pos,axiom,
    ! [I6: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I6))
     => ( ( I6 != bot_bo7653980558646680370d_enat )
       => ( ! [I4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_Extended_enat_nat(F,I4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups2027974829824023292at_nat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5358_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( pp(aa_set_real_bool(member_real(I4),I6))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,I4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5359_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I4: int] :
              ( pp(aa_set_int_bool(member_int(I4),I6))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,I4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5360_sum__pos,axiom,
    ! [I6: set_complex,F: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( pp(aa_set_complex_bool(member_complex(I4),I6))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,I4))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups1752964319039525884d_enat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5361_sum__pos,axiom,
    ! [I6: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I6))
     => ( ( I6 != bot_bo7653980558646680370d_enat )
       => ( ! [I4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,I4))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups2433450451889696826d_enat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5362_sum__pos,axiom,
    ! [I6: set_real,F: fun_re6065469302365115155d_enat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( pp(aa_set_real_bool(member_real(I4),I6))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,I4))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups2800946370649118462d_enat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5363_sum__pos,axiom,
    ! [I6: set_nat,F: fun_na8797796813480665199d_enat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I4: nat] :
              ( pp(aa_set_nat_bool(member_nat(I4),I6))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,I4))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups7108830773950497114d_enat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5364_sum__pos,axiom,
    ! [I6: set_int,F: fun_in5914218760682845459d_enat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I4: int] :
              ( pp(aa_set_int_bool(member_int(I4),I6))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,I4))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups4225252721152677374d_enat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5365_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( pp(aa_set_complex_bool(member_complex(I4),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5366_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups1935376822645274424al_nat(G,T3) = groups1935376822645274424al_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5367_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups5693394587270226106ex_nat(G,T3) = groups5693394587270226106ex_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5368_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Extended_enat_nat(G,X4) = aa_Extended_enat_nat(H,X4) ) )
           => ( groups2027974829824023292at_nat(G,T3) = groups2027974829824023292at_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5369_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,T3) = groups8097168146408367636l_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5370_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,T3) = groups5808333547571424918x_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5371_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Ext2541572767299617110t_real(G,X4) = aa_Ext2541572767299617110t_real(H,X4) ) )
           => ( groups4148127829035722712t_real(G,T3) = groups4148127829035722712t_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5372_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_int(G,X4) = zero_zero_int ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
           => ( groups1932886352136224148al_int(G,T3) = groups1932886352136224148al_int(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5373_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_int(G,X4) = aa_complex_int(H,X4) ) )
           => ( groups5690904116761175830ex_int(G,T3) = groups5690904116761175830ex_int(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5374_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex3482447378695272101at_int,H: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_int(G,X4) = zero_zero_int ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Extended_enat_int(G,X4) = aa_Extended_enat_int(H,X4) ) )
           => ( groups2025484359314973016at_int(G,T3) = groups2025484359314973016at_int(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5375_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_complex(G,X4) = zero_zero_complex ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups5754745047067104278omplex(G,T3) = groups5754745047067104278omplex(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5376_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups1935376822645274424al_nat(G,S2) = groups1935376822645274424al_nat(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5377_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5378_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Extended_enat_nat(G,X4) = aa_Extended_enat_nat(H,X4) ) )
           => ( groups2027974829824023292at_nat(G,S2) = groups2027974829824023292at_nat(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5379_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5380_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5381_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext2541572767299617110t_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Ext2541572767299617110t_real(G,X4) = aa_Ext2541572767299617110t_real(H,X4) ) )
           => ( groups4148127829035722712t_real(G,S2) = groups4148127829035722712t_real(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5382_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_int,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_int(H,X4) = zero_zero_int ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
           => ( groups1932886352136224148al_int(G,S2) = groups1932886352136224148al_int(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5383_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_int(H,X4) = zero_zero_int ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_int(G,X4) = aa_complex_int(H,X4) ) )
           => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5384_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H: fun_Ex3482447378695272101at_int,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_int(H,X4) = zero_zero_int ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Extended_enat_int(G,X4) = aa_Extended_enat_int(H,X4) ) )
           => ( groups2025484359314973016at_int(G,S2) = groups2025484359314973016at_int(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5385_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_complex(H,X4) = zero_zero_complex ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5386_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T3) = groups5693394587270226106ex_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5387_sum_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_nat(G,X4) = zero_zero_nat ) )
         => ( groups2027974829824023292at_nat(G,T3) = groups2027974829824023292at_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5388_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T3) = groups5808333547571424918x_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5389_sum_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = zero_zero_real ) )
         => ( groups4148127829035722712t_real(G,T3) = groups4148127829035722712t_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5390_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T3) = groups5690904116761175830ex_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5391_sum_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_int(G,X4) = zero_zero_int ) )
         => ( groups2025484359314973016at_int(G,T3) = groups2025484359314973016at_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5392_sum_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext8748583412541189208omplex(G,X4) = zero_zero_complex ) )
         => ( groups6818542070133387226omplex(G,T3) = groups6818542070133387226omplex(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5393_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_com3683005661447327866d_enat(G,X4) = zero_z5237406670263579293d_enat ) )
         => ( groups1752964319039525884d_enat(G,T3) = groups1752964319039525884d_enat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5394_sum_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext7969882492699152700d_enat(G,X4) = zero_z5237406670263579293d_enat ) )
         => ( groups2433450451889696826d_enat(G,T3) = groups2433450451889696826d_enat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5395_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_int(G,X4) = zero_zero_int ) )
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),T3) = aa_set_nat_int(groups3539618377306564664at_int(G),S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5396_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5397_sum_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_nat(G,X4) = zero_zero_nat ) )
         => ( groups2027974829824023292at_nat(G,S2) = groups2027974829824023292at_nat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5398_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5399_sum_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = zero_zero_real ) )
         => ( groups4148127829035722712t_real(G,S2) = groups4148127829035722712t_real(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5400_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5401_sum_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_int(G,X4) = zero_zero_int ) )
         => ( groups2025484359314973016at_int(G,S2) = groups2025484359314973016at_int(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5402_sum_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext8748583412541189208omplex(G,X4) = zero_zero_complex ) )
         => ( groups6818542070133387226omplex(G,S2) = groups6818542070133387226omplex(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5403_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_com3683005661447327866d_enat(G,X4) = zero_z5237406670263579293d_enat ) )
         => ( groups1752964319039525884d_enat(G,S2) = groups1752964319039525884d_enat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5404_sum_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext7969882492699152700d_enat(G,X4) = zero_z5237406670263579293d_enat ) )
         => ( groups2433450451889696826d_enat(G,S2) = groups2433450451889696826d_enat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5405_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_int(G,X4) = zero_zero_int ) )
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),S2) = aa_set_nat_int(groups3539618377306564664at_int(G),T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5406_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) )
               => ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5407_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B)))
                 => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,C4) = groups5693394587270226106ex_nat(H,C4) )
               => ( groups5693394587270226106ex_nat(G,A2) = groups5693394587270226106ex_nat(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5408_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Extended_enat_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B)))
                 => ( aa_Extended_enat_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups2027974829824023292at_nat(G,C4) = groups2027974829824023292at_nat(H,C4) )
               => ( groups2027974829824023292at_nat(G,A2) = groups2027974829824023292at_nat(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5409_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) )
               => ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5410_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) )
               => ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5411_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Ext2541572767299617110t_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B)))
                 => ( aa_Ext2541572767299617110t_real(H,B3) = zero_zero_real ) )
             => ( ( groups4148127829035722712t_real(G,C4) = groups4148127829035722712t_real(H,C4) )
               => ( groups4148127829035722712t_real(G,A2) = groups4148127829035722712t_real(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5412_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_int(H,B3) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,C4) = groups1932886352136224148al_int(H,C4) )
               => ( groups1932886352136224148al_int(G,A2) = groups1932886352136224148al_int(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5413_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B)))
                 => ( aa_complex_int(H,B3) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,C4) = groups5690904116761175830ex_int(H,C4) )
               => ( groups5690904116761175830ex_int(G,A2) = groups5690904116761175830ex_int(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5414_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: fun_Ex3482447378695272101at_int,H: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Extended_enat_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B)))
                 => ( aa_Extended_enat_int(H,B3) = zero_zero_int ) )
             => ( ( groups2025484359314973016at_int(G,C4) = groups2025484359314973016at_int(H,C4) )
               => ( groups2025484359314973016at_int(G,A2) = groups2025484359314973016at_int(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5415_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) )
               => ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5416_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B) )
              <=> ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5417_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B)))
                 => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,A2) = groups5693394587270226106ex_nat(H,B) )
              <=> ( groups5693394587270226106ex_nat(G,C4) = groups5693394587270226106ex_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5418_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Extended_enat_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B)))
                 => ( aa_Extended_enat_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups2027974829824023292at_nat(G,A2) = groups2027974829824023292at_nat(H,B) )
              <=> ( groups2027974829824023292at_nat(G,C4) = groups2027974829824023292at_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5419_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B) )
              <=> ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5420_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B) )
              <=> ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5421_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Ext2541572767299617110t_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B)))
                 => ( aa_Ext2541572767299617110t_real(H,B3) = zero_zero_real ) )
             => ( ( groups4148127829035722712t_real(G,A2) = groups4148127829035722712t_real(H,B) )
              <=> ( groups4148127829035722712t_real(G,C4) = groups4148127829035722712t_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5422_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_int(H,B3) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,A2) = groups1932886352136224148al_int(H,B) )
              <=> ( groups1932886352136224148al_int(G,C4) = groups1932886352136224148al_int(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5423_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B)))
                 => ( aa_complex_int(H,B3) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,A2) = groups5690904116761175830ex_int(H,B) )
              <=> ( groups5690904116761175830ex_int(G,C4) = groups5690904116761175830ex_int(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5424_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: fun_Ex3482447378695272101at_int,H: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Extended_enat_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B)))
                 => ( aa_Extended_enat_int(H,B3) = zero_zero_int ) )
             => ( ( groups2025484359314973016at_int(G,A2) = groups2025484359314973016at_int(H,B) )
              <=> ( groups2025484359314973016at_int(G,C4) = groups2025484359314973016at_int(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5425_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B) )
              <=> ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5426_sum_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,B))),groups5693394587270226106ex_nat(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_5427_sum_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
       => ( groups2027974829824023292at_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2027974829824023292at_nat(G,minus_925952699566721837d_enat(A2,B))),groups2027974829824023292at_nat(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_5428_sum_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,B))),groups5690904116761175830ex_int(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_5429_sum_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
       => ( groups2025484359314973016at_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups2025484359314973016at_int(G,minus_925952699566721837d_enat(A2,B))),groups2025484359314973016at_int(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_5430_sum_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(plus_plus_real(groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,B))),groups5808333547571424918x_real(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_5431_sum_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
       => ( groups4148127829035722712t_real(G,A2) = aa_real_real(plus_plus_real(groups4148127829035722712t_real(G,minus_925952699566721837d_enat(A2,B))),groups4148127829035722712t_real(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_5432_sum_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups1752964319039525884d_enat(G,A2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups1752964319039525884d_enat(G,minus_811609699411566653omplex(A2,B))),groups1752964319039525884d_enat(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_5433_sum_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
       => ( groups2433450451889696826d_enat(G,A2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups2433450451889696826d_enat(G,minus_925952699566721837d_enat(A2,B))),groups2433450451889696826d_enat(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_5434_sum_Osubset__diff,axiom,
    ! [B: set_nat,A2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),minus_minus_set_nat(A2,B))),aa_set_nat_int(groups3539618377306564664at_int(G),B)) ) ) ) ).

% sum.subset_diff
tff(fact_5435_sum_Osubset__diff,axiom,
    ! [B: set_nat,A2: set_nat,G: fun_na8797796813480665199d_enat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups7108830773950497114d_enat(G,A2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,minus_minus_set_nat(A2,B))),groups7108830773950497114d_enat(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_5436_sum__diff,axiom,
    ! [A2: set_complex,B: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A2))
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B)) ) ) ) ).

% sum_diff
tff(fact_5437_sum__diff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),A2))
       => ( groups2025484359314973016at_int(F,minus_925952699566721837d_enat(A2,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups2025484359314973016at_int(F,A2)),groups2025484359314973016at_int(F,B)) ) ) ) ).

% sum_diff
tff(fact_5438_sum__diff,axiom,
    ! [A2: set_complex,B: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A2))
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,B)) = aa_real_real(minus_minus_real(groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B)) ) ) ) ).

% sum_diff
tff(fact_5439_sum__diff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),A2))
       => ( groups4148127829035722712t_real(F,minus_925952699566721837d_enat(A2,B)) = aa_real_real(minus_minus_real(groups4148127829035722712t_real(F,A2)),groups4148127829035722712t_real(F,B)) ) ) ) ).

% sum_diff
tff(fact_5440_sum__diff,axiom,
    ! [A2: set_nat,B: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A2))
       => ( aa_set_nat_int(groups3539618377306564664at_int(F),minus_minus_set_nat(A2,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(F),A2)),aa_set_nat_int(groups3539618377306564664at_int(F),B)) ) ) ) ).

% sum_diff
tff(fact_5441_sum__diff,axiom,
    ! [A2: set_int,B: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A2))
       => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,B)) = aa_real_real(minus_minus_real(groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B)) ) ) ) ).

% sum_diff
tff(fact_5442_sum__diff,axiom,
    ! [A2: set_int,B: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A2))
       => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(F,B)) ) ) ) ).

% sum_diff
tff(fact_5443_sum__diff,axiom,
    ! [A2: set_complex,B: set_complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A2))
       => ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A2,B)) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(F,B)) ) ) ) ).

% sum_diff
tff(fact_5444_sum__diff,axiom,
    ! [A2: set_nat,B: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A2))
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),minus_minus_set_nat(A2,B)) = aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(F),A2)),aa_set_nat_real(groups6591440286371151544t_real(F),B)) ) ) ) ).

% sum_diff
tff(fact_5445_sum__mono2,axiom,
    ! [B: set_real,A2: set_real,F: fun_re6065469302365115155d_enat] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B,A2)))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,B3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups2800946370649118462d_enat(F,A2)),groups2800946370649118462d_enat(F,B))) ) ) ) ).

% sum_mono2
tff(fact_5446_sum__mono2,axiom,
    ! [B: set_complex,A2: set_complex,F: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B,A2)))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,B3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups1752964319039525884d_enat(F,A2)),groups1752964319039525884d_enat(F,B))) ) ) ) ).

% sum_mono2
tff(fact_5447_sum__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
       => ( ! [B3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B,A2)))
             => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,B3))) )
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups2433450451889696826d_enat(F,A2)),groups2433450451889696826d_enat(F,B))) ) ) ) ).

% sum_mono2
tff(fact_5448_sum__mono2,axiom,
    ! [B: set_real,A2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B))) ) ) ) ).

% sum_mono2
tff(fact_5449_sum__mono2,axiom,
    ! [B: set_complex,A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B))) ) ) ) ).

% sum_mono2
tff(fact_5450_sum__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
       => ( ! [B3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,A2)),groups4148127829035722712t_real(F,B))) ) ) ) ).

% sum_mono2
tff(fact_5451_sum__mono2,axiom,
    ! [B: set_real,A2: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B,A2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B))) ) ) ) ).

% sum_mono2
tff(fact_5452_sum__mono2,axiom,
    ! [B: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B,A2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B))) ) ) ) ).

% sum_mono2
tff(fact_5453_sum__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
       => ( ! [B3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B,A2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_Extended_enat_nat(F,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups2027974829824023292at_nat(F,A2)),groups2027974829824023292at_nat(F,B))) ) ) ) ).

% sum_mono2
tff(fact_5454_sum__mono2,axiom,
    ! [B: set_real,A2: set_real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B,A2)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,B3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,A2)),groups1932886352136224148al_int(F,B))) ) ) ) ).

% sum_mono2
tff(fact_5455_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_5456_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_5457_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(plus_plus_real(aa_complex_real(G,X)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_5458_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups1752964319039525884d_enat(G,A2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_com3683005661447327866d_enat(G,X)),groups1752964319039525884d_enat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_5459_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
       => ( groups2027974829824023292at_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(G,X)),groups2027974829824023292at_nat(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% sum.remove
tff(fact_5460_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
       => ( groups2025484359314973016at_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Extended_enat_int(G,X)),groups2025484359314973016at_int(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% sum.remove
tff(fact_5461_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
       => ( groups4148127829035722712t_real(G,A2) = aa_real_real(plus_plus_real(aa_Ext2541572767299617110t_real(G,X)),groups4148127829035722712t_real(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% sum.remove
tff(fact_5462_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
       => ( groups2433450451889696826d_enat(G,A2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(G,X)),groups2433450451889696826d_enat(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% sum.remove
tff(fact_5463_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1935376822645274424al_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_5464_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1932886352136224148al_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X)),groups1932886352136224148al_int(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_5465_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_nat,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5693394587270226106ex_nat(G,insert_complex(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_5466_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_int,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5690904116761175830ex_int(G,insert_complex(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_5467_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_real,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(plus_plus_real(aa_complex_real(G,X)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_5468_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_co6496365337113880337d_enat,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups1752964319039525884d_enat(G,insert_complex(X,A2)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_com3683005661447327866d_enat(G,X)),groups1752964319039525884d_enat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_5469_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2027974829824023292at_nat(G,insert_Extended_enat(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(G,X)),groups2027974829824023292at_nat(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.insert_remove
tff(fact_5470_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex3482447378695272101at_int,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2025484359314973016at_int(G,insert_Extended_enat(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Extended_enat_int(G,X)),groups2025484359314973016at_int(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.insert_remove
tff(fact_5471_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups4148127829035722712t_real(G,insert_Extended_enat(X,A2)) = aa_real_real(plus_plus_real(aa_Ext2541572767299617110t_real(G,X)),groups4148127829035722712t_real(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.insert_remove
tff(fact_5472_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex5852502975177633731d_enat,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2433450451889696826d_enat(G,insert_Extended_enat(X,A2)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(G,X)),groups2433450451889696826d_enat(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.insert_remove
tff(fact_5473_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_nat,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_5474_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_int,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1932886352136224148al_int(G,insert_real(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X)),groups1932886352136224148al_int(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_5475_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A2)),aa_complex_int(F,A)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups5690904116761175830ex_int(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5476_sum__diff1,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
         => ( groups2025484359314973016at_int(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups2025484359314973016at_int(F,A2)),aa_Extended_enat_int(F,A)) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
         => ( groups2025484359314973016at_int(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = groups2025484359314973016at_int(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5477_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(A),A2))
         => ( groups1932886352136224148al_int(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups1932886352136224148al_int(F,A2)),aa_real_int(F,A)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),A2))
         => ( groups1932886352136224148al_int(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups1932886352136224148al_int(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5478_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_real_real(minus_minus_real(groups5808333547571424918x_real(F,A2)),aa_complex_real(F,A)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups5808333547571424918x_real(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5479_sum__diff1,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
         => ( groups4148127829035722712t_real(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = aa_real_real(minus_minus_real(groups4148127829035722712t_real(F,A2)),aa_Ext2541572767299617110t_real(F,A)) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
         => ( groups4148127829035722712t_real(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = groups4148127829035722712t_real(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5480_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(A),A2))
         => ( groups8097168146408367636l_real(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_real_real(minus_minus_real(groups8097168146408367636l_real(F,A2)),aa_real_real(F,A)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),A2))
         => ( groups8097168146408367636l_real(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups8097168146408367636l_real(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5481_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(A),A2))
         => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_real_real(minus_minus_real(groups8778361861064173332t_real(F,A2)),aa_int_real(F,A)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),A2))
         => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups8778361861064173332t_real(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5482_sum__diff1,axiom,
    ! [A2: set_nat,A: nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),A2))
         => ( aa_set_nat_int(groups3539618377306564664at_int(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(F),A2)),aa_nat_int(F,A)) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
         => ( aa_set_nat_int(groups3539618377306564664at_int(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_set_nat_int(groups3539618377306564664at_int(F),A2) ) ) ) ) ).

% sum_diff1
tff(fact_5483_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(A),A2))
         => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),aa_int_int(F,A)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),A2))
         => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups4538972089207619220nt_int(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5484_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A2)),aa_complex_complex(F,A)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups7754918857620584856omplex(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5485_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_nat,C: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_ke(A),B2),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B2,A)),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_ke(A),B2),C),S2) = groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5486_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_int,C: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_kf(A),B2),C),S2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(B2,A)),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_kf(A),B2),C),S2) = groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5487_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_real,C: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_kg(A),B2),C),S2) = aa_real_real(plus_plus_real(aa_complex_real(B2,A)),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_kg(A),B2),C),S2) = groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5488_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: fun_co6496365337113880337d_enat,C: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups1752964319039525884d_enat(aa_fun4954905782667315452d_enat(aa_fun2568869125039355352d_enat(aTP_Lamm_kh(A),B2),C),S2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_com3683005661447327866d_enat(B2,A)),groups1752964319039525884d_enat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups1752964319039525884d_enat(aa_fun4954905782667315452d_enat(aa_fun2568869125039355352d_enat(aTP_Lamm_kh(A),B2),C),S2) = groups1752964319039525884d_enat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5489_sum_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex7660298398204468809at_nat,C: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2027974829824023292at_nat(aa_fun1810565495048290044at_nat(aa_fun5812849712608808208at_nat(aTP_Lamm_ki(A),B2),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(B2,A)),groups2027974829824023292at_nat(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2027974829824023292at_nat(aa_fun1810565495048290044at_nat(aa_fun5812849712608808208at_nat(aTP_Lamm_ki(A),B2),C),S2) = groups2027974829824023292at_nat(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5490_sum_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex3482447378695272101at_int,C: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2025484359314973016at_int(aa_fun1695976788104708604at_int(aa_fun8556764835550362732at_int(aTP_Lamm_kj(A),B2),C),S2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Extended_enat_int(B2,A)),groups2025484359314973016at_int(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2025484359314973016at_int(aa_fun1695976788104708604at_int(aa_fun8556764835550362732at_int(aTP_Lamm_kj(A),B2),C),S2) = groups2025484359314973016at_int(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5491_sum_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex1451620480498224165t_real,C: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aa_fun967516747568145260t_real(aTP_Lamm_kk(A),B2),C),S2) = aa_real_real(plus_plus_real(aa_Ext2541572767299617110t_real(B2,A)),groups4148127829035722712t_real(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aa_fun967516747568145260t_real(aTP_Lamm_kk(A),B2),C),S2) = groups4148127829035722712t_real(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5492_sum_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex5852502975177633731d_enat,C: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2433450451889696826d_enat(aa_fun7865826685190571204d_enat(aa_fun6128326792922203780d_enat(aTP_Lamm_kl(A),B2),C),S2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(B2,A)),groups2433450451889696826d_enat(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2433450451889696826d_enat(aa_fun7865826685190571204d_enat(aa_fun6128326792922203780d_enat(aTP_Lamm_kl(A),B2),C),S2) = groups2433450451889696826d_enat(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5493_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B2: fun_real_nat,C: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_km(A),B2),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(B2,A)),groups1935376822645274424al_nat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_km(A),B2),C),S2) = groups1935376822645274424al_nat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5494_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B2: fun_real_int,C: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_kn(A),B2),C),S2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(B2,A)),groups1932886352136224148al_int(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_kn(A),B2),C),S2) = groups1932886352136224148al_int(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5495_sum__strict__mono2,axiom,
    ! [B: set_real,A2: set_real,B2: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
       => ( pp(aa_set_real_bool(member_real(B2),minus_minus_set_real(B,A2)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,B2)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5496_sum__strict__mono2,axiom,
    ! [B: set_complex,A2: set_complex,B2: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
       => ( pp(aa_set_complex_bool(member_complex(B2),minus_811609699411566653omplex(B,A2)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,B2)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5497_sum__strict__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,B2: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B2),minus_925952699566721837d_enat(B,A2)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,B2)))
           => ( ! [X4: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),B))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups4148127829035722712t_real(F,A2)),groups4148127829035722712t_real(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5498_sum__strict__mono2,axiom,
    ! [B: set_real,A2: set_real,B2: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
       => ( pp(aa_set_real_bool(member_real(B2),minus_minus_set_real(B,A2)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,B2)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5499_sum__strict__mono2,axiom,
    ! [B: set_complex,A2: set_complex,B2: complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
       => ( pp(aa_set_complex_bool(member_complex(B2),minus_811609699411566653omplex(B,A2)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,B2)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5500_sum__strict__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,B2: extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B2),minus_925952699566721837d_enat(B,A2)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_Extended_enat_nat(F,B2)))
           => ( ! [X4: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),B))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_Extended_enat_nat(F,X4))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups2027974829824023292at_nat(F,A2)),groups2027974829824023292at_nat(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5501_sum__strict__mono2,axiom,
    ! [B: set_real,A2: set_real,B2: real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
       => ( pp(aa_set_real_bool(member_real(B2),minus_minus_set_real(B,A2)))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(F,B2)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B))
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups1932886352136224148al_int(F,A2)),groups1932886352136224148al_int(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5502_sum__strict__mono2,axiom,
    ! [B: set_complex,A2: set_complex,B2: complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
       => ( pp(aa_set_complex_bool(member_complex(B2),minus_811609699411566653omplex(B,A2)))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_complex_int(F,B2)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B))
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X4))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5503_sum__strict__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,B2: extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B2),minus_925952699566721837d_enat(B,A2)))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_Extended_enat_int(F,B2)))
           => ( ! [X4: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),B))
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_Extended_enat_int(F,X4))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups2025484359314973016at_int(F,A2)),groups2025484359314973016at_int(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5504_sum__strict__mono2,axiom,
    ! [B: set_nat,A2: set_nat,B2: nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
       => ( pp(aa_set_nat_bool(member_nat(B2),minus_minus_set_nat(B,A2)))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,B2)))
           => ( ! [X4: nat] :
                  ( pp(aa_set_nat_bool(member_nat(X4),B))
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),A2)),aa_set_nat_int(groups3539618377306564664at_int(F),B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5505_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(member_complex(I),A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A2,insert_complex(I,bot_bot_set_complex))))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,X4))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_com3683005661447327866d_enat(F,I)),groups1752964319039525884d_enat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5506_member__le__sum,axiom,
    ! [I: extended_enat,A2: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),A2))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(A2,insert_Extended_enat(I,bot_bo7653980558646680370d_enat))))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,X4))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(F,I)),groups2433450451889696826d_enat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5507_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_re6065469302365115155d_enat] :
      ( pp(aa_set_real_bool(member_real(I),A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A2,insert_real(I,bot_bot_set_real))))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,X4))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,I)),groups2800946370649118462d_enat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5508_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_in5914218760682845459d_enat] :
      ( pp(aa_set_int_bool(member_int(I),A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A2,insert_int(I,bot_bot_set_int))))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,X4))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,I)),groups4225252721152677374d_enat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5509_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: fun_na8797796813480665199d_enat] :
      ( pp(aa_set_nat_bool(member_nat(I),A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(A2,insert_nat(I,bot_bot_set_nat))))
           => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,X4))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,I)),groups7108830773950497114d_enat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5510_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(member_complex(I),A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A2,insert_complex(I,bot_bot_set_complex))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),groups5808333547571424918x_real(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5511_member__le__sum,axiom,
    ! [I: extended_enat,A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),A2))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(A2,insert_Extended_enat(I,bot_bo7653980558646680370d_enat))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X4))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,I)),groups4148127829035722712t_real(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5512_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(member_real(I),A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A2,insert_real(I,bot_bot_set_real))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),groups8097168146408367636l_real(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5513_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(member_int(I),A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A2,insert_int(I,bot_bot_set_int))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),groups8778361861064173332t_real(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5514_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(member_complex(I),A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A2,insert_complex(I,bot_bot_set_complex))))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I)),groups5693394587270226106ex_nat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5515_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa2) = Y )
     => ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(X,Xa2))
       => ~ ( ( ( ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                  & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y = aa_int_int(uminus_uminus_int,zero_n2684676970156552555ol_int(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),X)),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xa2))))) ) )
              & ( ~ ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_n2684676970156552555ol_int(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),X)),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xa2))))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(Xa2),numeral_numeral_int(bit0(one)))))) ) ) )
           => ~ accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(X,Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_5516_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(K,L))
     => ( ( ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(uminus_uminus_int,zero_n2684676970156552555ol_int(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),K)),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),L))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_n2684676970156552555ol_int(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),K)),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L),numeral_numeral_int(bit0(one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_5517_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),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,N)),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(suc,N))),zero_n2684676970156552555ol_int(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_5518_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cd),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% Sum_Icc_nat
tff(fact_5519_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)),X)) ) ).

% neg_numeral_le_ceiling
tff(fact_5520_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real))) ) ).

% ceiling_less_neg_numeral
tff(fact_5521_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ko(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D)))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% arith_series_nat
tff(fact_5522_bit__0__eq,axiom,
    aa_int_fun_nat_bool(bit_se1146084159140164899it_int,zero_zero_int) = bot_bot_fun_nat_bool ).

% bit_0_eq
tff(fact_5523_bit__0__eq,axiom,
    aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,zero_zero_nat) = bot_bot_fun_nat_bool ).

% bit_0_eq
tff(fact_5524_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_5525_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,bit0(X))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_5526_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,bit0(Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_5527_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,numeral_numeral_int(bit0(M))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,numeral_numeral_int(M)),N)) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_5528_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,bit0(M))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,M)),N)) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_5529_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,numeral_numeral_int(aa_num_num(bit1,M))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,numeral_numeral_int(M)),N)) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_5530_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,M)),N)) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_5531_ceiling__add__of__int,axiom,
    ! [X: real,Z2: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(X),aa_int_real(ring_1_of_int_real,Z2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),Z2) ).

% ceiling_add_of_int
tff(fact_5532_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_5533_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_5534_ceiling__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% ceiling_le_zero
tff(fact_5535_zero__less__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% zero_less_ceiling
tff(fact_5536_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),numeral_numeral_int(V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),numeral_numeral_real(V))) ) ).

% ceiling_le_numeral
tff(fact_5537_ceiling__less__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% ceiling_less_one
tff(fact_5538_one__le__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% one_le_ceiling
tff(fact_5539_numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,numeral_numeral_int(V)),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(V)),X)) ) ).

% numeral_less_ceiling
tff(fact_5540_ceiling__le__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ).

% ceiling_le_one
tff(fact_5541_one__less__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ).

% one_less_ceiling
tff(fact_5542_ceiling__add__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(X),numeral_numeral_real(V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),numeral_numeral_int(V)) ).

% ceiling_add_numeral
tff(fact_5543_ceiling__add__one,axiom,
    ! [X: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(X),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int) ).

% ceiling_add_one
tff(fact_5544_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(W2)))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(W2))),N)) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_5545_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(aa_num_num(bit1,W2)))),aa_nat_nat(suc,N)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,numeral_numeral_int(W2)),N)) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_5546_bit__0,axiom,
    ! [A: int] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,A),zero_zero_nat))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)) ) ).

% bit_0
tff(fact_5547_bit__0,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),zero_zero_nat))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)) ) ).

% bit_0
tff(fact_5548_and__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% and_Suc_0_eq
tff(fact_5549_Suc__0__and__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% Suc_0_and_eq
tff(fact_5550_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5551_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5552_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_na8797796813480665199d_enat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_z5237406670263579293d_enat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_Extended_enat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5553_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5554_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5555_ceiling__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,one_one_real))) ) ).

% ceiling_less_zero
tff(fact_5556_zero__le__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)) ) ).

% zero_le_ceiling
tff(fact_5557_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_kp(C),A2) = aa_nat_complex(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_kp(C),A2) = zero_zero_complex ) ) ) ).

% sum_zero_power
tff(fact_5558_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kq(C)),A2) = aa_nat_real(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kq(C)),A2) = zero_zero_real ) ) ) ).

% sum_zero_power
tff(fact_5559_bit__mod__2__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))),N))
    <=> ( ( N = zero_zero_nat )
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)) ) ) ).

% bit_mod_2_iff
tff(fact_5560_bit__mod__2__iff,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),N))
    <=> ( ( N = zero_zero_nat )
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)) ) ) ).

% bit_mod_2_iff
tff(fact_5561_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),numeral_numeral_int(V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(minus_minus_real(numeral_numeral_real(V)),one_one_real))) ) ).

% ceiling_less_numeral
tff(fact_5562_numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(V)),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(numeral_numeral_real(V)),one_one_real)),X)) ) ).

% numeral_le_ceiling
tff(fact_5563_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(V)))) ) ).

% ceiling_le_neg_numeral
tff(fact_5564_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),X)) ) ).

% neg_numeral_less_ceiling
tff(fact_5565_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_complex,D: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_kr(C),D),A2) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_kr(C),D),A2) = zero_zero_complex ) ) ) ).

% sum_zero_power'
tff(fact_5566_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_real,D: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ks(C),D)),A2) = aa_real_real(divide_divide_real(aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ks(C),D)),A2) = zero_zero_real ) ) ) ).

% sum_zero_power'
tff(fact_5567_bit__disjunctive__add__iff,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,A),N2))
          | ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,B2),N2)) )
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),N))
      <=> ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,A),N))
          | pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,B2),N)) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_5568_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N2))
          | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,B2),N2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),N))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,B2),N)) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_5569_not__bit__1__Suc,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,one_one_int),aa_nat_nat(suc,N))) ).

% not_bit_1_Suc
tff(fact_5570_not__bit__1__Suc,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,one_one_nat),aa_nat_nat(suc,N))) ).

% not_bit_1_Suc
tff(fact_5571_bit__1__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,one_one_int),N))
    <=> ( N = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_5572_bit__1__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,one_one_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_5573_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A2))
           => ( aa_nat_nat(F,aa_nat_nat(suc,X4)) = aa_nat_nat(G,aa_nat_nat(suc,X4)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_5574_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A2))
           => ( aa_nat_real(F,aa_nat_nat(suc,X4)) = aa_nat_real(G,aa_nat_nat(suc,X4)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),A2) = aa_set_nat_real(groups6591440286371151544t_real(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_5575_bit__take__bit__iff,axiom,
    ! [M: nat,A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),A)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N)) ) ) ).

% bit_take_bit_iff
tff(fact_5576_bit__take__bit__iff,axiom,
    ! [M: nat,A: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),A)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,A),N)) ) ) ).

% bit_take_bit_iff
tff(fact_5577_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,X))) ) ).

% ceiling_mono
tff(fact_5578_le__of__int__ceiling,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ).

% le_of_int_ceiling
tff(fact_5579_ceiling__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% ceiling_less_cancel
tff(fact_5580_bit__of__bool__iff,axiom,
    ! [B2: bool,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,zero_n2684676970156552555ol_int(B2)),N))
    <=> ( pp(B2)
        & ( N = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_5581_bit__of__bool__iff,axiom,
    ! [B2: bool,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,zero_n2687167440665602831ol_nat(B2)),N))
    <=> ( pp(B2)
        & ( N = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_5582_sum__subtractf__nat,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,F: fun_Ex7660298398204468809at_nat] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(G,X4)),aa_Extended_enat_nat(F,X4))) )
     => ( groups2027974829824023292at_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_kt(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups2027974829824023292at_nat(F,A2)),groups2027974829824023292at_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_5583_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(G,X4)),aa_real_nat(F,X4))) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ku(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_5584_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(G,X4)),aa_set_nat_nat(F,X4))) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_kv(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_5585_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(G,X4)),aa_int_nat(F,X4))) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_kw(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_5586_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(G,X4)),aa_nat_nat(F,X4))) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kx(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_5587_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_5588_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kz(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_5589_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_la(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_5590_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,M: nat,K: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_lb(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_5591_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups5693394587270226106ex_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A2))
            & ( aa_complex_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa3),A2))
               => ( ( X5 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5592_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups4541462559716669496nt_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A2))
            & ( aa_int_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A2))
               => ( ( X5 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5593_sum__eq__Suc0__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( groups2027974829824023292at_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
            & ( aa_Extended_enat_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa3),A2))
               => ( ( X5 != Xa3 )
                 => ( aa_Extended_enat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5594_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A2))
            & ( aa_nat_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A2))
               => ( ( X5 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5595_sum__SucD,axiom,
    ! [F: fun_nat_nat,A2: set_nat,N: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,N) )
     => ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4))) ) ) ).

% sum_SucD
tff(fact_5596_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups5693394587270226106ex_nat(F,A2) = one_one_nat )
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A2))
            & ( aa_complex_nat(F,X5) = one_one_nat )
            & ! [Xa3: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa3),A2))
               => ( ( X5 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5597_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups4541462559716669496nt_nat(F,A2) = one_one_nat )
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A2))
            & ( aa_int_nat(F,X5) = one_one_nat )
            & ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A2))
               => ( ( X5 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5598_sum__eq__1__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( groups2027974829824023292at_nat(F,A2) = one_one_nat )
      <=> ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
            & ( aa_Extended_enat_nat(F,X5) = one_one_nat )
            & ! [Xa3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa3),A2))
               => ( ( X5 != Xa3 )
                 => ( aa_Extended_enat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5599_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = one_one_nat )
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A2))
            & ( aa_nat_nat(F,X5) = one_one_nat )
            & ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A2))
               => ( ( X5 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5600_ceiling__le,axiom,
    ! [X: real,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),A)) ) ).

% ceiling_le
tff(fact_5601_ceiling__le__iff,axiom,
    ! [X: real,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),Z2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z2))) ) ).

% ceiling_le_iff
tff(fact_5602_less__ceiling__iff,axiom,
    ! [Z2: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),X)) ) ).

% less_ceiling_iff
tff(fact_5603_ceiling__add__le,axiom,
    ! [X: real,Y: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))) ).

% ceiling_add_le
tff(fact_5604_sum__power__add,axiom,
    ! [X: int,M: nat,I6: set_nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_lc(X),M)),I6) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),M)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),I6)) ).

% sum_power_add
tff(fact_5605_sum__power__add,axiom,
    ! [X: complex,M: nat,I6: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_ld(X),M),I6) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),groups2073611262835488442omplex(power_power_complex(X),I6)) ).

% sum_power_add
tff(fact_5606_sum__power__add,axiom,
    ! [X: real,M: nat,I6: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_le(X),M)),I6) = aa_real_real(times_times_real(aa_nat_real(power_power_real(X),M)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),I6)) ).

% sum_power_add
tff(fact_5607_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lf(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% sum.atLeastAtMost_rev
tff(fact_5608_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,N: nat,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_lg(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% sum.atLeastAtMost_rev
tff(fact_5609_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_cg,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_cu(N),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_5610_sum__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_cg,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_cw,N))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_5611_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),aa_real_real(plus_plus_real(R2),one_one_real))) ).

% of_int_ceiling_le_add_one
tff(fact_5612_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),one_one_real)),R2)) ).

% of_int_ceiling_diff_one_le
tff(fact_5613_sum__diff__nat,axiom,
    ! [B: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A2))
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B)) ) ) ) ).

% sum_diff_nat
tff(fact_5614_sum__diff__nat,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),A2))
       => ( groups2027974829824023292at_nat(F,minus_925952699566721837d_enat(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups2027974829824023292at_nat(F,A2)),groups2027974829824023292at_nat(F,B)) ) ) ) ).

% sum_diff_nat
tff(fact_5615_sum__diff__nat,axiom,
    ! [B: set_int,A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A2))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B)) ) ) ) ).

% sum_diff_nat
tff(fact_5616_sum__diff__nat,axiom,
    ! [B: set_nat,A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,B)) = 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),B)) ) ) ) ).

% sum_diff_nat
tff(fact_5617_sum__diff1__nat,axiom,
    ! [A: set_nat,A2: set_set_nat,F: fun_set_nat_nat] :
      ( ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),aa_set_nat_nat(F,A)) ) )
      & ( ~ pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = groups8294997508430121362at_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_5618_sum__diff1__nat,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
       => ( groups2027974829824023292at_nat(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups2027974829824023292at_nat(F,A2)),aa_Extended_enat_nat(F,A)) ) )
      & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
       => ( groups2027974829824023292at_nat(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = groups2027974829824023292at_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_5619_sum__diff1__nat,axiom,
    ! [A: real,A2: set_real,F: fun_real_nat] :
      ( ( pp(aa_set_real_bool(member_real(A),A2))
       => ( groups1935376822645274424al_nat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),aa_real_nat(F,A)) ) )
      & ( ~ pp(aa_set_real_bool(member_real(A),A2))
       => ( groups1935376822645274424al_nat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups1935376822645274424al_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_5620_sum__diff1__nat,axiom,
    ! [A: int,A2: set_int,F: fun_int_nat] :
      ( ( pp(aa_set_int_bool(member_int(A),A2))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),aa_int_nat(F,A)) ) )
      & ( ~ pp(aa_set_int_bool(member_int(A),A2))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups4541462559716669496nt_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_5621_sum__diff1__nat,axiom,
    ! [A: nat,A2: set_nat,F: fun_nat_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_nat_nat(F,A)) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) ) ) ) ).

% sum_diff1_nat
tff(fact_5622_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5623_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5624_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_na8797796813480665199d_enat,K: nat] :
      ( ( aa_nat_Extended_enat(F,zero_zero_nat) = zero_z5237406670263579293d_enat )
     => ( groups7108830773950497114d_enat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups7108830773950497114d_enat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5625_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5626_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5627_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5628_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_Extended_enat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5629_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5630_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5631_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,N))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5632_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_na8797796813480665199d_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,aa_nat_nat(suc,N))),groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5633_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5634_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_nat_real(G,aa_nat_nat(suc,N))),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5635_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5636_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_na8797796813480665199d_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,M)),groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5637_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5638_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(plus_plus_real(aa_nat_real(G,M)),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5639_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_5640_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_concat_bit(M,K),L)),N))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_5641_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lh(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5642_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_na8797796813480665199d_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_Extended_enat(G,aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,M)),groups7108830773950497114d_enat(aTP_Lamm_li(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5643_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5644_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_nat_real(G,M)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kz(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5645_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lj(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_5646_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lk(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(minus_minus_real(aa_nat_real(F,aa_nat_nat(suc,N))),aa_nat_real(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_5647_exp__eq__0__imp__not__bit,axiom,
    ! [N: nat,A: int] :
      ( ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N) = zero_zero_int )
     => ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,A),N)) ) ).

% exp_eq_0_imp_not_bit
tff(fact_5648_exp__eq__0__imp__not__bit,axiom,
    ! [N: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) = zero_zero_nat )
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N)) ) ).

% exp_eq_0_imp_not_bit
tff(fact_5649_bit__Suc,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,A),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))),N)) ) ).

% bit_Suc
tff(fact_5650_bit__Suc,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),N)) ) ).

% bit_Suc
tff(fact_5651_ceiling__correct,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X))),one_one_real)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ) ).

% ceiling_correct
tff(fact_5652_ceiling__unique,axiom,
    ! [Z2: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z2)),one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z2)))
       => ( aa_real_int(archim7802044766580827645g_real,X) = Z2 ) ) ) ).

% ceiling_unique
tff(fact_5653_ceiling__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,X) = A )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,A)),one_one_real)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A))) ) ) ).

% ceiling_eq_iff
tff(fact_5654_ceiling__split,axiom,
    ! [P: fun_int_bool,T: real] :
      ( pp(aa_int_bool(P,aa_real_int(archim7802044766580827645g_real,T)))
    <=> ! [I3: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,I3)),one_one_real)),T))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),aa_int_real(ring_1_of_int_real,I3))) )
         => pp(aa_int_bool(P,I3)) ) ) ).

% ceiling_split
tff(fact_5655_mult__ceiling__le,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(times_times_real(A),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B2)))) ) ) ).

% mult_ceiling_le
tff(fact_5656_ceiling__less__iff,axiom,
    ! [X: real,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),Z2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z2)),one_one_real))) ) ).

% ceiling_less_iff
tff(fact_5657_le__ceiling__iff,axiom,
    ! [Z2: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z2)),one_one_real)),X)) ) ).

% le_ceiling_iff
tff(fact_5658_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M5))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),M5))
              <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)))
              <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) ) ) ).

% int_bit_bound
tff(fact_5659_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_5660_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_na8797796813480665199d_enat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_5661_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_5662_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_real,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_5663_sum__count__set,axiom,
    ! [Xs: list_complex,X6: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Xs)),X6))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,X6))
       => ( groups5693394587270226106ex_nat(count_list_complex(Xs),X6) = size_s3451745648224563538omplex(Xs) ) ) ) ).

% sum_count_set
tff(fact_5664_sum__count__set,axiom,
    ! [Xs: list_Extended_enat,X6: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(set_Extended_enat2(Xs)),X6))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,X6))
       => ( groups2027974829824023292at_nat(count_101369445342291426d_enat(Xs),X6) = size_s3941691890525107288d_enat(Xs) ) ) ) ).

% sum_count_set
tff(fact_5665_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X6: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),X6)
     => ( finite5795047828879050333T_VEBT(X6)
       => ( groups771621172384141258BT_nat(count_list_VEBT_VEBT(Xs),X6) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) ) ) ) ).

% sum_count_set
tff(fact_5666_sum__count__set,axiom,
    ! [Xs: list_int,X6: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),X6))
     => ( pp(aa_set_int_bool(finite_finite_int,X6))
       => ( groups4541462559716669496nt_nat(count_list_int(Xs),X6) = aa_list_int_nat(size_size_list_int,Xs) ) ) ) ).

% sum_count_set
tff(fact_5667_sum__count__set,axiom,
    ! [Xs: list_nat,X6: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),X6))
     => ( pp(aa_set_nat_bool(finite_finite_nat,X6))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(count_list_nat(Xs)),X6) = aa_list_nat_nat(size_size_list_nat,Xs) ) ) ) ).

% sum_count_set
tff(fact_5668_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,N: nat] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = zero_zero_int )
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,A),N)) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_5669_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = zero_zero_nat )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N)) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_5670_ceiling__divide__upper,axiom,
    ! [Q3: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,P3),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(P3),Q3)))),Q3))) ) ).

% ceiling_divide_upper
tff(fact_5671_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2073611262835488442omplex(aTP_Lamm_ll(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(F,M)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2073611262835488442omplex(aTP_Lamm_ll(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
tff(fact_5672_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lm(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lm(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_int ) ) ) ).

% sum_natinterval_diff
tff(fact_5673_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ln(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(minus_minus_real(aa_nat_real(F,M)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ln(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_real ) ) ) ).

% sum_natinterval_diff
tff(fact_5674_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lo(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_5675_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lp(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_real_real(minus_minus_real(aa_nat_real(F,N)),aa_nat_real(F,M)) ) ) ).

% sum_telescope''
tff(fact_5676_even__bit__succ__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),N))
      <=> ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,A),N))
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_5677_even__bit__succ__iff,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),N))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N))
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_5678_odd__bit__iff__bit__pred,axiom,
    ! [A: int,N: nat] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,A),N))
      <=> ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)),N))
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_5679_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),one_one_nat)),N))
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_5680_ceiling__divide__lower,axiom,
    ! [Q3: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(P3),Q3)))),one_one_real)),Q3)),P3)) ) ).

% ceiling_divide_lower
tff(fact_5681_ceiling__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim7802044766580827645g_real,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_5682_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)),one_one_int) = aa_set_nat_int(groups3539618377306564664at_int(power_power_int(numeral_numeral_int(bit0(one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,N))) ).

% mask_eq_sum_exp
tff(fact_5683_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,N))) ).

% mask_eq_sum_exp
tff(fact_5684_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_5685_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_5686_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X),M)),aa_nat_real(power_power_real(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_5687_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lq(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_5688_sum_Oin__pairs,axiom,
    ! [G: fun_na8797796813480665199d_enat,M: nat,N: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = groups7108830773950497114d_enat(aTP_Lamm_lr(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_5689_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ls(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_5690_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lt(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_5691_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(A0,A1))
     => ( ! [K3: int,L3: int] :
            ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(K3,L3))
           => ( ( ~ ( pp(aa_set_int_bool(member_int(K3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(L3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(divide_divide_int(K3),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L3),numeral_numeral_int(bit0(one))))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,K3),L3)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_5692_bit__sum__mult__2__cases,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ! [J3: nat] : ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,A),aa_nat_nat(suc,J3)))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B2))),N))
      <=> ( ( ( N = zero_zero_nat )
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)) )
          & ( ( N != zero_zero_nat )
           => pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B2)),N)) ) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_5693_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ! [J3: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,J3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))),N))
      <=> ( ( ( N = zero_zero_nat )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)) )
          & ( ( N != zero_zero_nat )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2)),N)) ) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_5694_bit__rec,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,A),N))
    <=> ( ( ( N = zero_zero_nat )
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)) )
        & ( ( N != zero_zero_nat )
         => pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ) ).

% bit_rec
tff(fact_5695_bit__rec,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N))
    <=> ( ( ( N = zero_zero_nat )
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)) )
        & ( ( N != zero_zero_nat )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ) ).

% bit_rec
tff(fact_5696_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero_nat )
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( ( M = zero_zero_nat )
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ).

% and_nat_unfold
tff(fact_5697_and__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% and_nat_rec
tff(fact_5698_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,N))) ).

% mask_eq_sum_exp_nat
tff(fact_5699_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cd),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% gauss_sum_nat
tff(fact_5700_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)),zero_n2684676970156552555ol_int(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)) ).

% take_bit_Suc_from_most
tff(fact_5701_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(A0,A1))
     => ( ! [I4: int,J3: int] :
            ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I4,J3))
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I4),J3))
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I4),one_one_int)),J3)) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,I4),J3)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_5702_sum__gp,axiom,
    ! [N: nat,M: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X != one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N)))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_5703_sum__gp,axiom,
    ! [N: nat,M: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_real )
           => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X != one_one_real )
           => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X),M)),aa_nat_real(power_power_real(X),aa_nat_nat(suc,N)))),aa_real_real(minus_minus_real(one_one_real),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_5704_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),numeral_numeral_int(bit0(one))) ).

% gauss_sum_from_Suc_0
tff(fact_5705_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% gauss_sum_from_Suc_0
tff(fact_5706_sum__gp__offset,axiom,
    ! [X: complex,M: nat,N: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_offset
tff(fact_5707_sum__gp__offset,axiom,
    ! [X: real,M: nat,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(X),M)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X),aa_nat_nat(suc,N))))),aa_real_real(minus_minus_real(one_one_real),X)) ) ) ) ).

% sum_gp_offset
tff(fact_5708_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_5709_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)),one_on7984719198319812577d_enat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_5710_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_5711_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_5712_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_5713_arith__series,axiom,
    ! [A: int,D: int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_lu(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D)))),numeral_numeral_int(bit0(one))) ).

% arith_series
tff(fact_5714_arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lv(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D)))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% arith_series
tff(fact_5715_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_5716_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_5717_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_5718_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),aa_nat_real(semiri5074537144036343181t_real,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),M)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_5719_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = numeral_numeral_int(V) )
    <=> ( M = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_5720_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_5721_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% of_int_of_nat_eq
tff(fact_5722_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% of_int_of_nat_eq
tff(fact_5723_negative__zle,axiom,
    ! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zle
tff(fact_5724_int__dvd__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% int_dvd_int_iff
tff(fact_5725_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_5726_of__nat__0,axiom,
    aa_nat_Extended_enat(semiri4216267220026989637d_enat,zero_zero_nat) = zero_z5237406670263579293d_enat ).

% of_nat_0
tff(fact_5727_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_5728_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_5729_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_5730_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_5731_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_z5237406670263579293d_enat = aa_nat_Extended_enat(semiri4216267220026989637d_enat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_5732_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_5733_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_5734_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_5735_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = zero_zero_complex )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_5736_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,M) = zero_z5237406670263579293d_enat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_5737_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = zero_zero_real )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_5738_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = zero_zero_int )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_5739_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = zero_zero_nat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_5740_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(semiri4216267220026989637d_enat,M)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_5741_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_5742_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_5743_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_5744_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_5745_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_5746_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_5747_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(W2)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,W2)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_5748_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),numeral_numeral_real(W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,W2))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_5749_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,M)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)) ).

% of_nat_add
tff(fact_5750_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_add
tff(fact_5751_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_add
tff(fact_5752_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_add
tff(fact_5753_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N)) ).

% of_nat_mult
tff(fact_5754_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,M)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)) ).

% of_nat_mult
tff(fact_5755_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_mult
tff(fact_5756_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mult
tff(fact_5757_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mult
tff(fact_5758_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,N) = one_one_complex )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_5759_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,N) = one_one_real )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_5760_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,N) = one_one_int )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_5761_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,N) = one_one_nat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_5762_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_5763_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_5764_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_5765_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_5766_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_5767_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_5768_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_5769_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_5770_negative__zless,axiom,
    ! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zless
tff(fact_5771_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_real(semiri5074537144036343181t_real,zero_n2687167440665602831ol_nat(P)) = zero_n3304061248610475627l_real(P) ).

% of_nat_of_bool
tff(fact_5772_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_nat(semiri1316708129612266289at_nat,zero_n2687167440665602831ol_nat(P)) = zero_n2687167440665602831ol_nat(P) ).

% of_nat_of_bool
tff(fact_5773_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_int(semiri1314217659103216013at_int,zero_n2687167440665602831ol_nat(P)) = zero_n2684676970156552555ol_int(P) ).

% of_nat_of_bool
tff(fact_5774_of__nat__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_lw(F),A2) ).

% of_nat_sum
tff(fact_5775_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A2: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_lx(F),A2) ).

% of_nat_sum
tff(fact_5776_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_ly(F)),A2) ).

% of_nat_sum
tff(fact_5777_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lz(F)),A2) ).

% of_nat_sum
tff(fact_5778_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ma(F)),A2) ).

% of_nat_sum
tff(fact_5779_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(semiri4216267220026989637d_enat,M)),zero_z5237406670263579293d_enat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_5780_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_5781_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_5782_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_5783_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,M)) ).

% of_nat_Suc
tff(fact_5784_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(suc,M)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),aa_nat_Extended_enat(semiri4216267220026989637d_enat,M)) ).

% of_nat_Suc
tff(fact_5785_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M)) = aa_real_real(plus_plus_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,M)) ).

% of_nat_Suc
tff(fact_5786_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% of_nat_Suc
tff(fact_5787_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M)) ).

% of_nat_Suc
tff(fact_5788_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_5789_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_5790_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_5791_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_5792_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B2)),W2)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_5793_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W2)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_5794_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_5795_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B2)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_5796_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_5797_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_5798_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B2)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_5799_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_5800_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_5801_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B2)),W2)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_5802_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_5803_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W2)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_5804_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_5805_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_5806_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_5807_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(numeral_numeral_real(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5808_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(numeral_numeral_int(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5809_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5810_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(numeral_numeral_real(I)),N)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5811_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(numeral_numeral_int(I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5812_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5813_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(numeral_numeral_real(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5814_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5815_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(numeral_numeral_int(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5816_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(numeral_numeral_real(I)),N)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5817_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5818_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5819_real__arch__simple,axiom,
    ! [X: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% real_arch_simple
tff(fact_5820_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% reals_Archimedean2
tff(fact_5821_mult__of__nat__commute,axiom,
    ! [X: nat,Y: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(semiri8010041392384452111omplex,X)) ).

% mult_of_nat_commute
tff(fact_5822_mult__of__nat__commute,axiom,
    ! [X: nat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,X)),Y) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(Y),aa_nat_Extended_enat(semiri4216267220026989637d_enat,X)) ).

% mult_of_nat_commute
tff(fact_5823_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X)),Y) = aa_real_real(times_times_real(Y),aa_nat_real(semiri5074537144036343181t_real,X)) ).

% mult_of_nat_commute
tff(fact_5824_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).

% mult_of_nat_commute
tff(fact_5825_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X)) ).

% mult_of_nat_commute
tff(fact_5826_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% nat_less_real_le
tff(fact_5827_int__cases2,axiom,
    ! [Z2: int] :
      ( ! [N2: nat] : Z2 != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] : Z2 != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% int_cases2
tff(fact_5828_int__diff__cases,axiom,
    ! [Z2: int] :
      ~ ! [M3: nat,N2: nat] : Z2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M3)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_diff_cases
tff(fact_5829_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_5830_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_5831_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),N))
    <=> ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_5832_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_5833_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(semiri4216267220026989637d_enat,N))) ).

% of_nat_0_le_iff
tff(fact_5834_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% of_nat_0_le_iff
tff(fact_5835_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% of_nat_0_le_iff
tff(fact_5836_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% of_nat_0_le_iff
tff(fact_5837_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(semiri4216267220026989637d_enat,M)),zero_z5237406670263579293d_enat)) ).

% of_nat_less_0_iff
tff(fact_5838_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)) ).

% of_nat_less_0_iff
tff(fact_5839_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)) ).

% of_nat_less_0_iff
tff(fact_5840_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)) ).

% of_nat_less_0_iff
tff(fact_5841_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_5842_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(suc,N)) != zero_z5237406670263579293d_enat ).

% of_nat_neq_0
tff(fact_5843_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_5844_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_5845_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_5846_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(semiri4216267220026989637d_enat,M)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,N))) ) ).

% less_imp_of_nat_less
tff(fact_5847_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))) ) ).

% less_imp_of_nat_less
tff(fact_5848_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% less_imp_of_nat_less
tff(fact_5849_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) ) ).

% less_imp_of_nat_less
tff(fact_5850_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(semiri4216267220026989637d_enat,M)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_5851_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_5852_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_5853_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_5854_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J))) ) ).

% of_nat_mono
tff(fact_5855_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J))) ) ).

% of_nat_mono
tff(fact_5856_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J))) ) ).

% of_nat_mono
tff(fact_5857_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real))) ) ).

% nat_le_real_less
tff(fact_5858_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_5859_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2))) ) ).

% nat_int_comparison(2)
tff(fact_5860_int__cases,axiom,
    ! [Z2: int] :
      ( ! [N2: nat] : Z2 != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] : Z2 != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% int_cases
tff(fact_5861_int__of__nat__induct,axiom,
    ! [P: fun_int_bool,Z2: int] :
      ( ! [N2: nat] : pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)))))
       => pp(aa_int_bool(P,Z2)) ) ) ).

% int_of_nat_induct
tff(fact_5862_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% zle_int
tff(fact_5863_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2))) ) ).

% nat_int_comparison(3)
tff(fact_5864_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ~ ! [N2: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% nonneg_int_cases
tff(fact_5865_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ? [N2: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% zero_le_imp_eq_int
tff(fact_5866_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),Z2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))),Z2) ).

% zadd_int_left
tff(fact_5867_int__plus,axiom,
    ! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_5868_int__ops_I5_J,axiom,
    ! [A: nat,B2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% int_ops(5)
tff(fact_5869_zle__iff__zadd,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z2))
    <=> ? [N4: nat] : Z2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,N4)) ) ).

% zle_iff_zadd
tff(fact_5870_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% not_int_zless_negative
tff(fact_5871_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_real_real(ord_max_real(aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_5872_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_5873_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_5874_nat__less__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_less_as_int
tff(fact_5875_nat__leq__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_leq_as_int
tff(fact_5876_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,N))) ).

% not_bit_Suc_0_numeral
tff(fact_5877_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N2)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_5878_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% of_nat_diff
tff(fact_5879_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).

% of_nat_diff
tff(fact_5880_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ) ).

% of_nat_diff
tff(fact_5881_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M3)),X)),C)) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_5882_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).

% int_cases4
tff(fact_5883_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))))
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_5884_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_5885_int__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ).

% int_Suc
tff(fact_5886_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2))
    <=> ? [N4: nat] : Z2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_5887_negative__zle__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),zero_zero_int)) ).

% negative_zle_0
tff(fact_5888_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ~ ! [N2: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% nonpos_int_cases
tff(fact_5889_int__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_lw(F),A2) ).

% int_sum
tff(fact_5890_int__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_ly(F)),A2) ).

% int_sum
tff(fact_5891_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(modulo_modulo_int(A),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_5892_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_5893_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_5894_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ~ ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% pos_int_cases
tff(fact_5895_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) )
       => ~ ! [N2: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ) ).

% int_cases3
tff(fact_5896_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_5897_not__zle__0__negative,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))) ).

% not_zle_0_negative
tff(fact_5898_negD,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
     => ? [N2: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% negD
tff(fact_5899_negative__zless__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),zero_zero_int)) ).

% negative_zless_0
tff(fact_5900_nat__approx__posE,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ~ ! [N2: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))),E2)) ) ).

% nat_approx_posE
tff(fact_5901_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(numeral_numeral_real(bit0(one))),N))) ).

% of_nat_less_two_power
tff(fact_5902_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N))) ).

% of_nat_less_two_power
tff(fact_5903_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( N != zero_zero_nat )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,M))),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_5904_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ~ ! [N2: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% neg_int_cases
tff(fact_5905_zdiff__int__split,axiom,
    ! [P: fun_int_bool,X: nat,Y: nat] :
      ( pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y))))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)))) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% zdiff_int_split
tff(fact_5906_double__arith__series,axiom,
    ! [A: complex,D: complex,N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mb(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),A)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),D))) ).

% double_arith_series
tff(fact_5907_double__arith__series,axiom,
    ! [A: extended_enat,D: extended_enat,N: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),groups7108830773950497114d_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_mc(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)),one_on7984719198319812577d_enat)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),A)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)),D))) ).

% double_arith_series
tff(fact_5908_double__arith__series,axiom,
    ! [A: int,D: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_lu(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D))) ).

% double_arith_series
tff(fact_5909_double__arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lv(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D))) ).

% double_arith_series
tff(fact_5910_double__arith__series,axiom,
    ! [A: real,D: real,N: nat] : aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_md(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),A)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),D))) ).

% double_arith_series
tff(fact_5911_double__gauss__sum,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum
tff(fact_5912_double__gauss__sum,axiom,
    ! [N: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)),one_on7984719198319812577d_enat)) ).

% double_gauss_sum
tff(fact_5913_double__gauss__sum,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum
tff(fact_5914_double__gauss__sum,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum
tff(fact_5915_double__gauss__sum,axiom,
    ! [N: nat] : aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum
tff(fact_5916_gauss__sum,axiom,
    ! [N: nat] : aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),numeral_numeral_int(bit0(one))) ).

% gauss_sum
tff(fact_5917_gauss__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% gauss_sum
tff(fact_5918_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(semiri8010041392384452111omplex,N) = zero_zero_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(semiri8010041392384452111omplex,N) = produc1917071388513777916omplex(aTP_Lamm_me,divmod_nat(N,aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ) ).

% of_nat_code_if
tff(fact_5919_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,N) = zero_z5237406670263579293d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,N) = produc2676513652042109336d_enat(aTP_Lamm_mf,divmod_nat(N,aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ) ).

% of_nat_code_if
tff(fact_5920_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(semiri5074537144036343181t_real,N) = zero_zero_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_Pro6168199777849092593t_real(produc1703576794950452218t_real(aTP_Lamm_mg),divmod_nat(N,aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ) ).

% of_nat_code_if
tff(fact_5921_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(semiri1314217659103216013at_int,N) = zero_zero_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(produc6840382203811409530at_int(aTP_Lamm_mh),divmod_nat(N,aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ) ).

% of_nat_code_if
tff(fact_5922_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(semiri1316708129612266289at_nat,N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_mi),divmod_nat(N,aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ) ).

% of_nat_code_if
tff(fact_5923_lemma__termdiff2,axiom,
    ! [H: complex,Z2: complex,N: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z2),H)),N)),aa_nat_complex(power_power_complex(Z2),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(power_power_complex(Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mk(H),Z2),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_5924_lemma__termdiff2,axiom,
    ! [H: real,Z2: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Z2),H)),N)),aa_nat_real(power_power_real(Z2),N))),H)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(times_times_real(H),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mm(H),Z2),N)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_5925_lemma__termdiff3,axiom,
    ! [H: real,Z2: real,K5: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z2)),K5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Z2),H))),K5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Z2),H)),N)),aa_nat_real(power_power_real(Z2),N))),H)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(power_power_real(K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_real_real(real_V7735802525324610683m_real,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_5926_lemma__termdiff3,axiom,
    ! [H: complex,Z2: complex,K5: real,N: nat] :
      ( ( H != zero_zero_complex )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),K5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z2),H))),K5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z2),H)),N)),aa_nat_complex(power_power_complex(Z2),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(power_power_complex(Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(power_power_real(K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_complex_real(real_V1022390504157884413omplex,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_5927_pochhammer__double,axiom,
    ! [Z2: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),aa_nat_complex(comm_s2602460028002588243omplex(Z2),N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z2),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(bit0(one))))),N)) ).

% pochhammer_double
tff(fact_5928_pochhammer__double,axiom,
    ! [Z2: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),aa_nat_real(comm_s7457072308508201937r_real(Z2),N))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(Z2),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))),N)) ).

% pochhammer_double
tff(fact_5929_of__nat__code,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,N) = aa_complex_complex(aa_nat3829984526574172007omplex(semiri2816024913162550771omplex(aTP_Lamm_mn),N),zero_zero_complex) ).

% of_nat_code
tff(fact_5930_of__nat__code,axiom,
    ! [N: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,N) = aa_Ext7969882492699152700d_enat(semiri8563196900006977889d_enat(aTP_Lamm_mo,N),zero_z5237406670263579293d_enat) ).

% of_nat_code
tff(fact_5931_of__nat__code,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_mp),N),zero_zero_real) ).

% of_nat_code
tff(fact_5932_of__nat__code,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_int_int(aa_nat_fun_int_int(semiri8420488043553186161ux_int(aTP_Lamm_mq),N),zero_zero_int) ).

% of_nat_code
tff(fact_5933_of__nat__code,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_mr),N),zero_zero_nat) ).

% of_nat_code
tff(fact_5934_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_5935_lessThan__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,X) = aa_nat_set_nat(set_ord_lessThan_nat,Y) )
    <=> ( X = Y ) ) ).

% lessThan_eq_iff
tff(fact_5936_lessThan__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( set_ord_lessThan_int(X) = set_ord_lessThan_int(Y) )
    <=> ( X = Y ) ) ).

% lessThan_eq_iff
tff(fact_5937_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),set_or890127255671739683et_nat(K)))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,I),K)) ) ).

% lessThan_iff
tff(fact_5938_lessThan__iff,axiom,
    ! [I: extended_enat,K: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),set_or8419480210114673929d_enat(K)))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,I),K)) ) ).

% lessThan_iff
tff(fact_5939_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( pp(aa_set_real_bool(member_real(I),set_or5984915006950818249n_real(K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),K)) ) ).

% lessThan_iff
tff(fact_5940_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% lessThan_iff
tff(fact_5941_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( pp(aa_set_int_bool(member_int(I),set_ord_lessThan_int(K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) ) ).

% lessThan_iff
tff(fact_5942_finite__lessThan,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_lessThan_nat,K))) ).

% finite_lessThan
tff(fact_5943_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or5984915006950818249n_real(X)),set_or5984915006950818249n_real(Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_5944_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,X)),aa_nat_set_nat(set_ord_lessThan_nat,Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_5945_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_lessThan_int(X)),set_ord_lessThan_int(Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_5946_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_5947_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_5948_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_5949_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(comm_s7457072308508201937r_real(A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_5950_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_5951_single__Diff__lessThan,axiom,
    ! [K: extended_enat] : minus_925952699566721837d_enat(insert_Extended_enat(K,bot_bo7653980558646680370d_enat),set_or8419480210114673929d_enat(K)) = insert_Extended_enat(K,bot_bo7653980558646680370d_enat) ).

% single_Diff_lessThan
tff(fact_5952_single__Diff__lessThan,axiom,
    ! [K: real] : minus_minus_set_real(insert_real(K,bot_bot_set_real),set_or5984915006950818249n_real(K)) = insert_real(K,bot_bot_set_real) ).

% single_Diff_lessThan
tff(fact_5953_single__Diff__lessThan,axiom,
    ! [K: nat] : minus_minus_set_nat(insert_nat(K,bot_bot_set_nat),aa_nat_set_nat(set_ord_lessThan_nat,K)) = insert_nat(K,bot_bot_set_nat) ).

% single_Diff_lessThan
tff(fact_5954_single__Diff__lessThan,axiom,
    ! [K: int] : minus_minus_set_int(insert_int(K,bot_bot_set_int),set_ord_lessThan_int(K)) = insert_int(K,bot_bot_set_int) ).

% single_Diff_lessThan
tff(fact_5955_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G,N)) ).

% sum.lessThan_Suc
tff(fact_5956_sum_OlessThan__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_Extended_enat(G,N)) ).

% sum.lessThan_Suc
tff(fact_5957_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_nat(G,N)) ).

% sum.lessThan_Suc
tff(fact_5958_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G,N)) ).

% sum.lessThan_Suc
tff(fact_5959_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M = N ) ) ).

% int_int_eq
tff(fact_5960_lessThan__non__empty,axiom,
    ! [X: real] : set_or5984915006950818249n_real(X) != bot_bot_set_real ).

% lessThan_non_empty
tff(fact_5961_lessThan__non__empty,axiom,
    ! [X: int] : set_ord_lessThan_int(X) != bot_bot_set_int ).

% lessThan_non_empty
tff(fact_5962_infinite__Iio,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_lessThan_int(A))) ).

% infinite_Iio
tff(fact_5963_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_ms,U)) ).

% lessThan_def
tff(fact_5964_lessThan__def,axiom,
    ! [U: extended_enat] : set_or8419480210114673929d_enat(U) = collec4429806609662206161d_enat(aa_Ext291666158496884126t_bool(aTP_Lamm_mt,U)) ).

% lessThan_def
tff(fact_5965_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_mu,U)) ).

% lessThan_def
tff(fact_5966_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,U)) ).

% lessThan_def
tff(fact_5967_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_mv,U)) ).

% lessThan_def
tff(fact_5968_Iio__eq__empty__iff,axiom,
    ! [N: extended_enat] :
      ( ( set_or8419480210114673929d_enat(N) = bot_bo7653980558646680370d_enat )
    <=> ( N = bot_bo4199563552545308370d_enat ) ) ).

% Iio_eq_empty_iff
tff(fact_5969_Iio__eq__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_5970_lessThan__strict__subset__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(set_or8419480210114673929d_enat(M)),set_or8419480210114673929d_enat(N)))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_5971_lessThan__strict__subset__iff,axiom,
    ! [M: real,N: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or5984915006950818249n_real(M)),set_or5984915006950818249n_real(N)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_5972_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,M)),aa_nat_set_nat(set_ord_lessThan_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_5973_lessThan__strict__subset__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_ord_lessThan_int(M)),set_ord_lessThan_int(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_5974_pochhammer__pos,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N))) ) ).

% pochhammer_pos
tff(fact_5975_pochhammer__pos,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(X),N))) ) ).

% pochhammer_pos
tff(fact_5976_pochhammer__pos,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).

% pochhammer_pos
tff(fact_5977_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),M) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5978_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5979_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),M) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5980_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5981_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_5982_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_5983_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ? [K3: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_lessThan_nat,K3))) ) ).

% finite_nat_bounded
tff(fact_5984_finite__nat__iff__bounded,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ? [K2: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_lessThan_nat,K2))) ) ).

% finite_nat_iff_bounded
tff(fact_5985_pochhammer__nonneg,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(X),N))) ) ).

% pochhammer_nonneg
tff(fact_5986_pochhammer__nonneg,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N))) ) ).

% pochhammer_nonneg
tff(fact_5987_pochhammer__nonneg,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).

% pochhammer_nonneg
tff(fact_5988_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% pochhammer_0_left
tff(fact_5989_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(zero_zero_real),N) = zero_zero_real ) ) ) ).

% pochhammer_0_left
tff(fact_5990_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% pochhammer_0_left
tff(fact_5991_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% pochhammer_0_left
tff(fact_5992_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_Extended_enat(comm_s3181272606743183617d_enat(zero_z5237406670263579293d_enat),N) = one_on7984719198319812577d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_Extended_enat(comm_s3181272606743183617d_enat(zero_z5237406670263579293d_enat),N) = zero_z5237406670263579293d_enat ) ) ) ).

% pochhammer_0_left
tff(fact_5993_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_5994_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mw(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_5995_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_mx(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_5996_sum__diff__distrib,axiom,
    ! [Q: fun_int_nat,P: fun_int_nat,N: int] :
      ( ! [X4: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(Q,X4)),aa_int_nat(P,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(P,set_ord_lessThan_int(N))),groups4541462559716669496nt_nat(Q,set_ord_lessThan_int(N))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_kw(Q),P),set_ord_lessThan_int(N)) ) ) ).

% sum_diff_distrib
tff(fact_5997_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,N: nat] :
      ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Q,X4)),aa_nat_nat(P,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kx(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ) ).

% sum_diff_distrib
tff(fact_5998_log__of__power__le,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(power_power_real(B2),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log(B2),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_le
tff(fact_5999_log__of__power__less,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(power_power_real(B2),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log(B2),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_less
tff(fact_6000_pochhammer__rec,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),N)) ).

% pochhammer_rec
tff(fact_6001_pochhammer__rec,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),N)) ).

% pochhammer_rec
tff(fact_6002_pochhammer__rec,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = aa_real_real(times_times_real(A),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(A),one_one_real)),N)) ).

% pochhammer_rec
tff(fact_6003_pochhammer__rec,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),N)) ).

% pochhammer_rec
tff(fact_6004_pochhammer__rec,axiom,
    ! [A: extended_enat,N: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),aa_nat_nat(suc,N)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_nat_Extended_enat(comm_s3181272606743183617d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),one_on7984719198319812577d_enat)),N)) ).

% pochhammer_rec
tff(fact_6005_pochhammer__rec_H,axiom,
    ! [Z2: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z2),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Z2),aa_nat_complex(semiri8010041392384452111omplex,N))),aa_nat_complex(comm_s2602460028002588243omplex(Z2),N)) ).

% pochhammer_rec'
tff(fact_6006_pochhammer__rec_H,axiom,
    ! [Z2: extended_enat,N: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z2),aa_nat_nat(suc,N)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Z2),aa_nat_Extended_enat(semiri4216267220026989637d_enat,N))),aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z2),N)) ).

% pochhammer_rec'
tff(fact_6007_pochhammer__rec_H,axiom,
    ! [Z2: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z2),aa_nat_nat(suc,N)) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(Z2),aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(comm_s7457072308508201937r_real(Z2),N)) ).

% pochhammer_rec'
tff(fact_6008_pochhammer__rec_H,axiom,
    ! [Z2: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z2),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(comm_s4660882817536571857er_int(Z2),N)) ).

% pochhammer_rec'
tff(fact_6009_pochhammer__rec_H,axiom,
    ! [Z2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),aa_nat_nat(semiri1316708129612266289at_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),N)) ).

% pochhammer_rec'
tff(fact_6010_pochhammer__Suc,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(A),N)),aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,N))) ).

% pochhammer_Suc
tff(fact_6011_pochhammer__Suc,axiom,
    ! [A: extended_enat,N: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),aa_nat_nat(suc,N)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),N)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),aa_nat_Extended_enat(semiri4216267220026989637d_enat,N))) ).

% pochhammer_Suc
tff(fact_6012_pochhammer__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = aa_real_real(times_times_real(aa_nat_real(comm_s7457072308508201937r_real(A),N)),aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% pochhammer_Suc
tff(fact_6013_pochhammer__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% pochhammer_Suc
tff(fact_6014_pochhammer__Suc,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% pochhammer_Suc
tff(fact_6015_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6016_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6017_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6018_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6019_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6020_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6021_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
    <=> ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K2)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_6022_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = zero_zero_real )
    <=> ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K2)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_6023_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6024_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6025_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6026_pochhammer__product_H,axiom,
    ! [Z2: complex,N: nat,M: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z2),N)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z2),aa_nat_complex(semiri8010041392384452111omplex,N))),M)) ).

% pochhammer_product'
tff(fact_6027_pochhammer__product_H,axiom,
    ! [Z2: extended_enat,N: nat,M: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z2),N)),aa_nat_Extended_enat(comm_s3181272606743183617d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Z2),aa_nat_Extended_enat(semiri4216267220026989637d_enat,N))),M)) ).

% pochhammer_product'
tff(fact_6028_pochhammer__product_H,axiom,
    ! [Z2: real,N: nat,M: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_real_real(times_times_real(aa_nat_real(comm_s7457072308508201937r_real(Z2),N)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(Z2),aa_nat_real(semiri5074537144036343181t_real,N))),M)) ).

% pochhammer_product'
tff(fact_6029_pochhammer__product_H,axiom,
    ! [Z2: int,N: nat,M: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z2),N)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_nat_int(semiri1314217659103216013at_int,N))),M)) ).

% pochhammer_product'
tff(fact_6030_pochhammer__product_H,axiom,
    ! [Z2: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),N)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),aa_nat_nat(semiri1316708129612266289at_nat,N))),M)) ).

% pochhammer_product'
tff(fact_6031_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lh(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6032_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,zero_zero_nat)),groups7108830773950497114d_enat(aTP_Lamm_li(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6033_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6034_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kz(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6035_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lj(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6036_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lk(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(minus_minus_real(aa_nat_real(F,M)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6037_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_my(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,M)) ).

% sum_lessThan_telescope'
tff(fact_6038_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mz(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(minus_minus_real(aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,M)) ).

% sum_lessThan_telescope'
tff(fact_6039_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_6040_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kz(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_6041_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% le_log2_of_power
tff(fact_6042_one__diff__power__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6043_one__diff__power__eq,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6044_one__diff__power__eq,axiom,
    ! [X: real,N: nat] : aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X),N)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6045_power__diff__1__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6046_power__diff__1__eq,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),one_one_complex)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6047_power__diff__1__eq,axiom,
    ! [X: real,N: nat] : aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X),N)),one_one_real) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(X),one_one_real)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6048_geometric__sum,axiom,
    ! [X: complex,N: nat] :
      ( ( X != one_one_complex )
     => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_6049_geometric__sum,axiom,
    ! [X: real,N: nat] :
      ( ( X != one_one_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X),N)),one_one_real)),aa_real_real(minus_minus_real(X),one_one_real)) ) ) ).

% geometric_sum
tff(fact_6050_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(Z2),N) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z2),M)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z2),aa_nat_complex(semiri8010041392384452111omplex,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_6051_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z2: extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z2),N) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z2),M)),aa_nat_Extended_enat(comm_s3181272606743183617d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Z2),aa_nat_Extended_enat(semiri4216267220026989637d_enat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_6052_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_real(comm_s7457072308508201937r_real(Z2),N) = aa_real_real(times_times_real(aa_nat_real(comm_s7457072308508201937r_real(Z2),M)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(Z2),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_6053_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z2),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z2),M)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_6054_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),M)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_6055_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% less_log2_of_power
tff(fact_6056_sum__gp__strict,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_strict
tff(fact_6057_sum__gp__strict,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X),N))),aa_real_real(minus_minus_real(one_one_real),X)) ) ) ) ).

% sum_gp_strict
tff(fact_6058_lemma__termdiff1,axiom,
    ! [Z2: int,H: int,M: nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_na(Z2),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nb(Z2),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_6059_lemma__termdiff1,axiom,
    ! [Z2: complex,H: complex,M: nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nc(Z2),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nd(Z2),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_6060_lemma__termdiff1,axiom,
    ! [Z2: real,H: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ne(Z2),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nf(Z2),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_6061_diff__power__eq__sum,axiom,
    ! [X: int,N: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ng(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6062_diff__power__eq__sum,axiom,
    ! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nh(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6063_diff__power__eq__sum,axiom,
    ! [X: real,N: nat,Y: real] : aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X),aa_nat_nat(suc,N))),aa_nat_real(power_power_real(Y),aa_nat_nat(suc,N))) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ni(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6064_power__diff__sumr2,axiom,
    ! [X: int,N: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_nj(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6065_power__diff__sumr2,axiom,
    ! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nk(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6066_power__diff__sumr2,axiom,
    ! [X: real,N: nat,Y: real] : aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X),N)),aa_nat_real(power_power_real(Y),N)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_nl(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6067_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K)) = aa_complex_complex(times_times_complex(R2),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K)) ).

% pochhammer_absorb_comp
tff(fact_6068_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(R2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,R2)),K)) = aa_real_real(times_times_real(R2),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,R2)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_6069_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_6070_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_less
tff(fact_6071_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_le
tff(fact_6072_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_int,K5: int,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,P4)),K5)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K5))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6073_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_nat,K5: nat,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,P4)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),K5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6074_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_real,K5: real,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,P4)),K5)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),K5))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6075_one__diff__power__eq_H,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_nm(X),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6076_one__diff__power__eq_H,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_nn(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6077_one__diff__power__eq_H,axiom,
    ! [X: real,N: nat] : aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X),N)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_no(X),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6078_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_np(F),G)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nq(F)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nr(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum_split_even_odd
tff(fact_6079_pochhammer__minus,axiom,
    ! [B2: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B2)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B2),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% pochhammer_minus
tff(fact_6080_pochhammer__minus,axiom,
    ! [B2: real,K: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B2)),K) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(B2),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_6081_pochhammer__minus,axiom,
    ! [B2: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B2)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).

% pochhammer_minus
tff(fact_6082_pochhammer__minus_H,axiom,
    ! [B2: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B2),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B2)),K)) ).

% pochhammer_minus'
tff(fact_6083_pochhammer__minus_H,axiom,
    ! [B2: real,K: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(B2),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B2)),K)) ).

% pochhammer_minus'
tff(fact_6084_pochhammer__minus_H,axiom,
    ! [B2: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B2)),K)) ).

% pochhammer_minus'
tff(fact_6085_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_6086_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_6087_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),zero_zero_real))
    <=> ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_6088_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real))
    <=> ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_6089_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,X)))
    <=> ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_6090_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X)))
    <=> ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_6091_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_6092_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,X) = zero_zero_real )
    <=> ( X = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_6093_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_6094_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_6095_nonzero__norm__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A),B2)) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B2)) ) ) ).

% nonzero_norm_divide
tff(fact_6096_nonzero__norm__divide,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B2)) ) ) ).

% nonzero_norm_divide
tff(fact_6097_norm__diff__ineq,axiom,
    ! [A: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(A),B2)))) ).

% norm_diff_ineq
tff(fact_6098_norm__diff__ineq,axiom,
    ! [A: complex,B2: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B2)))) ).

% norm_diff_ineq
tff(fact_6099_norm__uminus__minus,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,X)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X),Y)) ).

% norm_uminus_minus
tff(fact_6100_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y)) ).

% norm_uminus_minus
tff(fact_6101_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N: nat,Z2: real] :
      ( ( aa_nat_real(power_power_real(W2),N) = aa_nat_real(power_power_real(Z2),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_real_real(real_V7735802525324610683m_real,W2) = aa_real_real(real_V7735802525324610683m_real,Z2) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6102_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,N: nat,Z2: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),N) = aa_nat_complex(power_power_complex(Z2),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_complex_real(real_V1022390504157884413omplex,W2) = aa_complex_real(real_V1022390504157884413omplex,Z2) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6103_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_6104_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_6105_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X),Y))),aa_real_real(plus_plus_real(R2),S))) ) ) ).

% norm_add_less
tff(fact_6106_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(plus_plus_real(R2),S))) ) ) ).

% norm_add_less
tff(fact_6107_norm__add__leD,axiom,
    ! [A: real,B2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(A),B2))),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B2)),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,A)),C))) ) ).

% norm_add_leD
tff(fact_6108_norm__add__leD,axiom,
    ! [A: complex,B2: complex,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B2))),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B2)),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,A)),C))) ) ).

% norm_add_leD
tff(fact_6109_norm__triangle__le,axiom,
    ! [X: real,Y: real,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X),Y))),E2)) ) ).

% norm_triangle_le
tff(fact_6110_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2)) ) ).

% norm_triangle_le
tff(fact_6111_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X),Y))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).

% norm_triangle_ineq
tff(fact_6112_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).

% norm_triangle_ineq
tff(fact_6113_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B2: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B2)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(A),B2))),aa_real_real(plus_plus_real(R2),S))) ) ) ).

% norm_triangle_mono
tff(fact_6114_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B2: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B2)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B2))),aa_real_real(plus_plus_real(R2),S))) ) ) ).

% norm_triangle_mono
tff(fact_6115_power__eq__1__iff,axiom,
    ! [W2: real,N: nat] :
      ( ( aa_nat_real(power_power_real(W2),N) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W2) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6116_power__eq__1__iff,axiom,
    ! [W2: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(W2),N) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W2) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6117_norm__diff__triangle__ineq,axiom,
    ! [A: real,B2: real,C: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B2)),aa_real_real(plus_plus_real(C),D)))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(B2),D))))) ).

% norm_diff_triangle_ineq
tff(fact_6118_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B2: complex,C: complex,D: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B2)),aa_complex_complex(plus_plus_complex(C),D)))),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B2),D))))) ).

% norm_diff_triangle_ineq
tff(fact_6119_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ky(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_6120_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kz(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_6121_sumr__cos__zero__one,axiom,
    ! [N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ns),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_6122_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z2),aa_nat_nat(suc,N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z2),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(bit0(one))))),aa_nat_nat(suc,N))) = groups6464643781859351333omplex(aTP_Lamm_nt(Z2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6123_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: real,N: nat] : aa_real_real(times_times_real(aa_nat_real(comm_s7457072308508201937r_real(Z2),aa_nat_nat(suc,N))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(Z2),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))),aa_nat_nat(suc,N))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_nu(Z2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6124_pochhammer__code,axiom,
    ! [N: nat,A: complex] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_nv(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_complex) ) ) ) ).

% pochhammer_code
tff(fact_6125_pochhammer__code,axiom,
    ! [N: nat,A: extended_enat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),N) = one_on7984719198319812577d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),N) = aa_Ext7969882492699152700d_enat(set_fo2538466533108834004d_enat(aTP_Lamm_nw(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_on7984719198319812577d_enat) ) ) ) ).

% pochhammer_code
tff(fact_6126_pochhammer__code,axiom,
    ! [N: nat,A: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_nx(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_real) ) ) ) ).

% pochhammer_code
tff(fact_6127_pochhammer__code,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_ny(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_int) ) ) ) ).

% pochhammer_code
tff(fact_6128_pochhammer__code,axiom,
    ! [N: nat,A: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_nz(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_nat) ) ) ) ).

% pochhammer_code
tff(fact_6129_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(B2),X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(B2),aa_nat_real(semiri5074537144036343181t_real,K))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(powr_real(B2),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_6130_geometric__deriv__sums,axiom,
    ! [Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z2)),one_one_real))
     => pp(aa_real_bool(sums_real(aTP_Lamm_oa(Z2)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(one_one_real),Z2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ).

% geometric_deriv_sums
tff(fact_6131_geometric__deriv__sums,axiom,
    ! [Z2: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),one_one_real))
     => pp(aa_complex_bool(sums_complex(aTP_Lamm_ob(Z2)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Z2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ).

% geometric_deriv_sums
tff(fact_6132_powr__eq__0__iff,axiom,
    ! [W2: real,Z2: real] :
      ( ( aa_real_real(powr_real(W2),Z2) = zero_zero_real )
    <=> ( W2 = zero_zero_real ) ) ).

% powr_eq_0_iff
tff(fact_6133_powr__0,axiom,
    ! [Z2: real] : aa_real_real(powr_real(zero_zero_real),Z2) = zero_zero_real ).

% powr_0
tff(fact_6134_of__nat__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_lw(F),A2) ).

% of_nat_prod
tff(fact_6135_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_ma(F)),A2) ).

% of_nat_prod
tff(fact_6136_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ly(F)),A2) ).

% of_nat_prod
tff(fact_6137_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_lz(F)),A2) ).

% of_nat_prod
tff(fact_6138_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,aa_set_nat_int(groups705719431365010083at_int(F),A2)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_hk(F)),A2) ).

% of_int_prod
tff(fact_6139_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_int(ring_1_of_int_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_oc(F)),A2) ).

% of_int_prod
tff(fact_6140_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups1705073143266064639nt_int(F,A2)) = groups2316167850115554303t_real(aTP_Lamm_hl(F),A2) ).

% of_int_prod
tff(fact_6141_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_hm(F),A2) ).

% of_int_prod
tff(fact_6142_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups861055069439313189ex_nat(F,A2) = zero_zero_nat )
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A2))
            & ( aa_complex_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_6143_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups1707563613775114915nt_nat(F,A2) = zero_zero_nat )
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A2))
            & ( aa_int_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_6144_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( groups2880970938130013265at_nat(F,A2) = zero_zero_nat )
      <=> ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
            & ( aa_Extended_enat_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_6145_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_real(groups129246275422532515t_real(F),A2) = zero_zero_real )
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A2))
            & ( aa_nat_real(F,X5) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6146_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups766887009212190081x_real(F,A2) = zero_zero_real )
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A2))
            & ( aa_complex_real(F,X5) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6147_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups2316167850115554303t_real(F,A2) = zero_zero_real )
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A2))
            & ( aa_int_real(F,X5) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6148_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( groups97031904164794029t_real(F,A2) = zero_zero_real )
      <=> ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
            & ( aa_Ext2541572767299617110t_real(F,X5) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6149_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups858564598930262913ex_int(F,A2) = zero_zero_int )
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A2))
            & ( aa_complex_int(F,X5) = zero_zero_int ) ) ) ) ).

% prod_zero_iff
tff(fact_6150_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( groups2878480467620962989at_int(F,A2) = zero_zero_int )
      <=> ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
            & ( aa_Extended_enat_int(F,X5) = zero_zero_int ) ) ) ) ).

% prod_zero_iff
tff(fact_6151_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( groups6464643781859351333omplex(F,A2) = zero_zero_complex )
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A2))
            & ( aa_nat_complex(F,X5) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_6152_prod_Oempty,axiom,
    ! [G: fun_Ex7660298398204468809at_nat] : groups2880970938130013265at_nat(G,bot_bo7653980558646680370d_enat) = one_one_nat ).

% prod.empty
tff(fact_6153_prod_Oempty,axiom,
    ! [G: fun_Ex3482447378695272101at_int] : groups2878480467620962989at_int(G,bot_bo7653980558646680370d_enat) = one_one_int ).

% prod.empty
tff(fact_6154_prod_Oempty,axiom,
    ! [G: fun_Ex3741802288643809319omplex] : groups4622424608036095791omplex(G,bot_bo7653980558646680370d_enat) = one_one_complex ).

% prod.empty
tff(fact_6155_prod_Oempty,axiom,
    ! [G: fun_Ex1451620480498224165t_real] : groups97031904164794029t_real(G,bot_bo7653980558646680370d_enat) = one_one_real ).

% prod.empty
tff(fact_6156_prod_Oempty,axiom,
    ! [G: fun_real_nat] : groups4696554848551431203al_nat(G,bot_bot_set_real) = one_one_nat ).

% prod.empty
tff(fact_6157_prod_Oempty,axiom,
    ! [G: fun_real_int] : groups4694064378042380927al_int(G,bot_bot_set_real) = one_one_int ).

% prod.empty
tff(fact_6158_prod_Oempty,axiom,
    ! [G: fun_real_complex] : groups713298508707869441omplex(G,bot_bot_set_real) = one_one_complex ).

% prod.empty
tff(fact_6159_prod_Oempty,axiom,
    ! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).

% prod.empty
tff(fact_6160_prod_Oempty,axiom,
    ! [G: fun_nat_complex] : groups6464643781859351333omplex(G,bot_bot_set_nat) = one_one_complex ).

% prod.empty
tff(fact_6161_prod_Oempty,axiom,
    ! [G: fun_nat_real] : aa_set_nat_real(groups129246275422532515t_real(G),bot_bot_set_nat) = one_one_real ).

% prod.empty
tff(fact_6162_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups861055069439313189ex_nat(G,A2) = one_one_nat ) ) ).

% prod.infinite
tff(fact_6163_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups1707563613775114915nt_nat(G,A2) = one_one_nat ) ) ).

% prod.infinite
tff(fact_6164_prod_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2880970938130013265at_nat(G,A2) = one_one_nat ) ) ).

% prod.infinite
tff(fact_6165_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_int] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups858564598930262913ex_int(G,A2) = one_one_int ) ) ).

% prod.infinite
tff(fact_6166_prod_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2878480467620962989at_int(G,A2) = one_one_int ) ) ).

% prod.infinite
tff(fact_6167_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_complex] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups6464643781859351333omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_6168_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups3708469109370488835omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_6169_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups7440179247065528705omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_6170_prod_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex3741802288643809319omplex] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups4622424608036095791omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_6171_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_6172_powr__zero__eq__one,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(powr_real(X),zero_zero_real) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( aa_real_real(powr_real(X),zero_zero_real) = one_one_real ) ) ) ).

% powr_zero_eq_one
tff(fact_6173_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_real_nat(F,A)),groups4696554848551431203al_nat(F,A2))) ) ) ).

% dvd_prodI
tff(fact_6174_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_complex_nat(F,A)),groups861055069439313189ex_nat(F,A2))) ) ) ).

% dvd_prodI
tff(fact_6175_dvd__prodI,axiom,
    ! [A2: set_int,A: int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(F,A)),groups1707563613775114915nt_nat(F,A2))) ) ) ).

% dvd_prodI
tff(fact_6176_dvd__prodI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_Extended_enat_nat(F,A)),groups2880970938130013265at_nat(F,A2))) ) ) ).

% dvd_prodI
tff(fact_6177_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_real_int(F,A)),groups4694064378042380927al_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_6178_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_complex_int(F,A)),groups858564598930262913ex_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_6179_dvd__prodI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_Extended_enat_int(F,A)),groups2878480467620962989at_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_6180_dvd__prodI,axiom,
    ! [A2: set_nat,A: nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(F,A)),aa_set_nat_int(groups705719431365010083at_int(F),A2))) ) ) ).

% dvd_prodI
tff(fact_6181_dvd__prodI,axiom,
    ! [A2: set_int,A: int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(F,A)),groups1705073143266064639nt_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_6182_dvd__prodI,axiom,
    ! [A2: set_nat,A: nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(F,A)),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))) ) ) ).

% dvd_prodI
tff(fact_6183_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B2: nat,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ( ( B2 = aa_real_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),groups4696554848551431203al_nat(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_6184_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B2: nat,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(A),A2))
       => ( ( B2 = aa_complex_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),groups861055069439313189ex_nat(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_6185_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B2: nat,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ( ( B2 = aa_int_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),groups1707563613775114915nt_nat(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_6186_dvd__prod__eqI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B2: nat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
       => ( ( B2 = aa_Extended_enat_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),groups2880970938130013265at_nat(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_6187_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B2: int,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ( ( B2 = aa_real_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),groups4694064378042380927al_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_6188_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B2: int,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(A),A2))
       => ( ( B2 = aa_complex_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),groups858564598930262913ex_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_6189_dvd__prod__eqI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B2: int,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
       => ( ( B2 = aa_Extended_enat_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),groups2878480467620962989at_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_6190_dvd__prod__eqI,axiom,
    ! [A2: set_nat,A: nat,B2: int,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ( ( B2 = aa_nat_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),aa_set_nat_int(groups705719431365010083at_int(F),A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_6191_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B2: int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ( ( B2 = aa_int_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),groups1705073143266064639nt_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_6192_dvd__prod__eqI,axiom,
    ! [A2: set_nat,A: nat,B2: nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ( ( B2 = aa_nat_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_6193_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_6194_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4696554848551431203al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_od(A),B2),S2) = aa_real_nat(B2,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4696554848551431203al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_od(A),B2),S2) = one_one_nat ) ) ) ) ).

% prod.delta
tff(fact_6195_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups861055069439313189ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_oe(A),B2),S2) = aa_complex_nat(B2,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups861055069439313189ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_oe(A),B2),S2) = one_one_nat ) ) ) ) ).

% prod.delta
tff(fact_6196_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_of(A),B2),S2) = aa_int_nat(B2,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_of(A),B2),S2) = one_one_nat ) ) ) ) ).

% prod.delta
tff(fact_6197_prod_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2880970938130013265at_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_og(A),B2),S2) = aa_Extended_enat_nat(B2,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2880970938130013265at_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_og(A),B2),S2) = one_one_nat ) ) ) ) ).

% prod.delta
tff(fact_6198_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B2: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4694064378042380927al_int(aa_fun8270430325945277160al_int(aTP_Lamm_oh(A),B2),S2) = aa_real_int(B2,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4694064378042380927al_int(aa_fun8270430325945277160al_int(aTP_Lamm_oh(A),B2),S2) = one_one_int ) ) ) ) ).

% prod.delta
tff(fact_6199_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups858564598930262913ex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_oi(A),B2),S2) = aa_complex_int(B2,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups858564598930262913ex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_oi(A),B2),S2) = one_one_int ) ) ) ) ).

% prod.delta
tff(fact_6200_prod_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2878480467620962989at_int(aa_fun1695976788104708604at_int(aTP_Lamm_oj(A),B2),S2) = aa_Extended_enat_int(B2,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2878480467620962989at_int(aa_fun1695976788104708604at_int(aTP_Lamm_oj(A),B2),S2) = one_one_int ) ) ) ) ).

% prod.delta
tff(fact_6201_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B2: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_ok(A),B2),S2) = aa_real_complex(B2,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_ok(A),B2),S2) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_6202_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ol(A),B2),S2) = aa_nat_complex(B2,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ol(A),B2),S2) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_6203_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_om(A),B2),S2) = aa_complex_complex(B2,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_om(A),B2),S2) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_6204_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4696554848551431203al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_on(A),B2),S2) = aa_real_nat(B2,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4696554848551431203al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_on(A),B2),S2) = one_one_nat ) ) ) ) ).

% prod.delta'
tff(fact_6205_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups861055069439313189ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_oo(A),B2),S2) = aa_complex_nat(B2,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups861055069439313189ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_oo(A),B2),S2) = one_one_nat ) ) ) ) ).

% prod.delta'
tff(fact_6206_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_op(A),B2),S2) = aa_int_nat(B2,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_op(A),B2),S2) = one_one_nat ) ) ) ) ).

% prod.delta'
tff(fact_6207_prod_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2880970938130013265at_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_oq(A),B2),S2) = aa_Extended_enat_nat(B2,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2880970938130013265at_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_oq(A),B2),S2) = one_one_nat ) ) ) ) ).

% prod.delta'
tff(fact_6208_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B2: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4694064378042380927al_int(aa_fun8270430325945277160al_int(aTP_Lamm_or(A),B2),S2) = aa_real_int(B2,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4694064378042380927al_int(aa_fun8270430325945277160al_int(aTP_Lamm_or(A),B2),S2) = one_one_int ) ) ) ) ).

% prod.delta'
tff(fact_6209_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups858564598930262913ex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_os(A),B2),S2) = aa_complex_int(B2,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups858564598930262913ex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_os(A),B2),S2) = one_one_int ) ) ) ) ).

% prod.delta'
tff(fact_6210_prod_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2878480467620962989at_int(aa_fun1695976788104708604at_int(aTP_Lamm_ot(A),B2),S2) = aa_Extended_enat_int(B2,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2878480467620962989at_int(aa_fun1695976788104708604at_int(aTP_Lamm_ot(A),B2),S2) = one_one_int ) ) ) ) ).

% prod.delta'
tff(fact_6211_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B2: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_ou(A),B2),S2) = aa_real_complex(B2,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_ou(A),B2),S2) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_6212_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ov(A),B2),S2) = aa_nat_complex(B2,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ov(A),B2),S2) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_6213_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ow(A),B2),S2) = aa_complex_complex(B2,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ow(A),B2),S2) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_6214_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups4696554848551431203al_nat(G,insert_real(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(G,X)),groups4696554848551431203al_nat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_6215_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups861055069439313189ex_nat(G,insert_complex(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(G,X)),groups861055069439313189ex_nat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_6216_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X),A2))
       => ( groups1707563613775114915nt_nat(G,insert_int(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(G,X)),groups1707563613775114915nt_nat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_6217_prod_Oinsert,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
       => ( groups2880970938130013265at_nat(G,insert_Extended_enat(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Extended_enat_nat(G,X)),groups2880970938130013265at_nat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_6218_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups4694064378042380927al_int(G,insert_real(X,A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(G,X)),groups4694064378042380927al_int(G,A2)) ) ) ) ).

% prod.insert
tff(fact_6219_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups858564598930262913ex_int(G,insert_complex(X,A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_complex_int(G,X)),groups858564598930262913ex_int(G,A2)) ) ) ) ).

% prod.insert
tff(fact_6220_prod_Oinsert,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
       => ( groups2878480467620962989at_int(G,insert_Extended_enat(X,A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Extended_enat_int(G,X)),groups2878480467620962989at_int(G,A2)) ) ) ) ).

% prod.insert
tff(fact_6221_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1681761925125756287l_real(G,insert_real(X,A2)) = aa_real_real(times_times_real(aa_real_real(G,X)),groups1681761925125756287l_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_6222_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),insert_nat(X,A2)) = aa_real_real(times_times_real(aa_nat_real(G,X)),aa_set_nat_real(groups129246275422532515t_real(G),A2)) ) ) ) ).

% prod.insert
tff(fact_6223_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups766887009212190081x_real(G,insert_complex(X,A2)) = aa_real_real(times_times_real(aa_complex_real(G,X)),groups766887009212190081x_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_6224_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G,N)) ).

% prod.lessThan_Suc
tff(fact_6225_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_complex(G,N)) ).

% prod.lessThan_Suc
tff(fact_6226_prod_OlessThan__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_Extended_enat(G,N)) ).

% prod.lessThan_Suc
tff(fact_6227_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G,N)) ).

% prod.lessThan_Suc
tff(fact_6228_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_nat(G,N)) ).

% prod.lessThan_Suc
tff(fact_6229_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,X: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_kq(A)),X))
    <=> ( aa_nat_real(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_6230_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,X: complex] :
      ( pp(aa_complex_bool(sums_complex(aTP_Lamm_kp(A)),X))
    <=> ( aa_nat_complex(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_6231_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6232_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6233_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_na8797796813480665199d_enat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_on7984719198319812577d_enat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_Extended_enat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6234_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6235_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6236_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B: set_nat,G: fun_real_fun_nat_int,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups4694064378042380927al_int(aa_fun1626300625876492518al_int(aa_fun1560246819506852821al_int(aTP_Lamm_ox(B),G),R),A2) = aa_set_nat_int(groups705719431365010083at_int(aa_fun1494899438858969354at_int(aa_fun6942549246588167161at_int(aTP_Lamm_oz(A2),G),R)),B) ) ) ) ).

% prod.swap_restrict
tff(fact_6237_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B: set_nat,G: fun_co7370231399432144932at_int,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups858564598930262913ex_int(aa_fun4872694516007147366ex_int(aa_fun2110580046701903059ex_int(aTP_Lamm_pa(B),G),R),A2) = aa_set_nat_int(groups705719431365010083at_int(aa_fun6041977265744554632at_int(aa_fun3475035881565545205at_int(aTP_Lamm_pc(A2),G),R)),B) ) ) ) ).

% prod.swap_restrict
tff(fact_6238_prod_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B: set_nat,G: fun_Ex4633621507750988018at_int,R: fun_Ex7682446853002821117t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups2878480467620962989at_int(aa_fun2769206225242773418at_int(aa_fun1492286042523573003at_int(aTP_Lamm_pd(B),G),R),A2) = aa_set_nat_int(groups705719431365010083at_int(aa_fun6943296219937830324at_int(aa_fun3819077064105665321at_int(aTP_Lamm_pf(A2),G),R)),B) ) ) ) ).

% prod.swap_restrict
tff(fact_6239_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B: set_int,G: fun_real_fun_int_int,R: fun_re8661203372455305201t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( groups4694064378042380927al_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_pg(B),G),R),A2) = groups1705073143266064639nt_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_ph(A2),G),R),B) ) ) ) ).

% prod.swap_restrict
tff(fact_6240_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B: set_int,G: fun_co333780937270390912nt_int,R: fun_co2381122952212249071t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( groups858564598930262913ex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_pi(B),G),R),A2) = groups1705073143266064639nt_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_pj(A2),G),R),B) ) ) ) ).

% prod.swap_restrict
tff(fact_6241_prod_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B: set_int,G: fun_Ex6820543082444009806nt_int,R: fun_Ex7222163901356174497t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( groups2878480467620962989at_int(aa_fun1926933649907246598at_int(aa_fun1321938580149485067at_int(aTP_Lamm_pk(B),G),R),A2) = groups1705073143266064639nt_int(aa_fun1757119979448242868nt_int(aa_fun547143500490336389nt_int(aTP_Lamm_pl(A2),G),R),B) ) ) ) ).

% prod.swap_restrict
tff(fact_6242_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B: set_nat,G: fun_real_fun_nat_nat,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups4696554848551431203al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_pm(B),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_pn(A2),G),R)),B) ) ) ) ).

% prod.swap_restrict
tff(fact_6243_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B: set_nat,G: fun_co636942507509095880at_nat,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups861055069439313189ex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_po(B),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_pp(A2),G),R)),B) ) ) ) ).

% prod.swap_restrict
tff(fact_6244_prod_Oswap__restrict,axiom,
    ! [A2: set_int,B: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups1707563613775114915nt_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_pq(B),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_pr(A2),G),R)),B) ) ) ) ).

% prod.swap_restrict
tff(fact_6245_prod_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B: set_nat,G: fun_Ex7123704652682714774at_nat,R: fun_Ex7682446853002821117t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups2880970938130013265at_nat(aa_fun6947057244751970126at_nat(aa_fun635179117725900299at_nat(aTP_Lamm_ps(B),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun1897775202592251224at_nat(aa_fun594951741515554929at_nat(aTP_Lamm_pt(A2),G),R)),B) ) ) ) ).

% prod.swap_restrict
tff(fact_6246_prod__mono,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( ! [I4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,I4)),aa_Ext2541572767299617110t_real(G,I4))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups97031904164794029t_real(F,A2)),groups97031904164794029t_real(G,A2))) ) ).

% prod_mono
tff(fact_6247_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I4: real] :
          ( pp(aa_set_real_bool(member_real(I4),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I4)),aa_real_real(G,I4))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(G,A2))) ) ).

% prod_mono
tff(fact_6248_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I4: nat] :
          ( pp(aa_set_nat_bool(member_nat(I4),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I4)),aa_nat_real(G,I4))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A2)),aa_set_nat_real(groups129246275422532515t_real(G),A2))) ) ).

% prod_mono
tff(fact_6249_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I4: int] :
          ( pp(aa_set_int_bool(member_int(I4),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I4)),aa_int_real(G,I4))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(G,A2))) ) ).

% prod_mono
tff(fact_6250_prod__mono,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( ! [I4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_Extended_enat_nat(F,I4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,I4)),aa_Extended_enat_nat(G,I4))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups2880970938130013265at_nat(F,A2)),groups2880970938130013265at_nat(G,A2))) ) ).

% prod_mono
tff(fact_6251_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I4: real] :
          ( pp(aa_set_real_bool(member_real(I4),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I4)),aa_real_nat(G,I4))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F,A2)),groups4696554848551431203al_nat(G,A2))) ) ).

% prod_mono
tff(fact_6252_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I4: int] :
          ( pp(aa_set_int_bool(member_int(I4),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I4)),aa_int_nat(G,I4))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(G,A2))) ) ).

% prod_mono
tff(fact_6253_prod__mono,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int,G: fun_Ex3482447378695272101at_int] :
      ( ! [I4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_Extended_enat_int(F,I4)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Extended_enat_int(F,I4)),aa_Extended_enat_int(G,I4))) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups2878480467620962989at_int(F,A2)),groups2878480467620962989at_int(G,A2))) ) ).

% prod_mono
tff(fact_6254_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I4: real] :
          ( pp(aa_set_real_bool(member_real(I4),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,I4)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I4)),aa_real_int(G,I4))) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4694064378042380927al_int(F,A2)),groups4694064378042380927al_int(G,A2))) ) ).

% prod_mono
tff(fact_6255_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I4: nat] :
          ( pp(aa_set_nat_bool(member_nat(I4),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I4)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I4)),aa_nat_int(G,I4))) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)),aa_set_nat_int(groups705719431365010083at_int(G),A2))) ) ).

% prod_mono
tff(fact_6256_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_set_nat_int(groups705719431365010083at_int(F),A2))) ) ).

% prod_nonneg
tff(fact_6257_prod__nonneg,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A2))) ) ).

% prod_nonneg
tff(fact_6258_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))) ) ).

% prod_nonneg
tff(fact_6259_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_set_nat_int(groups705719431365010083at_int(F),A2))) ) ).

% prod_pos
tff(fact_6260_prod__pos,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A2))) ) ).

% prod_pos
tff(fact_6261_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))) ) ).

% prod_pos
tff(fact_6262_prod__ge__1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_Ext2541572767299617110t_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups97031904164794029t_real(F,A2))) ) ).

% prod_ge_1
tff(fact_6263_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A2))) ) ).

% prod_ge_1
tff(fact_6264_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),A2))) ) ).

% prod_ge_1
tff(fact_6265_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A2))) ) ).

% prod_ge_1
tff(fact_6266_prod__ge__1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_Extended_enat_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups2880970938130013265at_nat(F,A2))) ) ).

% prod_ge_1
tff(fact_6267_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A2))) ) ).

% prod_ge_1
tff(fact_6268_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_int_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups1707563613775114915nt_nat(F,A2))) ) ).

% prod_ge_1
tff(fact_6269_prod__ge__1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_Extended_enat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),groups2878480467620962989at_int(F,A2))) ) ).

% prod_ge_1
tff(fact_6270_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),groups4694064378042380927al_int(F,A2))) ) ).

% prod_ge_1
tff(fact_6271_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_set_nat_int(groups705719431365010083at_int(F),A2))) ) ).

% prod_ge_1
tff(fact_6272_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ? [X2: complex] :
            ( pp(aa_set_complex_bool(member_complex(X2),A2))
            & ( aa_complex_nat(F,X2) = zero_zero_nat ) )
       => ( groups861055069439313189ex_nat(F,A2) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_6273_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
            & ( aa_int_nat(F,X2) = zero_zero_nat ) )
       => ( groups1707563613775114915nt_nat(F,A2) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_6274_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ? [X2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A2))
            & ( aa_Extended_enat_nat(F,X2) = zero_zero_nat ) )
       => ( groups2880970938130013265at_nat(F,A2) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_6275_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ? [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
            & ( aa_nat_real(F,X2) = zero_zero_real ) )
       => ( aa_set_nat_real(groups129246275422532515t_real(F),A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_6276_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ? [X2: complex] :
            ( pp(aa_set_complex_bool(member_complex(X2),A2))
            & ( aa_complex_real(F,X2) = zero_zero_real ) )
       => ( groups766887009212190081x_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_6277_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
            & ( aa_int_real(F,X2) = zero_zero_real ) )
       => ( groups2316167850115554303t_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_6278_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ? [X2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A2))
            & ( aa_Ext2541572767299617110t_real(F,X2) = zero_zero_real ) )
       => ( groups97031904164794029t_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_6279_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ? [X2: complex] :
            ( pp(aa_set_complex_bool(member_complex(X2),A2))
            & ( aa_complex_int(F,X2) = zero_zero_int ) )
       => ( groups858564598930262913ex_int(F,A2) = zero_zero_int ) ) ) ).

% prod_zero
tff(fact_6280_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ? [X2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A2))
            & ( aa_Extended_enat_int(F,X2) = zero_zero_int ) )
       => ( groups2878480467620962989at_int(F,A2) = zero_zero_int ) ) ) ).

% prod_zero
tff(fact_6281_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ? [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
            & ( aa_nat_complex(F,X2) = zero_zero_complex ) )
       => ( groups6464643781859351333omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_6282_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B2: nat] : aa_set_nat_real(groups129246275422532515t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_pu(F),A),B2),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_6283_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B2: nat] : groups6464643781859351333omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_pv(F),A),B2),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_6284_prod__atLeastAtMost__code,axiom,
    ! [F: fun_na8797796813480665199d_enat,A: nat,B2: nat] : groups7961826882256487087d_enat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_Ext7969882492699152700d_enat(set_fo2538466533108834004d_enat(aTP_Lamm_pw(F),A,B2),one_on7984719198319812577d_enat) ).

% prod_atLeastAtMost_code
tff(fact_6285_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B2: nat] : aa_set_nat_int(groups705719431365010083at_int(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_px(F),A),B2),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_6286_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_py(F),A),B2),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_6287_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_nat,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups4696554848551431203al_nat(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bt(A2),P))) = groups4696554848551431203al_nat(aa_fun1504556115071737857al_nat(aTP_Lamm_pz(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_6288_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_nat,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups861055069439313189ex_nat(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_jk(A2),P))) = groups861055069439313189ex_nat(aa_fun9038227978352879617ex_nat(aTP_Lamm_qa(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_6289_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_nat,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups1707563613775114915nt_nat(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bx(A2),P))) = groups1707563613775114915nt_nat(aa_fun7908831221058585985nt_nat(aTP_Lamm_qb(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_6290_prod_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2880970938130013265at_nat(G,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_bs(A2),P))) = groups2880970938130013265at_nat(aa_fun943526657271735063at_nat(aTP_Lamm_qc(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_6291_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_int,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups4694064378042380927al_int(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bt(A2),P))) = groups4694064378042380927al_int(aa_fun6550077132417316957al_int(aTP_Lamm_qd(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_6292_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_int,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups858564598930262913ex_int(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_jk(A2),P))) = groups858564598930262913ex_int(aa_fun4860376958843682909ex_int(aTP_Lamm_qe(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_6293_prod_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex3482447378695272101at_int,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2878480467620962989at_int(G,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_bs(A2),P))) = groups2878480467620962989at_int(aa_fun5989047674617314163at_int(aTP_Lamm_qf(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_6294_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_complex,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups713298508707869441omplex(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bt(A2),P))) = groups713298508707869441omplex(aa_fun6813637976755515871omplex(aTP_Lamm_qg(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_6295_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_complex,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups6464643781859351333omplex(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bw(A2),P))) = groups6464643781859351333omplex(aa_fun8588439219086927967omplex(aTP_Lamm_qh(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_6296_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups3708469109370488835omplex(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_jk(A2),P))) = groups3708469109370488835omplex(aa_fun1034237174562290143omplex(aTP_Lamm_qi(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_6297_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_lh(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_6298_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_6299_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qj(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_6300_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_la(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_6301_prod__le__1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,X4)))
            & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(F,X4)),one_on7984719198319812577d_enat)) ) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups8932437906259616549d_enat(F,A2)),one_on7984719198319812577d_enat)) ) ).

% prod_le_1
tff(fact_6302_prod__le__1,axiom,
    ! [A2: set_real,F: fun_re6065469302365115155d_enat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,X4)))
            & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,X4)),one_on7984719198319812577d_enat)) ) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups7973222482632965587d_enat(F,A2)),one_on7984719198319812577d_enat)) ) ).

% prod_le_1
tff(fact_6303_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_na8797796813480665199d_enat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,X4)))
            & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),one_on7984719198319812577d_enat)) ) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups7961826882256487087d_enat(F,A2)),one_on7984719198319812577d_enat)) ) ).

% prod_le_1
tff(fact_6304_prod__le__1,axiom,
    ! [A2: set_int,F: fun_in5914218760682845459d_enat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,X4)))
            & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,X4)),one_on7984719198319812577d_enat)) ) )
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,groups5078248829458667347d_enat(F,A2)),one_on7984719198319812577d_enat)) ) ).

% prod_le_1
tff(fact_6305_prod__le__1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups97031904164794029t_real(F,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_6306_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_6307_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_6308_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_6309_prod__le__1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( ! [X4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_Extended_enat_nat(F,X4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,X4)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups2880970938130013265at_nat(F,A2)),one_one_nat)) ) ).

% prod_le_1
tff(fact_6310_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F,A2)),one_one_nat)) ) ).

% prod_le_1
tff(fact_6311_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,one_one_nat),one_one_nat))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y22)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X23),Y22))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_complex_nat(H,X4)),aa_complex_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups861055069439313189ex_nat(H,S2)),groups861055069439313189ex_nat(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_6312_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,one_one_nat),one_one_nat))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y22)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X23),Y22))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_int_nat(H,X4)),aa_int_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups1707563613775114915nt_nat(H,S2)),groups1707563613775114915nt_nat(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_6313_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S2: set_Extended_enat,H: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,one_one_nat),one_one_nat))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y22)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X23),Y22))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_Extended_enat_nat(H,X4)),aa_Extended_enat_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups2880970938130013265at_nat(H,S2)),groups2880970938130013265at_nat(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_6314_prod_Orelated,axiom,
    ! [R: fun_int_fun_int_bool,S2: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(R,one_one_int),one_one_int))
     => ( ! [X1: int,Y1: int,X23: int,Y22: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(R,X1),X23))
              & pp(aa_int_bool(aa_int_fun_int_bool(R,Y1),Y22)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_int_int(aa_int_fun_int_int(times_times_int,X1),Y1)),aa_int_int(aa_int_fun_int_int(times_times_int,X23),Y22))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_complex_int(H,X4)),aa_complex_int(G,X4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,groups858564598930262913ex_int(H,S2)),groups858564598930262913ex_int(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_6315_prod_Orelated,axiom,
    ! [R: fun_int_fun_int_bool,S2: set_Extended_enat,H: fun_Ex3482447378695272101at_int,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(R,one_one_int),one_one_int))
     => ( ! [X1: int,Y1: int,X23: int,Y22: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(R,X1),X23))
              & pp(aa_int_bool(aa_int_fun_int_bool(R,Y1),Y22)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_int_int(aa_int_fun_int_int(times_times_int,X1),Y1)),aa_int_int(aa_int_fun_int_int(times_times_int,X23),Y22))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_Extended_enat_int(H,X4)),aa_Extended_enat_int(G,X4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,groups2878480467620962989at_int(H,S2)),groups2878480467620962989at_int(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_6316_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: set_nat,H: fun_nat_real,G: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(times_times_real(X1),Y1)),aa_real_real(times_times_real(X23),Y22))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_nat_real(H,X4)),aa_nat_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_set_nat_real(groups129246275422532515t_real(H),S2)),aa_set_nat_real(groups129246275422532515t_real(G),S2))) ) ) ) ) ).

% prod.related
tff(fact_6317_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(times_times_real(X1),Y1)),aa_real_real(times_times_real(X23),Y22))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_complex_real(H,X4)),aa_complex_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups766887009212190081x_real(H,S2)),groups766887009212190081x_real(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_6318_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(times_times_real(X1),Y1)),aa_real_real(times_times_real(X23),Y22))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_int_real(H,X4)),aa_int_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups2316167850115554303t_real(H,S2)),groups2316167850115554303t_real(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_6319_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(times_times_real(X1),Y1)),aa_real_real(times_times_real(X23),Y22))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_Ext2541572767299617110t_real(H,X4)),aa_Ext2541572767299617110t_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups97031904164794029t_real(H,S2)),groups97031904164794029t_real(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_6320_prod_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,one_one_complex),one_one_complex))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y22)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),Y22))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_nat_complex(H,X4)),aa_nat_complex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups6464643781859351333omplex(H,S2)),groups6464643781859351333omplex(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_6321_prod_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups4696554848551431203al_nat(G,insert_real(X,A2)) = groups4696554848551431203al_nat(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups4696554848551431203al_nat(G,insert_real(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(G,X)),groups4696554848551431203al_nat(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_6322_prod_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups861055069439313189ex_nat(G,insert_complex(X,A2)) = groups861055069439313189ex_nat(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups861055069439313189ex_nat(G,insert_complex(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(G,X)),groups861055069439313189ex_nat(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_6323_prod_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(X),A2))
         => ( groups1707563613775114915nt_nat(G,insert_int(X,A2)) = groups1707563613775114915nt_nat(G,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A2))
         => ( groups1707563613775114915nt_nat(G,insert_int(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(G,X)),groups1707563613775114915nt_nat(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_6324_prod_Oinsert__if,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
         => ( groups2880970938130013265at_nat(G,insert_Extended_enat(X,A2)) = groups2880970938130013265at_nat(G,A2) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
         => ( groups2880970938130013265at_nat(G,insert_Extended_enat(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Extended_enat_nat(G,X)),groups2880970938130013265at_nat(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_6325_prod_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups4694064378042380927al_int(G,insert_real(X,A2)) = groups4694064378042380927al_int(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups4694064378042380927al_int(G,insert_real(X,A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(G,X)),groups4694064378042380927al_int(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_6326_prod_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups858564598930262913ex_int(G,insert_complex(X,A2)) = groups858564598930262913ex_int(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups858564598930262913ex_int(G,insert_complex(X,A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_complex_int(G,X)),groups858564598930262913ex_int(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_6327_prod_Oinsert__if,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
         => ( groups2878480467620962989at_int(G,insert_Extended_enat(X,A2)) = groups2878480467620962989at_int(G,A2) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
         => ( groups2878480467620962989at_int(G,insert_Extended_enat(X,A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Extended_enat_int(G,X)),groups2878480467620962989at_int(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_6328_prod_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1681761925125756287l_real(G,insert_real(X,A2)) = groups1681761925125756287l_real(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1681761925125756287l_real(G,insert_real(X,A2)) = aa_real_real(times_times_real(aa_real_real(G,X)),groups1681761925125756287l_real(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_6329_prod_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A2))
         => ( aa_set_nat_real(groups129246275422532515t_real(G),insert_nat(X,A2)) = aa_set_nat_real(groups129246275422532515t_real(G),A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
         => ( aa_set_nat_real(groups129246275422532515t_real(G),insert_nat(X,A2)) = aa_real_real(times_times_real(aa_nat_real(G,X)),aa_set_nat_real(groups129246275422532515t_real(G),A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_6330_prod_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups766887009212190081x_real(G,insert_complex(X,A2)) = groups766887009212190081x_real(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups766887009212190081x_real(G,insert_complex(X,A2)) = aa_real_real(times_times_real(aa_complex_real(G,X)),groups766887009212190081x_real(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_6331_prod__dvd__prod__subset,axiom,
    ! [B: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_6332_prod__dvd__prod__subset,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups2880970938130013265at_nat(F,A2)),groups2880970938130013265at_nat(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_6333_prod__dvd__prod__subset,axiom,
    ! [B: set_complex,A2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_6334_prod__dvd__prod__subset,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups2878480467620962989at_int(F,A2)),groups2878480467620962989at_int(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_6335_prod__dvd__prod__subset,axiom,
    ! [B: set_int,A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_6336_prod__dvd__prod__subset,axiom,
    ! [B: set_nat,A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)),aa_set_nat_int(groups705719431365010083at_int(F),B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_6337_prod__dvd__prod__subset,axiom,
    ! [B: set_int,A2: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups1705073143266064639nt_int(F,A2)),groups1705073143266064639nt_int(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_6338_prod__dvd__prod__subset,axiom,
    ! [B: set_nat,A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),aa_set_nat_nat(groups708209901874060359at_nat(F),B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_6339_prod__dvd__prod__subset2,axiom,
    ! [B: set_real,A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_real_nat(F,A3)),aa_real_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups4696554848551431203al_nat(F,A2)),groups4696554848551431203al_nat(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6340_prod__dvd__prod__subset2,axiom,
    ! [B: set_complex,A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_complex_nat(F,A3)),aa_complex_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6341_prod__dvd__prod__subset2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
       => ( ! [A3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_Extended_enat_nat(F,A3)),aa_Extended_enat_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups2880970938130013265at_nat(F,A2)),groups2880970938130013265at_nat(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6342_prod__dvd__prod__subset2,axiom,
    ! [B: set_real,A2: set_real,F: fun_real_int,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_real_int(F,A3)),aa_real_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups4694064378042380927al_int(F,A2)),groups4694064378042380927al_int(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6343_prod__dvd__prod__subset2,axiom,
    ! [B: set_complex,A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_complex_int(F,A3)),aa_complex_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6344_prod__dvd__prod__subset2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
       => ( ! [A3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_Extended_enat_int(F,A3)),aa_Extended_enat_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups2878480467620962989at_int(F,A2)),groups2878480467620962989at_int(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6345_prod__dvd__prod__subset2,axiom,
    ! [B: set_int,A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(F,A3)),aa_int_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6346_prod__dvd__prod__subset2,axiom,
    ! [B: set_nat,A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
       => ( ! [A3: nat] :
              ( pp(aa_set_nat_bool(member_nat(A3),A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(F,A3)),aa_nat_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)),aa_set_nat_int(groups705719431365010083at_int(G),B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6347_prod__dvd__prod__subset2,axiom,
    ! [B: set_int,A2: set_int,F: fun_int_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(F,A3)),aa_int_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups1705073143266064639nt_int(F,A2)),groups1705073143266064639nt_int(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6348_prod__dvd__prod__subset2,axiom,
    ! [B: set_nat,A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
       => ( ! [A3: nat] :
              ( pp(aa_set_nat_bool(member_nat(A3),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(F,A3)),aa_nat_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),aa_set_nat_nat(groups708209901874060359at_nat(G),B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6349_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T3: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T5))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A3)),minus_minus_set_real(T3,T5))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T5)))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T5)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T5))
                       => ( aa_real_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_real_nat(H,aa_real_real(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups4696554848551431203al_nat(G,S2) = groups4696554848551431203al_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6350_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T3: set_complex,G: fun_real_nat,H: fun_complex_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T3,T5))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T5)))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T5)))
                   => pp(aa_set_real_bool(member_real(aa_complex_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T5))
                       => ( aa_complex_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_complex_nat(H,aa_real_complex(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups4696554848551431203al_nat(G,S2) = groups861055069439313189ex_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6351_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T3: set_int,G: fun_real_nat,H: fun_int_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T5))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T3,T5))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T5)))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T5)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T5))
                       => ( aa_int_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_int_nat(H,aa_real_int(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups4696554848551431203al_nat(G,S2) = groups1707563613775114915nt_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6352_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_Extended_enat,S2: set_real,I: fun_Ex1451620480498224165t_real,J: fun_re6065469302365115155d_enat,T3: set_Extended_enat,G: fun_real_nat,H: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T5))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_Ext2541572767299617110t_real(I,aa_rea1194391308913012860d_enat(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set3347977610044113549t_bool(member_Extended_enat(aa_rea1194391308913012860d_enat(J,A3)),minus_925952699566721837d_enat(T3,T5))) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T3,T5)))
                 => ( aa_rea1194391308913012860d_enat(J,aa_Ext2541572767299617110t_real(I,B3)) = B3 ) )
             => ( ! [B3: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T3,T5)))
                   => pp(aa_set_real_bool(member_real(aa_Ext2541572767299617110t_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),T5))
                       => ( aa_Extended_enat_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_Extended_enat_nat(H,aa_rea1194391308913012860d_enat(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups4696554848551431203al_nat(G,S2) = groups2880970938130013265at_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6353_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_real,S2: set_complex,I: fun_real_complex,J: fun_complex_real,T3: set_real,G: fun_complex_nat,H: fun_real_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T5))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_real_complex(I,aa_complex_real(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_complex_real(J,A3)),minus_minus_set_real(T3,T5))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T5)))
                 => ( aa_complex_real(J,aa_real_complex(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T5)))
                   => pp(aa_set_complex_bool(member_complex(aa_real_complex(I,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T5))
                       => ( aa_real_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_real_nat(H,aa_complex_real(J,A3)) = aa_complex_nat(G,A3) ) )
                     => ( groups861055069439313189ex_nat(G,S2) = groups4696554848551431203al_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6354_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_complex,S2: set_complex,I: fun_complex_complex,J: fun_complex_complex,T3: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_complex_complex(I,aa_complex_complex(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_complex_complex(J,A3)),minus_811609699411566653omplex(T3,T5))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T5)))
                 => ( aa_complex_complex(J,aa_complex_complex(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T5)))
                   => pp(aa_set_complex_bool(member_complex(aa_complex_complex(I,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T5))
                       => ( aa_complex_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_complex_nat(H,aa_complex_complex(J,A3)) = aa_complex_nat(G,A3) ) )
                     => ( groups861055069439313189ex_nat(G,S2) = groups861055069439313189ex_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6355_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_int,S2: set_complex,I: fun_int_complex,J: fun_complex_int,T3: set_int,G: fun_complex_nat,H: fun_int_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T5))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_int_complex(I,aa_complex_int(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_complex_int(J,A3)),minus_minus_set_int(T3,T5))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T5)))
                 => ( aa_complex_int(J,aa_int_complex(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T5)))
                   => pp(aa_set_complex_bool(member_complex(aa_int_complex(I,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T5))
                       => ( aa_int_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_int_nat(H,aa_complex_int(J,A3)) = aa_complex_nat(G,A3) ) )
                     => ( groups861055069439313189ex_nat(G,S2) = groups1707563613775114915nt_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6356_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_Extended_enat,S2: set_complex,I: fun_Ex3741802288643809319omplex,J: fun_co6496365337113880337d_enat,T3: set_Extended_enat,G: fun_complex_nat,H: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T5))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_Ext8748583412541189208omplex(I,aa_com3683005661447327866d_enat(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set3347977610044113549t_bool(member_Extended_enat(aa_com3683005661447327866d_enat(J,A3)),minus_925952699566721837d_enat(T3,T5))) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T3,T5)))
                 => ( aa_com3683005661447327866d_enat(J,aa_Ext8748583412541189208omplex(I,B3)) = B3 ) )
             => ( ! [B3: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T3,T5)))
                   => pp(aa_set_complex_bool(member_complex(aa_Ext8748583412541189208omplex(I,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),T5))
                       => ( aa_Extended_enat_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_Extended_enat_nat(H,aa_com3683005661447327866d_enat(J,A3)) = aa_complex_nat(G,A3) ) )
                     => ( groups861055069439313189ex_nat(G,S2) = groups2880970938130013265at_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6357_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T3: set_real,G: fun_int_nat,H: fun_real_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T5))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_int_real(J,A3)),minus_minus_set_real(T3,T5))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T5)))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T5)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T5))
                       => ( aa_real_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_real_nat(H,aa_int_real(J,A3)) = aa_int_nat(G,A3) ) )
                     => ( groups1707563613775114915nt_nat(G,S2) = groups4696554848551431203al_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6358_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_complex,S2: set_int,I: fun_complex_int,J: fun_int_complex,T3: set_complex,G: fun_int_nat,H: fun_complex_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_complex_int(I,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_int_complex(J,A3)),minus_811609699411566653omplex(T3,T5))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T5)))
                 => ( aa_int_complex(J,aa_complex_int(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T5)))
                   => pp(aa_set_int_bool(member_int(aa_complex_int(I,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T5))
                       => ( aa_complex_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_complex_nat(H,aa_int_complex(J,A3)) = aa_int_nat(G,A3) ) )
                     => ( groups1707563613775114915nt_nat(G,S2) = groups861055069439313189ex_nat(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6359_powr__add,axiom,
    ! [X: real,A: real,B2: real] : aa_real_real(powr_real(X),aa_real_real(plus_plus_real(A),B2)) = aa_real_real(times_times_real(aa_real_real(powr_real(X),A)),aa_real_real(powr_real(X),B2)) ).

% powr_add
tff(fact_6360_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups4696554848551431203al_nat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_qk(G)))) = groups4696554848551431203al_nat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6361_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_ql(G)))) = groups861055069439313189ex_nat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6362_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_qm(G)))) = groups1707563613775114915nt_nat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6363_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2880970938130013265at_nat(G,minus_925952699566721837d_enat(A2,collec4429806609662206161d_enat(aTP_Lamm_qn(G)))) = groups2880970938130013265at_nat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6364_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups4694064378042380927al_int(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_qo(G)))) = groups4694064378042380927al_int(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6365_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups858564598930262913ex_int(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_qp(G)))) = groups858564598930262913ex_int(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6366_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2878480467620962989at_int(G,minus_925952699566721837d_enat(A2,collec4429806609662206161d_enat(aTP_Lamm_qq(G)))) = groups2878480467620962989at_int(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6367_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups713298508707869441omplex(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_qr(G)))) = groups713298508707869441omplex(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6368_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups3708469109370488835omplex(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_qs(G)))) = groups3708469109370488835omplex(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6369_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups7440179247065528705omplex(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_qt(G)))) = groups7440179247065528705omplex(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6370_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qu(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nat_diff_reindex
tff(fact_6371_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mw(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nat_diff_reindex
tff(fact_6372_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,N: nat,M: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_qv(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% prod.atLeastAtMost_rev
tff(fact_6373_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lf(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% prod.atLeastAtMost_rev
tff(fact_6374_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(F,I)))
         => ( ! [I4: real] :
                ( pp(aa_set_real_bool(member_real(I4),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,I4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_6375_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( pp(aa_set_nat_bool(member_nat(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(F,I)))
         => ( ! [I4: nat] :
                ( pp(aa_set_nat_bool(member_nat(I4),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,I4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_6376_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_complex_real(F,I)))
         => ( ! [I4: complex] :
                ( pp(aa_set_complex_bool(member_complex(I4),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,I4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups766887009212190081x_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_6377_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(F,I)))
         => ( ! [I4: int] :
                ( pp(aa_set_int_bool(member_int(I4),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,I4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_6378_less__1__prod2,axiom,
    ! [I6: set_Extended_enat,I: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I6))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_Ext2541572767299617110t_real(F,I)))
         => ( ! [I4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_Ext2541572767299617110t_real(F,I4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups97031904164794029t_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_6379_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(F,I)))
         => ( ! [I4: real] :
                ( pp(aa_set_real_bool(member_real(I4),I6))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(F,I4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),groups4694064378042380927al_int(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_6380_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I),I6))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_complex_int(F,I)))
         => ( ! [I4: complex] :
                ( pp(aa_set_complex_bool(member_complex(I4),I6))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_complex_int(F,I4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),groups858564598930262913ex_int(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_6381_less__1__prod2,axiom,
    ! [I6: set_Extended_enat,I: extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I6))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I6))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_Extended_enat_int(F,I)))
         => ( ! [I4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_Extended_enat_int(F,I4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),groups2878480467620962989at_int(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_6382_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( pp(aa_set_nat_bool(member_nat(I),I6))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(F,I)))
         => ( ! [I4: nat] :
                ( pp(aa_set_nat_bool(member_nat(I4),I6))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(F,I4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_set_nat_int(groups705719431365010083at_int(F),I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_6383_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I),I6))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(F,I)))
         => ( ! [I4: int] :
                ( pp(aa_set_int_bool(member_int(I4),I6))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(F,I4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),groups1705073143266064639nt_int(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_6384_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( pp(aa_set_complex_bool(member_complex(I4),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_complex_real(F,I4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups766887009212190081x_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_6385_less__1__prod,axiom,
    ! [I6: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I6))
     => ( ( I6 != bot_bo7653980558646680370d_enat )
       => ( ! [I4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_Ext2541572767299617110t_real(F,I4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups97031904164794029t_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_6386_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( pp(aa_set_real_bool(member_real(I4),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(F,I4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_6387_less__1__prod,axiom,
    ! [I6: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I4: nat] :
              ( pp(aa_set_nat_bool(member_nat(I4),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(F,I4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I6))) ) ) ) ).

% less_1_prod
tff(fact_6388_less__1__prod,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I4: int] :
              ( pp(aa_set_int_bool(member_int(I4),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(F,I4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_6389_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( pp(aa_set_complex_bool(member_complex(I4),I6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_complex_int(F,I4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),groups858564598930262913ex_int(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_6390_less__1__prod,axiom,
    ! [I6: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I6))
     => ( ( I6 != bot_bo7653980558646680370d_enat )
       => ( ! [I4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_Extended_enat_int(F,I4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),groups2878480467620962989at_int(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_6391_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( pp(aa_set_real_bool(member_real(I4),I6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(F,I4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),groups4694064378042380927al_int(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_6392_less__1__prod,axiom,
    ! [I6: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I4: nat] :
              ( pp(aa_set_nat_bool(member_nat(I4),I6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(F,I4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_set_nat_int(groups705719431365010083at_int(F),I6))) ) ) ) ).

% less_1_prod
tff(fact_6393_less__1__prod,axiom,
    ! [I6: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I4: int] :
              ( pp(aa_set_int_bool(member_int(I4),I6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(F,I4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),groups1705073143266064639nt_int(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_6394_prod_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups861055069439313189ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A2,B))),groups861055069439313189ex_nat(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_6395_prod_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
       => ( groups2880970938130013265at_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups2880970938130013265at_nat(G,minus_925952699566721837d_enat(A2,B))),groups2880970938130013265at_nat(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_6396_prod_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups858564598930262913ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G,minus_811609699411566653omplex(A2,B))),groups858564598930262913ex_int(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_6397_prod_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
       => ( groups2878480467620962989at_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups2878480467620962989at_int(G,minus_925952699566721837d_enat(A2,B))),groups2878480467620962989at_int(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_6398_prod_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups766887009212190081x_real(G,A2) = aa_real_real(times_times_real(groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,B))),groups766887009212190081x_real(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_6399_prod_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
       => ( groups97031904164794029t_real(G,A2) = aa_real_real(times_times_real(groups97031904164794029t_real(G,minus_925952699566721837d_enat(A2,B))),groups97031904164794029t_real(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_6400_prod_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups3708469109370488835omplex(G,A2) = aa_complex_complex(times_times_complex(groups3708469109370488835omplex(G,minus_811609699411566653omplex(A2,B))),groups3708469109370488835omplex(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_6401_prod_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
       => ( groups4622424608036095791omplex(G,A2) = aa_complex_complex(times_times_complex(groups4622424608036095791omplex(G,minus_925952699566721837d_enat(A2,B))),groups4622424608036095791omplex(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_6402_prod_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: fun_co6496365337113880337d_enat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups8780218893797010257d_enat(G,A2) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups8780218893797010257d_enat(G,minus_811609699411566653omplex(A2,B))),groups8780218893797010257d_enat(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_6403_prod_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex5852502975177633731d_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
       => ( groups8932437906259616549d_enat(G,A2) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups8932437906259616549d_enat(G,minus_925952699566721837d_enat(A2,B))),groups8932437906259616549d_enat(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_6404_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_nat(H,B3) = one_one_nat ) )
             => ( ( groups4696554848551431203al_nat(G,A2) = groups4696554848551431203al_nat(H,B) )
              <=> ( groups4696554848551431203al_nat(G,C4) = groups4696554848551431203al_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6405_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B)))
                 => ( aa_complex_nat(H,B3) = one_one_nat ) )
             => ( ( groups861055069439313189ex_nat(G,A2) = groups861055069439313189ex_nat(H,B) )
              <=> ( groups861055069439313189ex_nat(G,C4) = groups861055069439313189ex_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6406_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Extended_enat_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B)))
                 => ( aa_Extended_enat_nat(H,B3) = one_one_nat ) )
             => ( ( groups2880970938130013265at_nat(G,A2) = groups2880970938130013265at_nat(H,B) )
              <=> ( groups2880970938130013265at_nat(G,C4) = groups2880970938130013265at_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6407_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_int(G,A3) = one_one_int ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_int(H,B3) = one_one_int ) )
             => ( ( groups4694064378042380927al_int(G,A2) = groups4694064378042380927al_int(H,B) )
              <=> ( groups4694064378042380927al_int(G,C4) = groups4694064378042380927al_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6408_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_int(G,A3) = one_one_int ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B)))
                 => ( aa_complex_int(H,B3) = one_one_int ) )
             => ( ( groups858564598930262913ex_int(G,A2) = groups858564598930262913ex_int(H,B) )
              <=> ( groups858564598930262913ex_int(G,C4) = groups858564598930262913ex_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6409_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: fun_Ex3482447378695272101at_int,H: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Extended_enat_int(G,A3) = one_one_int ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B)))
                 => ( aa_Extended_enat_int(H,B3) = one_one_int ) )
             => ( ( groups2878480467620962989at_int(G,A2) = groups2878480467620962989at_int(H,B) )
              <=> ( groups2878480467620962989at_int(G,C4) = groups2878480467620962989at_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6410_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,A2) = groups713298508707869441omplex(H,B) )
              <=> ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6411_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B)))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,A2) = groups3708469109370488835omplex(H,B) )
              <=> ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6412_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: fun_Ex3741802288643809319omplex,H: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Ext8748583412541189208omplex(G,A3) = one_one_complex ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B)))
                 => ( aa_Ext8748583412541189208omplex(H,B3) = one_one_complex ) )
             => ( ( groups4622424608036095791omplex(G,A2) = groups4622424608036095791omplex(H,B) )
              <=> ( groups4622424608036095791omplex(G,C4) = groups4622424608036095791omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6413_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,A2) = groups1681761925125756287l_real(H,B) )
              <=> ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6414_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_nat(H,B3) = one_one_nat ) )
             => ( ( groups4696554848551431203al_nat(G,C4) = groups4696554848551431203al_nat(H,C4) )
               => ( groups4696554848551431203al_nat(G,A2) = groups4696554848551431203al_nat(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6415_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B)))
                 => ( aa_complex_nat(H,B3) = one_one_nat ) )
             => ( ( groups861055069439313189ex_nat(G,C4) = groups861055069439313189ex_nat(H,C4) )
               => ( groups861055069439313189ex_nat(G,A2) = groups861055069439313189ex_nat(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6416_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Extended_enat_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B)))
                 => ( aa_Extended_enat_nat(H,B3) = one_one_nat ) )
             => ( ( groups2880970938130013265at_nat(G,C4) = groups2880970938130013265at_nat(H,C4) )
               => ( groups2880970938130013265at_nat(G,A2) = groups2880970938130013265at_nat(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6417_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_int(G,A3) = one_one_int ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_int(H,B3) = one_one_int ) )
             => ( ( groups4694064378042380927al_int(G,C4) = groups4694064378042380927al_int(H,C4) )
               => ( groups4694064378042380927al_int(G,A2) = groups4694064378042380927al_int(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6418_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_int(G,A3) = one_one_int ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B)))
                 => ( aa_complex_int(H,B3) = one_one_int ) )
             => ( ( groups858564598930262913ex_int(G,C4) = groups858564598930262913ex_int(H,C4) )
               => ( groups858564598930262913ex_int(G,A2) = groups858564598930262913ex_int(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6419_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: fun_Ex3482447378695272101at_int,H: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Extended_enat_int(G,A3) = one_one_int ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B)))
                 => ( aa_Extended_enat_int(H,B3) = one_one_int ) )
             => ( ( groups2878480467620962989at_int(G,C4) = groups2878480467620962989at_int(H,C4) )
               => ( groups2878480467620962989at_int(G,A2) = groups2878480467620962989at_int(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6420_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) )
               => ( groups713298508707869441omplex(G,A2) = groups713298508707869441omplex(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6421_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B)))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) )
               => ( groups3708469109370488835omplex(G,A2) = groups3708469109370488835omplex(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6422_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: fun_Ex3741802288643809319omplex,H: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Ext8748583412541189208omplex(G,A3) = one_one_complex ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B)))
                 => ( aa_Ext8748583412541189208omplex(H,B3) = one_one_complex ) )
             => ( ( groups4622424608036095791omplex(G,C4) = groups4622424608036095791omplex(H,C4) )
               => ( groups4622424608036095791omplex(G,A2) = groups4622424608036095791omplex(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6423_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B)))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) )
               => ( groups1681761925125756287l_real(G,A2) = groups1681761925125756287l_real(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6424_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,S2) = groups861055069439313189ex_nat(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6425_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_nat(G,X4) = one_one_nat ) )
         => ( groups2880970938130013265at_nat(G,S2) = groups2880970938130013265at_nat(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6426_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,S2) = groups858564598930262913ex_int(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6427_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_int(G,X4) = one_one_int ) )
         => ( groups2878480467620962989at_int(G,S2) = groups2878480467620962989at_int(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6428_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,S2) = groups3708469109370488835omplex(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6429_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext8748583412541189208omplex(G,X4) = one_one_complex ) )
         => ( groups4622424608036095791omplex(G,S2) = groups4622424608036095791omplex(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6430_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,S2) = groups766887009212190081x_real(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6431_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = one_one_real ) )
         => ( groups97031904164794029t_real(G,S2) = groups97031904164794029t_real(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6432_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_complex(G,X4) = one_one_complex ) )
         => ( groups6464643781859351333omplex(G,S2) = groups6464643781859351333omplex(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6433_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_real(G,X4) = one_one_real ) )
         => ( aa_set_nat_real(groups129246275422532515t_real(G),S2) = aa_set_nat_real(groups129246275422532515t_real(G),T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6434_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,T3) = groups861055069439313189ex_nat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6435_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_nat(G,X4) = one_one_nat ) )
         => ( groups2880970938130013265at_nat(G,T3) = groups2880970938130013265at_nat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6436_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,T3) = groups858564598930262913ex_int(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6437_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_int(G,X4) = one_one_int ) )
         => ( groups2878480467620962989at_int(G,T3) = groups2878480467620962989at_int(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6438_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,T3) = groups3708469109370488835omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6439_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext8748583412541189208omplex(G,X4) = one_one_complex ) )
         => ( groups4622424608036095791omplex(G,T3) = groups4622424608036095791omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6440_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,T3) = groups766887009212190081x_real(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6441_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = one_one_real ) )
         => ( groups97031904164794029t_real(G,T3) = groups97031904164794029t_real(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6442_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_complex(G,X4) = one_one_complex ) )
         => ( groups6464643781859351333omplex(G,T3) = groups6464643781859351333omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6443_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_real(G,X4) = one_one_real ) )
         => ( aa_set_nat_real(groups129246275422532515t_real(G),T3) = aa_set_nat_real(groups129246275422532515t_real(G),S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6444_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_nat(H,X4) = one_one_nat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups4696554848551431203al_nat(G,S2) = groups4696554848551431203al_nat(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6445_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_nat(H,X4) = one_one_nat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups861055069439313189ex_nat(G,S2) = groups861055069439313189ex_nat(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6446_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_nat(H,X4) = one_one_nat ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Extended_enat_nat(G,X4) = aa_Extended_enat_nat(H,X4) ) )
           => ( groups2880970938130013265at_nat(G,S2) = groups2880970938130013265at_nat(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6447_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_int,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_int(H,X4) = one_one_int ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
           => ( groups4694064378042380927al_int(G,S2) = groups4694064378042380927al_int(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6448_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_int(H,X4) = one_one_int ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_int(G,X4) = aa_complex_int(H,X4) ) )
           => ( groups858564598930262913ex_int(G,S2) = groups858564598930262913ex_int(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6449_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H: fun_Ex3482447378695272101at_int,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_int(H,X4) = one_one_int ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Extended_enat_int(G,X4) = aa_Extended_enat_int(H,X4) ) )
           => ( groups2878480467620962989at_int(G,S2) = groups2878480467620962989at_int(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6450_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_complex(H,X4) = one_one_complex ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups713298508707869441omplex(G,S2) = groups713298508707869441omplex(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6451_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_complex(H,X4) = one_one_complex ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_complex(G,X4) = aa_complex_complex(H,X4) ) )
           => ( groups3708469109370488835omplex(G,S2) = groups3708469109370488835omplex(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6452_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H: fun_Ex3741802288643809319omplex,G: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext8748583412541189208omplex(H,X4) = one_one_complex ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Ext8748583412541189208omplex(G,X4) = aa_Ext8748583412541189208omplex(H,X4) ) )
           => ( groups4622424608036095791omplex(G,S2) = groups4622424608036095791omplex(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6453_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_real(H,X4) = one_one_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups1681761925125756287l_real(G,S2) = groups1681761925125756287l_real(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6454_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_nat(G,X4) = one_one_nat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups4696554848551431203al_nat(G,T3) = groups4696554848551431203al_nat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6455_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups861055069439313189ex_nat(G,T3) = groups861055069439313189ex_nat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6456_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_nat(G,X4) = one_one_nat ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Extended_enat_nat(G,X4) = aa_Extended_enat_nat(H,X4) ) )
           => ( groups2880970938130013265at_nat(G,T3) = groups2880970938130013265at_nat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6457_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_int(G,X4) = one_one_int ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
           => ( groups4694064378042380927al_int(G,T3) = groups4694064378042380927al_int(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6458_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_int(G,X4) = aa_complex_int(H,X4) ) )
           => ( groups858564598930262913ex_int(G,T3) = groups858564598930262913ex_int(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6459_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex3482447378695272101at_int,H: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Extended_enat_int(G,X4) = one_one_int ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Extended_enat_int(G,X4) = aa_Extended_enat_int(H,X4) ) )
           => ( groups2878480467620962989at_int(G,T3) = groups2878480467620962989at_int(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6460_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_complex(G,X4) = one_one_complex ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups713298508707869441omplex(G,T3) = groups713298508707869441omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6461_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_complex(G,X4) = aa_complex_complex(H,X4) ) )
           => ( groups3708469109370488835omplex(G,T3) = groups3708469109370488835omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6462_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex3741802288643809319omplex,H: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T3))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T3,S2)))
             => ( aa_Ext8748583412541189208omplex(G,X4) = one_one_complex ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Ext8748583412541189208omplex(G,X4) = aa_Ext8748583412541189208omplex(H,X4) ) )
           => ( groups4622424608036095791omplex(G,T3) = groups4622424608036095791omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6463_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,S2)))
             => ( aa_real_real(G,X4) = one_one_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups1681761925125756287l_real(G,T3) = groups1681761925125756287l_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6464_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6465_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6466_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_Extended_enat(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6467_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6468_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6469_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : pp(aa_real_bool(sums_real(aTP_Lamm_kq(A)),aa_nat_real(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_6470_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aTP_Lamm_kp(A)),aa_nat_complex(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_6471_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(times_times_real(aa_nat_real(G,M)),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6472_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,M)),groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6473_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_na8797796813480665199d_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,M)),groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6474_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6475_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6476_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(times_times_real(aa_nat_real(G,aa_nat_nat(suc,N))),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6477_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,aa_nat_nat(suc,N))),groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6478_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_na8797796813480665199d_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,aa_nat_nat(suc,N))),groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6479_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,N))),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6480_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,N))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6481_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(times_times_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_kz(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_6482_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,zero_zero_nat)),groups6464643781859351333omplex(aTP_Lamm_qw(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_6483_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,zero_zero_nat)),groups7961826882256487087d_enat(aTP_Lamm_li(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_6484_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_lh(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_6485_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_6486_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) = aa_real_real(times_times_real(aa_nat_real(G,M)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_kz(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6487_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,M)),groups6464643781859351333omplex(aTP_Lamm_qw(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6488_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_na8797796813480665199d_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_Extended_enat(G,aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,M)),groups7961826882256487087d_enat(aTP_Lamm_li(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6489_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_lh(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6490_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6491_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_lh(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_6492_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_6493_prod__mono__strict,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [I4: complex] :
            ( pp(aa_set_complex_bool(member_complex(I4),A2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I4)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,I4)),aa_complex_real(G,I4))) ) )
       => ( ( A2 != bot_bot_set_complex )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups766887009212190081x_real(F,A2)),groups766887009212190081x_real(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_6494_prod__mono__strict,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [I4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),A2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I4)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,I4)),aa_Ext2541572767299617110t_real(G,I4))) ) )
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups97031904164794029t_real(F,A2)),groups97031904164794029t_real(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_6495_prod__mono__strict,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [I4: real] :
            ( pp(aa_set_real_bool(member_real(I4),A2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I4)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,I4)),aa_real_real(G,I4))) ) )
       => ( ( A2 != bot_bot_set_real )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_6496_prod__mono__strict,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [I4: nat] :
            ( pp(aa_set_nat_bool(member_nat(I4),A2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I4)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,I4)),aa_nat_real(G,I4))) ) )
       => ( ( A2 != bot_bot_set_nat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups129246275422532515t_real(F),A2)),aa_set_nat_real(groups129246275422532515t_real(G),A2))) ) ) ) ).

% prod_mono_strict
tff(fact_6497_prod__mono__strict,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [I4: int] :
            ( pp(aa_set_int_bool(member_int(I4),A2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I4)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,I4)),aa_int_real(G,I4))) ) )
       => ( ( A2 != bot_bot_set_int )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_6498_prod__mono__strict,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [I4: complex] :
            ( pp(aa_set_complex_bool(member_complex(I4),A2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I4)))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,I4)),aa_complex_nat(G,I4))) ) )
       => ( ( A2 != bot_bot_set_complex )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_6499_prod__mono__strict,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [I4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),A2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_Extended_enat_nat(F,I4)))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,I4)),aa_Extended_enat_nat(G,I4))) ) )
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups2880970938130013265at_nat(F,A2)),groups2880970938130013265at_nat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_6500_prod__mono__strict,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [I4: real] :
            ( pp(aa_set_real_bool(member_real(I4),A2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I4)))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,I4)),aa_real_nat(G,I4))) ) )
       => ( ( A2 != bot_bot_set_real )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4696554848551431203al_nat(F,A2)),groups4696554848551431203al_nat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_6501_prod__mono__strict,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [I4: int] :
            ( pp(aa_set_int_bool(member_int(I4),A2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I4)))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,I4)),aa_int_nat(G,I4))) ) )
       => ( ( A2 != bot_bot_set_int )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_6502_prod__mono__strict,axiom,
    ! [A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [I4: complex] :
            ( pp(aa_set_complex_bool(member_complex(I4),A2))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_complex_int(F,I4)))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_complex_int(F,I4)),aa_complex_int(G,I4))) ) )
       => ( ( A2 != bot_bot_set_complex )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_6503_even__prod__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),groups861055069439313189ex_nat(F,A2)))
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_complex_nat(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_6504_even__prod__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),groups1707563613775114915nt_nat(F,A2)))
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_int_nat(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_6505_even__prod__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),groups2880970938130013265at_nat(F,A2)))
      <=> ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_Extended_enat_nat(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_6506_even__prod__iff,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),groups858564598930262913ex_int(F,A2)))
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_complex_int(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_6507_even__prod__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),groups2878480467620962989at_int(F,A2)))
      <=> ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_Extended_enat_int(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_6508_even__prod__iff,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_set_nat_int(groups705719431365010083at_int(F),A2)))
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_nat_int(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_6509_even__prod__iff,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),groups1705073143266064639nt_int(F,A2)))
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_6510_even__prod__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)))
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_6511_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_nat,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups861055069439313189ex_nat(G,insert_complex(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(G,X)),groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_6512_prod_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2880970938130013265at_nat(G,insert_Extended_enat(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Extended_enat_nat(G,X)),groups2880970938130013265at_nat(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ).

% prod.insert_remove
tff(fact_6513_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_nat,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups4696554848551431203al_nat(G,insert_real(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(G,X)),groups4696554848551431203al_nat(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_6514_prod_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_nat,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups1707563613775114915nt_nat(G,insert_int(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(G,X)),groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_6515_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_int,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups858564598930262913ex_int(G,insert_complex(X,A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_complex_int(G,X)),groups858564598930262913ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_6516_prod_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex3482447378695272101at_int,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2878480467620962989at_int(G,insert_Extended_enat(X,A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Extended_enat_int(G,X)),groups2878480467620962989at_int(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ).

% prod.insert_remove
tff(fact_6517_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_int,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups4694064378042380927al_int(G,insert_real(X,A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(G,X)),groups4694064378042380927al_int(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_6518_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_real,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups766887009212190081x_real(G,insert_complex(X,A2)) = aa_real_real(times_times_real(aa_complex_real(G,X)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_6519_prod_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups97031904164794029t_real(G,insert_Extended_enat(X,A2)) = aa_real_real(times_times_real(aa_Ext2541572767299617110t_real(G,X)),groups97031904164794029t_real(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ).

% prod.insert_remove
tff(fact_6520_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1681761925125756287l_real(G,insert_real(X,A2)) = aa_real_real(times_times_real(aa_real_real(G,X)),groups1681761925125756287l_real(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_6521_prod_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups861055069439313189ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(G,X)),groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_6522_prod_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
       => ( groups2880970938130013265at_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Extended_enat_nat(G,X)),groups2880970938130013265at_nat(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% prod.remove
tff(fact_6523_prod_Oremove,axiom,
    ! [A2: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => ( groups4696554848551431203al_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(G,X)),groups4696554848551431203al_nat(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_6524_prod_Oremove,axiom,
    ! [A2: set_int,X: int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(X),A2))
       => ( groups1707563613775114915nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(G,X)),groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_6525_prod_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups858564598930262913ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_complex_int(G,X)),groups858564598930262913ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_6526_prod_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
       => ( groups2878480467620962989at_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Extended_enat_int(G,X)),groups2878480467620962989at_int(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% prod.remove
tff(fact_6527_prod_Oremove,axiom,
    ! [A2: set_real,X: real,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => ( groups4694064378042380927al_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(G,X)),groups4694064378042380927al_int(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_6528_prod_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups766887009212190081x_real(G,A2) = aa_real_real(times_times_real(aa_complex_real(G,X)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_6529_prod_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A2))
       => ( groups97031904164794029t_real(G,A2) = aa_real_real(times_times_real(aa_Ext2541572767299617110t_real(G,X)),groups97031904164794029t_real(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% prod.remove
tff(fact_6530_prod_Oremove,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1681761925125756287l_real(G,A2) = aa_real_real(times_times_real(aa_real_real(G,X)),groups1681761925125756287l_real(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_6531_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_real,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_6532_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_complex,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_6533_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_na8797796813480665199d_enat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_6534_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_6535_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_6536_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_nat,C: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups861055069439313189ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_ke(A),B2),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(B2,A)),groups861055069439313189ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups861055069439313189ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_ke(A),B2),C),S2) = groups861055069439313189ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6537_prod_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex7660298398204468809at_nat,C: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2880970938130013265at_nat(aa_fun1810565495048290044at_nat(aa_fun5812849712608808208at_nat(aTP_Lamm_ki(A),B2),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Extended_enat_nat(B2,A)),groups2880970938130013265at_nat(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2880970938130013265at_nat(aa_fun1810565495048290044at_nat(aa_fun5812849712608808208at_nat(aTP_Lamm_ki(A),B2),C),S2) = groups2880970938130013265at_nat(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6538_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B2: fun_real_nat,C: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4696554848551431203al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_km(A),B2),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(B2,A)),groups4696554848551431203al_nat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4696554848551431203al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_km(A),B2),C),S2) = groups4696554848551431203al_nat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6539_prod_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B2: fun_int_nat,C: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_qx(A),B2),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(B2,A)),groups1707563613775114915nt_nat(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_qx(A),B2),C),S2) = groups1707563613775114915nt_nat(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6540_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_int,C: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups858564598930262913ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_kf(A),B2),C),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_complex_int(B2,A)),groups858564598930262913ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups858564598930262913ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_kf(A),B2),C),S2) = groups858564598930262913ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6541_prod_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex3482447378695272101at_int,C: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2878480467620962989at_int(aa_fun1695976788104708604at_int(aa_fun8556764835550362732at_int(aTP_Lamm_kj(A),B2),C),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Extended_enat_int(B2,A)),groups2878480467620962989at_int(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups2878480467620962989at_int(aa_fun1695976788104708604at_int(aa_fun8556764835550362732at_int(aTP_Lamm_kj(A),B2),C),S2) = groups2878480467620962989at_int(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6542_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B2: fun_real_int,C: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4694064378042380927al_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_kn(A),B2),C),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(B2,A)),groups4694064378042380927al_int(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4694064378042380927al_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_kn(A),B2),C),S2) = groups4694064378042380927al_int(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6543_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: fun_complex_real,C: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_kg(A),B2),C),S2) = aa_real_real(times_times_real(aa_complex_real(B2,A)),groups766887009212190081x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_kg(A),B2),C),S2) = groups766887009212190081x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6544_prod_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: fun_Ex1451620480498224165t_real,C: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups97031904164794029t_real(aa_fun1749635019461158908t_real(aa_fun967516747568145260t_real(aTP_Lamm_kk(A),B2),C),S2) = aa_real_real(times_times_real(aa_Ext2541572767299617110t_real(B2,A)),groups97031904164794029t_real(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups97031904164794029t_real(aa_fun1749635019461158908t_real(aa_fun967516747568145260t_real(aTP_Lamm_kk(A),B2),C),S2) = groups97031904164794029t_real(C,minus_925952699566721837d_enat(S2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6545_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B2: fun_real_real,C: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_qy(A),B2),C),S2) = aa_real_real(times_times_real(aa_real_real(B2,A)),groups1681761925125756287l_real(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_qy(A),B2),C),S2) = groups1681761925125756287l_real(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6546_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa2),Xb),Xc) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa2))
         => ( Y = Xc ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa2))
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa2),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa2),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_6547_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [B2: nat,A: nat,F: fun_nat_fun_nat_nat,Acc2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B2),Acc2) = Acc2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B2),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)),B2),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc2)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_6548_prod__mono2,axiom,
    ! [B: set_real,A2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,B3))) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),A2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_6549_prod__mono2,axiom,
    ! [B: set_complex,A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,B3))) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),A2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F,A2)),groups766887009212190081x_real(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_6550_prod__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
       => ( ! [B3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_Ext2541572767299617110t_real(F,B3))) )
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),A2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups97031904164794029t_real(F,A2)),groups97031904164794029t_real(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_6551_prod__mono2,axiom,
    ! [B: set_real,A2: set_real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B,A2)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(F,B3))) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),A2))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,A3))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4694064378042380927al_int(F,A2)),groups4694064378042380927al_int(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_6552_prod__mono2,axiom,
    ! [B: set_complex,A2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B,A2)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_complex_int(F,B3))) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),A2))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_complex_int(F,A3))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_6553_prod__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B))
       => ( ! [B3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B,A2)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_Extended_enat_int(F,B3))) )
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),A2))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_Extended_enat_int(F,A3))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups2878480467620962989at_int(F,A2)),groups2878480467620962989at_int(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_6554_prod__mono2,axiom,
    ! [B: set_nat,A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,B3))) )
         => ( ! [A3: nat] :
                ( pp(aa_set_nat_bool(member_nat(A3),A2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A2)),aa_set_nat_real(groups129246275422532515t_real(F),B))) ) ) ) ) ).

% prod_mono2
tff(fact_6555_prod__mono2,axiom,
    ! [B: set_int,A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,B3))) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),A2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_6556_prod__mono2,axiom,
    ! [B: set_nat,A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B,A2)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(F,B3))) )
         => ( ! [A3: nat] :
                ( pp(aa_set_nat_bool(member_nat(A3),A2))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,A3))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)),aa_set_nat_int(groups705719431365010083at_int(F),B))) ) ) ) ) ).

% prod_mono2
tff(fact_6557_prod__mono2,axiom,
    ! [B: set_int,A2: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B,A2)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(F,B3))) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),A2))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(F,A3))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1705073143266064639nt_int(F,A2)),groups1705073143266064639nt_int(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_6558_prod__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_complex,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( aa_complex_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
           => ( groups3708469109370488835omplex(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_complex_complex(divide1717551699836669952omplex(groups3708469109370488835omplex(F,A2)),aa_complex_complex(F,A)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
           => ( groups3708469109370488835omplex(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups3708469109370488835omplex(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_6559_prod__diff1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex3741802288643809319omplex,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( aa_Ext8748583412541189208omplex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
           => ( groups4622424608036095791omplex(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = aa_complex_complex(divide1717551699836669952omplex(groups4622424608036095791omplex(F,A2)),aa_Ext8748583412541189208omplex(F,A)) ) )
          & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
           => ( groups4622424608036095791omplex(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = groups4622424608036095791omplex(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_6560_prod__diff1,axiom,
    ! [A2: set_real,F: fun_real_complex,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( aa_real_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_real_bool(member_real(A),A2))
           => ( groups713298508707869441omplex(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_complex_complex(divide1717551699836669952omplex(groups713298508707869441omplex(F,A2)),aa_real_complex(F,A)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(A),A2))
           => ( groups713298508707869441omplex(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups713298508707869441omplex(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_6561_prod__diff1,axiom,
    ! [A2: set_int,F: fun_int_complex,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( aa_int_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_int_bool(member_int(A),A2))
           => ( groups7440179247065528705omplex(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_complex_complex(divide1717551699836669952omplex(groups7440179247065528705omplex(F,A2)),aa_int_complex(F,A)) ) )
          & ( ~ pp(aa_set_int_bool(member_int(A),A2))
           => ( groups7440179247065528705omplex(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups7440179247065528705omplex(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_6562_prod__diff1,axiom,
    ! [A2: set_nat,F: fun_nat_complex,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_nat_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_nat_bool(member_nat(A),A2))
           => ( groups6464643781859351333omplex(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_complex_complex(divide1717551699836669952omplex(groups6464643781859351333omplex(F,A2)),aa_nat_complex(F,A)) ) )
          & ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
           => ( groups6464643781859351333omplex(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = groups6464643781859351333omplex(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_6563_prod__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_nat,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( aa_complex_nat(F,A) != zero_zero_nat )
       => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
           => ( groups861055069439313189ex_nat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups861055069439313189ex_nat(F,A2)),aa_complex_nat(F,A)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
           => ( groups861055069439313189ex_nat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups861055069439313189ex_nat(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_6564_prod__diff1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( aa_Extended_enat_nat(F,A) != zero_zero_nat )
       => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
           => ( groups2880970938130013265at_nat(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups2880970938130013265at_nat(F,A2)),aa_Extended_enat_nat(F,A)) ) )
          & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
           => ( groups2880970938130013265at_nat(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = groups2880970938130013265at_nat(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_6565_prod__diff1,axiom,
    ! [A2: set_real,F: fun_real_nat,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( aa_real_nat(F,A) != zero_zero_nat )
       => ( ( pp(aa_set_real_bool(member_real(A),A2))
           => ( groups4696554848551431203al_nat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups4696554848551431203al_nat(F,A2)),aa_real_nat(F,A)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(A),A2))
           => ( groups4696554848551431203al_nat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups4696554848551431203al_nat(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_6566_prod__diff1,axiom,
    ! [A2: set_int,F: fun_int_nat,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( aa_int_nat(F,A) != zero_zero_nat )
       => ( ( pp(aa_set_int_bool(member_int(A),A2))
           => ( groups1707563613775114915nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups1707563613775114915nt_nat(F,A2)),aa_int_nat(F,A)) ) )
          & ( ~ pp(aa_set_int_bool(member_int(A),A2))
           => ( groups1707563613775114915nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups1707563613775114915nt_nat(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_6567_prod__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_int,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( aa_complex_int(F,A) != zero_zero_int )
       => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
           => ( groups858564598930262913ex_int(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_int_int(divide_divide_int(groups858564598930262913ex_int(F,A2)),aa_complex_int(F,A)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
           => ( groups858564598930262913ex_int(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups858564598930262913ex_int(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_6568_pochhammer__Suc__prod,axiom,
    ! [A: extended_enat,N: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),aa_nat_nat(suc,N)) = groups7961826882256487087d_enat(aTP_Lamm_qz(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_6569_pochhammer__Suc__prod,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_ra(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_6570_pochhammer__Suc__prod,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_rb(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_6571_pochhammer__Suc__prod,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rc,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_6572_pochhammer__prod__rev,axiom,
    ! [A: extended_enat,N: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),N) = groups7961826882256487087d_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_rd(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_6573_pochhammer__prod__rev,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),N) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_re(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_6574_pochhammer__prod__rev,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),N) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_rf(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_6575_pochhammer__prod__rev,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rg(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_6576_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_rh(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_6577_prod_Oin__pairs,axiom,
    ! [G: fun_nat_complex,M: nat,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = groups6464643781859351333omplex(aTP_Lamm_ri(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_6578_prod_Oin__pairs,axiom,
    ! [G: fun_na8797796813480665199d_enat,M: nat,N: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = groups7961826882256487087d_enat(aTP_Lamm_rj(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_6579_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_rk(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_6580_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_rl(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_6581_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B2: nat] : aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_rm(F),A),B2),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_6582_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B2: nat] : groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_rn(F),A),B2),zero_zero_complex) ).

% sum_atLeastAtMost_code
tff(fact_6583_sum__atLeastAtMost__code,axiom,
    ! [F: fun_na8797796813480665199d_enat,A: nat,B2: nat] : groups7108830773950497114d_enat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_Ext7969882492699152700d_enat(set_fo2538466533108834004d_enat(aTP_Lamm_ro(F),A,B2),zero_z5237406670263579293d_enat) ).

% sum_atLeastAtMost_code
tff(fact_6584_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_rp(F),A),B2),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_6585_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B2: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_rq(F),A),B2),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_6586_pochhammer__Suc__prod__rev,axiom,
    ! [A: extended_enat,N: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),aa_nat_nat(suc,N)) = groups7961826882256487087d_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_rd(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_6587_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_re(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_6588_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_rf(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_6589_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rg(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_6590_power__half__series,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_rr),one_one_real)) ).

% power_half_series
tff(fact_6591_sums__zero,axiom,
    pp(aa_nat_bool(sums_nat(aTP_Lamm_by),zero_zero_nat)) ).

% sums_zero
tff(fact_6592_sums__zero,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_go),zero_zero_real)) ).

% sums_zero
tff(fact_6593_sums__zero,axiom,
    pp(aa_int_bool(sums_int(aTP_Lamm_rs),zero_zero_int)) ).

% sums_zero
tff(fact_6594_sums__zero,axiom,
    pp(aa_complex_bool(sums_complex(aTP_Lamm_rt),zero_zero_complex)) ).

% sums_zero
tff(fact_6595_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S2: real,A2: set_nat,S5: real,F: fun_nat_real] :
      ( pp(aa_real_bool(sums_real(G),S2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( ( S5 = aa_real_real(plus_plus_real(S2),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ru(G),F)),A2)) )
         => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_rv(G),A2),F)),S5)) ) ) ) ).

% sums_If_finite_set'
tff(fact_6596_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,N: nat,S: real] :
      ( pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_lb(F),N)),aa_real_real(minus_minus_real(S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N)))))
    <=> pp(aa_real_bool(sums_real(F),S)) ) ).

% sums_iff_shift'
tff(fact_6597_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,S: real,N: nat] :
      ( pp(aa_real_bool(sums_real(F),S))
     => pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_lb(F),N)),aa_real_real(minus_minus_real(S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))))) ) ).

% sums_split_initial_segment
tff(fact_6598_sums__iff__shift,axiom,
    ! [F: fun_nat_real,N: nat,S: real] :
      ( pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_lb(F),N)),S))
    <=> pp(aa_real_bool(sums_real(F),aa_real_real(plus_plus_real(S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))))) ) ).

% sums_iff_shift
tff(fact_6599_prod__eq__1__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups861055069439313189ex_nat(F,A2) = one_one_nat )
      <=> ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A2))
           => ( aa_complex_nat(F,X5) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_6600_prod__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups1707563613775114915nt_nat(F,A2) = one_one_nat )
      <=> ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A2))
           => ( aa_int_nat(F,X5) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_6601_prod__eq__1__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( groups2880970938130013265at_nat(F,A2) = one_one_nat )
      <=> ! [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
           => ( aa_Extended_enat_nat(F,X5) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_6602_prod__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(groups708209901874060359at_nat(F),A2) = one_one_nat )
      <=> ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A2))
           => ( aa_nat_nat(F,X5) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_6603_prod__pos__nat__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups861055069439313189ex_nat(F,A2)))
      <=> ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,X5))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6604_prod__pos__nat__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F,A2)))
      <=> ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,X5))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6605_prod__pos__nat__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups2880970938130013265at_nat(F,A2)))
      <=> ! [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_Extended_enat_nat(F,X5))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6606_prod__pos__nat__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)))
      <=> ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X5))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6607_int__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_lw(F),A2) ).

% int_prod
tff(fact_6608_int__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ly(F)),A2) ).

% int_prod
tff(fact_6609_prod__int__eq,axiom,
    ! [I: nat,J: nat] : aa_set_nat_int(groups705719431365010083at_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = groups1705073143266064639nt_int(aTP_Lamm_ce,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_6610_ln__prod,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ! [I4: real] :
            ( pp(aa_set_real_bool(member_real(I4),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I4))) )
       => ( aa_real_real(ln_ln_real,groups1681761925125756287l_real(F,I6)) = groups8097168146408367636l_real(aTP_Lamm_rw(F),I6) ) ) ) ).

% ln_prod
tff(fact_6611_ln__prod,axiom,
    ! [I6: set_set_nat,F: fun_set_nat_real] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,I6))
     => ( ! [I4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(I4),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_set_nat_real(F,I4))) )
       => ( aa_real_real(ln_ln_real,groups3619160379726066777t_real(F,I6)) = groups5107569545109728110t_real(aTP_Lamm_rx(F),I6) ) ) ) ).

% ln_prod
tff(fact_6612_ln__prod,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ! [I4: complex] :
            ( pp(aa_set_complex_bool(member_complex(I4),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I4))) )
       => ( aa_real_real(ln_ln_real,groups766887009212190081x_real(F,I6)) = groups5808333547571424918x_real(aTP_Lamm_ry(F),I6) ) ) ) ).

% ln_prod
tff(fact_6613_ln__prod,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( ! [I4: int] :
            ( pp(aa_set_int_bool(member_int(I4),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I4))) )
       => ( aa_real_real(ln_ln_real,groups2316167850115554303t_real(F,I6)) = groups8778361861064173332t_real(aTP_Lamm_rz(F),I6) ) ) ) ).

% ln_prod
tff(fact_6614_ln__prod,axiom,
    ! [I6: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I6))
     => ( ! [I4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I4))) )
       => ( aa_real_real(ln_ln_real,groups97031904164794029t_real(F,I6)) = groups4148127829035722712t_real(aTP_Lamm_sa(F),I6) ) ) ) ).

% ln_prod
tff(fact_6615_ln__prod,axiom,
    ! [I6: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( ! [I4: nat] :
            ( pp(aa_set_nat_bool(member_nat(I4),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I4))) )
       => ( aa_real_real(ln_ln_real,aa_set_nat_real(groups129246275422532515t_real(F),I6)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_sb(F)),I6) ) ) ) ).

% ln_prod
tff(fact_6616_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : aa_set_nat_int(groups705719431365010083at_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_or1269000886237332187st_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_ce,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)))) ).

% prod_int_plus_eq
tff(fact_6617_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S: real,T: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2)))
     => ( pp(aa_real_bool(sums_real(F),S))
       => ( pp(aa_real_bool(sums_real(G),T))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,S),T)) ) ) ) ).

% sums_le
tff(fact_6618_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S: nat,T: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G,N2)))
     => ( pp(aa_nat_bool(sums_nat(F),S))
       => ( pp(aa_nat_bool(sums_nat(G),T))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S),T)) ) ) ) ).

% sums_le
tff(fact_6619_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S: int,T: int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G,N2)))
     => ( pp(aa_int_bool(sums_int(F),S))
       => ( pp(aa_int_bool(sums_int(G),T))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,S),T)) ) ) ) ).

% sums_le
tff(fact_6620_sums__0,axiom,
    ! [F: fun_nat_nat] :
      ( ! [N2: nat] : aa_nat_nat(F,N2) = zero_zero_nat
     => pp(aa_nat_bool(sums_nat(F),zero_zero_nat)) ) ).

% sums_0
tff(fact_6621_sums__0,axiom,
    ! [F: fun_nat_real] :
      ( ! [N2: nat] : aa_nat_real(F,N2) = zero_zero_real
     => pp(aa_real_bool(sums_real(F),zero_zero_real)) ) ).

% sums_0
tff(fact_6622_sums__0,axiom,
    ! [F: fun_nat_int] :
      ( ! [N2: nat] : aa_nat_int(F,N2) = zero_zero_int
     => pp(aa_int_bool(sums_int(F),zero_zero_int)) ) ).

% sums_0
tff(fact_6623_sums__0,axiom,
    ! [F: fun_nat_complex] :
      ( ! [N2: nat] : aa_nat_complex(F,N2) = zero_zero_complex
     => pp(aa_complex_bool(sums_complex(F),zero_zero_complex)) ) ).

% sums_0
tff(fact_6624_sums__single,axiom,
    ! [I: nat,F: fun_nat_nat] : pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sc(I),F)),aa_nat_nat(F,I))) ).

% sums_single
tff(fact_6625_sums__single,axiom,
    ! [I: nat,F: fun_nat_real] : pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_sd(I),F)),aa_nat_real(F,I))) ).

% sums_single
tff(fact_6626_sums__single,axiom,
    ! [I: nat,F: fun_nat_int] : pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_gy(I),F)),aa_nat_int(F,I))) ).

% sums_single
tff(fact_6627_sums__single,axiom,
    ! [I: nat,F: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_se(I),F)),aa_nat_complex(F,I))) ).

% sums_single
tff(fact_6628_sums__add,axiom,
    ! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B2: nat] :
      ( pp(aa_nat_bool(sums_nat(F),A))
     => ( pp(aa_nat_bool(sums_nat(G),B2))
       => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ho(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2))) ) ) ).

% sums_add
tff(fact_6629_sums__add,axiom,
    ! [F: fun_nat_int,A: int,G: fun_nat_int,B2: int] :
      ( pp(aa_int_bool(sums_int(F),A))
     => ( pp(aa_int_bool(sums_int(G),B2))
       => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_sf(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))) ) ) ).

% sums_add
tff(fact_6630_sums__add,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B2: real] :
      ( pp(aa_real_bool(sums_real(F),A))
     => ( pp(aa_real_bool(sums_real(G),B2))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_hq(F),G)),aa_real_real(plus_plus_real(A),B2))) ) ) ).

% sums_add
tff(fact_6631_sums__mult__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_sg(C),F)),aa_real_real(times_times_real(C),D)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult_iff
tff(fact_6632_sums__mult__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sh(C),F)),aa_complex_complex(times_times_complex(C),D)))
      <=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).

% sums_mult_iff
tff(fact_6633_sums__mult2__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_si(C),F)),aa_real_real(times_times_real(D),C)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult2_iff
tff(fact_6634_sums__mult2__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sj(C),F)),aa_complex_complex(times_times_complex(D),C)))
      <=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).

% sums_mult2_iff
tff(fact_6635_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A: complex] :
      ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sh(C),F)),A))
     => ( ( C != zero_zero_complex )
       => pp(aa_complex_bool(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C))) ) ) ).

% sums_mult_D
tff(fact_6636_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_sg(C),F)),A))
     => ( ( C != zero_zero_real )
       => pp(aa_real_bool(sums_real(F),aa_real_real(divide_divide_real(A),C))) ) ) ).

% sums_mult_D
tff(fact_6637_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aTP_Lamm_kz(F)),S))
       => pp(aa_real_bool(sums_real(F),S)) ) ) ).

% sums_Suc_imp
tff(fact_6638_sums__Suc__imp,axiom,
    ! [F: fun_nat_complex,S: complex] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aTP_Lamm_qw(F)),S))
       => pp(aa_complex_bool(sums_complex(F),S)) ) ) ).

% sums_Suc_imp
tff(fact_6639_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_kz(F)),S))
    <=> pp(aa_real_bool(sums_real(F),aa_real_real(plus_plus_real(S),aa_nat_real(F,zero_zero_nat)))) ) ).

% sums_Suc_iff
tff(fact_6640_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( pp(aa_nat_bool(sums_nat(aTP_Lamm_ky(F)),L))
     => pp(aa_nat_bool(sums_nat(F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),aa_nat_nat(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_6641_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( pp(aa_int_bool(sums_int(aTP_Lamm_lh(F)),L))
     => pp(aa_int_bool(sums_int(F),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),aa_nat_int(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_6642_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_kz(F)),L))
     => pp(aa_real_bool(sums_real(F),aa_real_real(plus_plus_real(L),aa_nat_real(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_6643_sums__zero__iff__shift,axiom,
    ! [N: nat,F: fun_nat_real,S: real] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
         => ( aa_nat_real(F,I4) = zero_zero_real ) )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_sk(N),F)),S))
      <=> pp(aa_real_bool(sums_real(F),S)) ) ) ).

% sums_zero_iff_shift
tff(fact_6644_sums__zero__iff__shift,axiom,
    ! [N: nat,F: fun_nat_complex,S: complex] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
         => ( aa_nat_complex(F,I4) = zero_zero_complex ) )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sl(N),F)),S))
      <=> pp(aa_complex_bool(sums_complex(F),S)) ) ) ).

% sums_zero_iff_shift
tff(fact_6645_sums__finite,axiom,
    ! [N5: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N5))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => pp(aa_int_bool(sums_int(F),aa_set_nat_int(groups3539618377306564664at_int(F),N5))) ) ) ).

% sums_finite
tff(fact_6646_sums__finite,axiom,
    ! [N5: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N5))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => pp(aa_complex_bool(sums_complex(F),groups2073611262835488442omplex(F,N5))) ) ) ).

% sums_finite
tff(fact_6647_sums__finite,axiom,
    ! [N5: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N5))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => pp(aa_nat_bool(sums_nat(F),aa_set_nat_nat(groups3542108847815614940at_nat(F),N5))) ) ) ).

% sums_finite
tff(fact_6648_sums__finite,axiom,
    ! [N5: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N5))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => pp(aa_real_bool(sums_real(F),aa_set_nat_real(groups6591440286371151544t_real(F),N5))) ) ) ).

% sums_finite
tff(fact_6649_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_sm(P),F)),aa_set_nat_int(groups3539618377306564664at_int(F),collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_6650_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sn(P),F)),groups2073611262835488442omplex(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_6651_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_so(P),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_6652_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_sp(P),F)),aa_set_nat_real(groups6591440286371151544t_real(F),collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_6653_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_sq(A2),F)),aa_set_nat_int(groups3539618377306564664at_int(F),A2))) ) ).

% sums_If_finite_set
tff(fact_6654_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sr(A2),F)),groups2073611262835488442omplex(F,A2))) ) ).

% sums_If_finite_set
tff(fact_6655_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ss(A2),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2))) ) ).

% sums_If_finite_set
tff(fact_6656_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_st(A2),F)),aa_set_nat_real(groups6591440286371151544t_real(F),A2))) ) ).

% sums_If_finite_set
tff(fact_6657_powser__sums__if,axiom,
    ! [M: nat,Z2: int] : pp(aa_int_bool(sums_int(aa_int_fun_nat_int(aTP_Lamm_su(M),Z2)),aa_nat_int(power_power_int(Z2),M))) ).

% powser_sums_if
tff(fact_6658_powser__sums__if,axiom,
    ! [M: nat,Z2: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_sv(M),Z2)),aa_nat_real(power_power_real(Z2),M))) ).

% powser_sums_if
tff(fact_6659_powser__sums__if,axiom,
    ! [M: nat,Z2: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_sw(M),Z2)),aa_nat_complex(power_power_complex(Z2),M))) ).

% powser_sums_if
tff(fact_6660_ln__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),numeral_numeral_real(bit0(one))))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aTP_Lamm_sx(X)) ) ) ) ).

% ln_series
tff(fact_6661_arcosh__def,axiom,
    ! [X: real] : aa_real_real(arcosh_real,X) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(X),aa_real_real(powr_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))))) ).

% arcosh_def
tff(fact_6662_arsinh__def,axiom,
    ! [X: real] : aa_real_real(arsinh_real,X) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(X),aa_real_real(powr_real(aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))))) ).

% arsinh_def
tff(fact_6663_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_6664_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M: nat] : groups2073611262835488442omplex(aTP_Lamm_sy(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M))),numera6690914467698888265omplex(bit0(one)))),aa_nat_complex(gbinomial_complex(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_6665_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_sz(R2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),numeral_numeral_real(bit0(one)))),aa_nat_real(gbinomial_real(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_6666_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(aa_nat_real(power_power_real(numeral_numeral_real(bit0(bit0(one)))),N)),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,N)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_6667_binomial__Suc__n,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),N) = aa_nat_nat(suc,N) ).

% binomial_Suc_n
tff(fact_6668_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_by) = zero_zero_nat ).

% suminf_zero
tff(fact_6669_suminf__zero,axiom,
    suminf_real(aTP_Lamm_go) = zero_zero_real ).

% suminf_zero
tff(fact_6670_suminf__zero,axiom,
    suminf_int(aTP_Lamm_rs) = zero_zero_int ).

% suminf_zero
tff(fact_6671_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_rt) = zero_zero_complex ).

% suminf_zero
tff(fact_6672_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_6673_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,X) = zero_zero_complex )
    <=> ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_6674_of__real__0,axiom,
    aa_real_real(real_V1803761363581548252l_real,zero_zero_real) = zero_zero_real ).

% of_real_0
tff(fact_6675_of__real__0,axiom,
    aa_real_complex(real_V4546457046886955230omplex,zero_zero_real) = zero_zero_complex ).

% of_real_0
tff(fact_6676_floor__zero,axiom,
    aa_real_int(archim6058952711729229775r_real,zero_zero_real) = zero_zero_int ).

% floor_zero
tff(fact_6677_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_6678_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(gbinomial_real(zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_6679_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_6680_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_complex(gbinomial_complex(zero_zero_complex),aa_nat_nat(suc,K)) = zero_zero_complex ).

% gbinomial_0(2)
tff(fact_6681_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_6682_binomial__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,zero_zero_nat)) = N ).

% binomial_1
tff(fact_6683_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% binomial_eq_0_iff
tff(fact_6684_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_6685_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_6686_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_6687_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(gbinomial_real(A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_6688_of__real__add,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(plus_plus_real(X),Y)) = aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_add
tff(fact_6689_of__real__add,axiom,
    ! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(plus_plus_real(X),Y)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_add
tff(fact_6690_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_6691_binomial__n__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_6692_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% zero_less_binomial_iff
tff(fact_6693_zero__le__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_floor
tff(fact_6694_floor__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% floor_less_zero
tff(fact_6695_numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(V)),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(V)),X)) ) ).

% numeral_le_floor
tff(fact_6696_zero__less__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ).

% zero_less_floor
tff(fact_6697_floor__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% floor_le_zero
tff(fact_6698_floor__less__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),numeral_numeral_int(V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),numeral_numeral_real(V))) ) ).

% floor_less_numeral
tff(fact_6699_one__le__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ).

% one_le_floor
tff(fact_6700_floor__less__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% floor_less_one
tff(fact_6701_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_kq(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_6702_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_kp(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_6703_numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,numeral_numeral_int(V)),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(numeral_numeral_real(V)),one_one_real)),X)) ) ).

% numeral_less_floor
tff(fact_6704_floor__le__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),numeral_numeral_int(V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(plus_plus_real(numeral_numeral_real(V)),one_one_real))) ) ).

% floor_le_numeral
tff(fact_6705_one__less__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(bit0(one))),X)) ) ).

% one_less_floor
tff(fact_6706_floor__le__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),numeral_numeral_real(bit0(one)))) ) ).

% floor_le_one
tff(fact_6707_neg__numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),X)) ) ).

% neg_numeral_le_floor
tff(fact_6708_floor__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(V)))) ) ).

% floor_less_neg_numeral
tff(fact_6709_neg__numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)),X)) ) ).

% neg_numeral_less_floor
tff(fact_6710_floor__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real))) ) ).

% floor_le_neg_numeral
tff(fact_6711_floor__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))) ) ).

% floor_mono
tff(fact_6712_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_6713_of__int__floor__le,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X))),X)) ).

% of_int_floor_le
tff(fact_6714_floor__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% floor_less_cancel
tff(fact_6715_Suc__times__binomial,axiom,
    ! [K: nat,N: 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,N)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) ).

% Suc_times_binomial
tff(fact_6716_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),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,N)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_6717_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% binomial_symmetric
tff(fact_6718_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),M)) ).

% choose_mult_lemma
tff(fact_6719_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R2))) ) ).

% binomial_le_pow
tff(fact_6720_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ).

% zero_less_binomial
tff(fact_6721_le__floor__iff,axiom,
    ! [Z2: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),X)) ) ).

% le_floor_iff
tff(fact_6722_floor__less__iff,axiom,
    ! [X: real,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),Z2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z2))) ) ).

% floor_less_iff
tff(fact_6723_Suc__times__binomial__add,axiom,
    ! [A: nat,B2: 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),B2))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B2)),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),B2))),A)) ).

% Suc_times_binomial_add
tff(fact_6724_le__floor__add,axiom,
    ! [X: real,Y: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))),aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X),Y)))) ).

% le_floor_add
tff(fact_6725_int__add__floor,axiom,
    ! [Z2: int,X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_real_int(archim6058952711729229775r_real,X)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z2)),X)) ).

% int_add_floor
tff(fact_6726_floor__add__int,axiom,
    ! [X: real,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),Z2) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X),aa_int_real(ring_1_of_int_real,Z2))) ).

% floor_add_int
tff(fact_6727_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_6728_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_6729_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_6730_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(plus_plus_real(aa_nat_real(gbinomial_real(A),K)),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_6731_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,N)),K) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_6732_suminf__finite,axiom,
    ! [N5: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N5))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => ( suminf_int(F) = aa_set_nat_int(groups3539618377306564664at_int(F),N5) ) ) ) ).

% suminf_finite
tff(fact_6733_suminf__finite,axiom,
    ! [N5: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N5))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => ( suminf_complex(F) = groups2073611262835488442omplex(F,N5) ) ) ) ).

% suminf_finite
tff(fact_6734_suminf__finite,axiom,
    ! [N5: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N5))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => ( suminf_nat(F) = aa_set_nat_nat(groups3542108847815614940at_nat(F),N5) ) ) ) ).

% suminf_finite
tff(fact_6735_suminf__finite,axiom,
    ! [N5: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N5))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => ( suminf_real(F) = aa_set_nat_real(groups6591440286371151544t_real(F),N5) ) ) ) ).

% suminf_finite
tff(fact_6736_norm__less__p1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,X))),one_one_real)))) ).

% norm_less_p1
tff(fact_6737_norm__less__p1,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,X))),one_one_complex)))) ).

% norm_less_p1
tff(fact_6738_one__add__floor,axiom,
    ! [X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X),one_one_real)) ).

% one_add_floor
tff(fact_6739_binomial__absorption,axiom,
    ! [K: nat,N: 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,N),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_6740_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_addition_formula
tff(fact_6741_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)) = aa_real_real(plus_plus_real(aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),K)) ).

% gbinomial_addition_formula
tff(fact_6742_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(divide_divide_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(gbinomial_real(A),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_6743_gbinomial__mult__1_H,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),A) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_6744_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] : aa_real_real(times_times_real(aa_nat_real(gbinomial_real(A),K)),A) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(gbinomial_real(A),K))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_6745_gbinomial__mult__1,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(A),K)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_6746_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] : aa_real_real(times_times_real(A),aa_nat_real(gbinomial_real(A),K)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(gbinomial_real(A),K))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_6747_floor__split,axiom,
    ! [P: fun_int_bool,T: real] :
      ( pp(aa_int_bool(P,aa_real_int(archim6058952711729229775r_real,T)))
    <=> ! [I3: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I3)),T))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,I3)),one_one_real))) )
         => pp(aa_int_bool(P,I3)) ) ) ).

% floor_split
tff(fact_6748_floor__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,X) = A )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,A)),one_one_real))) ) ) ).

% floor_eq_iff
tff(fact_6749_floor__unique,axiom,
    ! [Z2: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z2)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = Z2 ) ) ) ).

% floor_unique
tff(fact_6750_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_6751_le__mult__floor,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B2))),aa_real_int(archim6058952711729229775r_real,aa_real_real(times_times_real(A),B2)))) ) ) ).

% le_mult_floor
tff(fact_6752_less__floor__iff,axiom,
    ! [Z2: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z2)),one_one_real)),X)) ) ).

% less_floor_iff
tff(fact_6753_floor__le__iff,axiom,
    ! [X: real,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),Z2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z2)),one_one_real))) ) ).

% floor_le_iff
tff(fact_6754_binomial__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K6)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6))) ) ) ).

% binomial_mono
tff(fact_6755_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),N))) ).

% binomial_maximum'
tff(fact_6756_binomial__maximum,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% binomial_maximum
tff(fact_6757_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))),K))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_antimono
tff(fact_6758_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% binomial_le_pow2
tff(fact_6759_floor__correct,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X))),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int)))) ) ).

% floor_correct
tff(fact_6760_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_6761_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_6762_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(gbinomial_complex(A),K)) ).

% Suc_times_gbinomial
tff(fact_6763_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(gbinomial_real(A),K)) ).

% Suc_times_gbinomial
tff(fact_6764_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_absorption
tff(fact_6765_gbinomial__absorption,axiom,
    ! [K: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K))) = aa_real_real(times_times_real(A),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),K)) ).

% gbinomial_absorption
tff(fact_6766_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),M)),aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_6767_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_real_real(times_times_real(aa_nat_real(gbinomial_real(A),M)),aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,M)),K)) = aa_real_real(times_times_real(aa_nat_real(gbinomial_real(A),K)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_6768_floor__divide__lower,axiom,
    ! [Q3: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(P3),Q3)))),Q3)),P3)) ) ).

% floor_divide_lower
tff(fact_6769_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)))) ) ).

% binomial_less_binomial_Suc
tff(fact_6770_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_6771_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K6)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6))) ) ) ).

% binomial_strict_mono
tff(fact_6772_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ).

% central_binomial_odd
tff(fact_6773_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_6774_gbinomial__rec,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_6775_gbinomial__rec,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(times_times_real(aa_nat_real(gbinomial_real(A),K)),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_6776_gbinomial__factors,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))),aa_nat_complex(gbinomial_complex(A),K)) ).

% gbinomial_factors
tff(fact_6777_gbinomial__factors,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(gbinomial_real(A),K)) ).

% gbinomial_factors
tff(fact_6778_floor__divide__upper,axiom,
    ! [Q3: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,P3),aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(P3),Q3)))),one_one_real)),Q3))) ) ).

% floor_divide_upper
tff(fact_6779_round__def,axiom,
    ! [X: real] : aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))) ).

% round_def
tff(fact_6780_gbinomial__minus,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% gbinomial_minus
tff(fact_6781_gbinomial__minus,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(uminus_uminus_real,A)),K) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% gbinomial_minus
tff(fact_6782_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_6783_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(plus_plus_real(aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_6784_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups2073611262835488442omplex(aTP_Lamm_ta(K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_6785_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_tb,K)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_6786_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_6787_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_6788_floor__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_6789_floor__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B2))
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_6790_pi__series,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit0(one)))) = suminf_real(aTP_Lamm_tc) ).

% pi_series
tff(fact_6791_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M: nat] : groups2073611262835488442omplex(aTP_Lamm_sy(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),one_one_complex)),numera6690914467698888265omplex(bit0(one)))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_6792_gbinomial__partial__row__sum,axiom,
    ! [A: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_sz(A)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)),numeral_numeral_real(bit0(one)))),aa_nat_real(gbinomial_real(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_6793_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),groups2073611262835488442omplex(aTP_Lamm_td(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(bit0(one))),N) ) ) ).

% choose_even_sum
tff(fact_6794_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_te,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N) ) ) ).

% choose_even_sum
tff(fact_6795_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_tf,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_real(power_power_real(numeral_numeral_real(bit0(one))),N) ) ) ).

% choose_even_sum
tff(fact_6796_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),groups2073611262835488442omplex(aTP_Lamm_tg(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(bit0(one))),N) ) ) ).

% choose_odd_sum
tff(fact_6797_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_th,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N) ) ) ).

% choose_odd_sum
tff(fact_6798_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ti,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_real(power_power_real(numeral_numeral_real(bit0(one))),N) ) ) ).

% choose_odd_sum
tff(fact_6799_round__altdef,axiom,
    ! [X: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_real_real(archim2898591450579166408c_real,X)))
       => ( aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim7802044766580827645g_real,X) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_real_real(archim2898591450579166408c_real,X)))
       => ( aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim6058952711729229775r_real,X) ) ) ) ).

% round_altdef
tff(fact_6800_gbinomial__r__part__sum,axiom,
    ! [M: nat] : groups2073611262835488442omplex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_nat_complex(semiri8010041392384452111omplex,M))),one_one_complex)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) ).

% gbinomial_r_part_sum
tff(fact_6801_gbinomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_real(groups6591440286371151544t_real(gbinomial_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))),one_one_real))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_real(power_power_real(numeral_numeral_real(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) ).

% gbinomial_r_part_sum
tff(fact_6802_atMost__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_set_nat(set_ord_atMost_nat,X) = aa_nat_set_nat(set_ord_atMost_nat,Y) )
    <=> ( X = Y ) ) ).

% atMost_eq_iff
tff(fact_6803_atMost__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( set_ord_atMost_int(X) = set_ord_atMost_int(Y) )
    <=> ( X = Y ) ) ).

% atMost_eq_iff
tff(fact_6804_atMost__iff,axiom,
    ! [I: extended_enat,K: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),set_or8332593352340944941d_enat(K)))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,I),K)) ) ).

% atMost_iff
tff(fact_6805_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( pp(aa_set_real_bool(member_real(I),set_ord_atMost_real(K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),K)) ) ).

% atMost_iff
tff(fact_6806_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),set_or4236626031148496127et_nat(K)))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I),K)) ) ).

% atMost_iff
tff(fact_6807_atMost__iff,axiom,
    ! [I: set_int,K: set_int] :
      ( member_set_int(I,set_or58775011639299419et_int(K))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(I),K)) ) ).

% atMost_iff
tff(fact_6808_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_atMost_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ).

% atMost_iff
tff(fact_6809_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( pp(aa_set_int_bool(member_int(I),set_ord_atMost_int(K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K)) ) ).

% atMost_iff
tff(fact_6810_finite__atMost,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_atMost_nat,K))) ).

% finite_atMost
tff(fact_6811_atMost__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_ord_atMost_real(X)),set_ord_atMost_real(Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% atMost_subset_iff
tff(fact_6812_atMost__subset__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_or4236626031148496127et_nat(X)),set_or4236626031148496127et_nat(Y)))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y)) ) ).

% atMost_subset_iff
tff(fact_6813_atMost__subset__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ord_le4403425263959731960et_int(set_or58775011639299419et_int(X),set_or58775011639299419et_int(Y))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% atMost_subset_iff
tff(fact_6814_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,X)),aa_nat_set_nat(set_ord_atMost_nat,Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% atMost_subset_iff
tff(fact_6815_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(X)),set_ord_atMost_int(Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% atMost_subset_iff
tff(fact_6816_frac__of__int,axiom,
    ! [Z2: int] : aa_real_real(archim2898591450579166408c_real,aa_int_real(ring_1_of_int_real,Z2)) = zero_zero_real ).

% frac_of_int
tff(fact_6817_Icc__subset__Iic__iff,axiom,
    ! [L: set_nat,H: set_nat,H2: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(L,H)),set_or4236626031148496127et_nat(H2)))
    <=> ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,L),H))
        | pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6818_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,H: set_int,H2: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(L,H),set_or58775011639299419et_int(H2))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(L),H))
        | pp(aa_set_int_bool(ord_less_eq_set_int(H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6819_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),H)),aa_nat_set_nat(set_ord_atMost_nat,H2)))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),H))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6820_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(L,H)),set_ord_atMost_int(H2)))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),H))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6821_Icc__subset__Iic__iff,axiom,
    ! [L: real,H: real,H2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(L,H)),set_ord_atMost_real(H2)))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6822_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_6823_sum_OatMost__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_Extended_enat(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_6824_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_6825_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_6826_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_6827_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_6828_prod_OatMost__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_Extended_enat(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_6829_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_6830_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_6831_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = insert_nat(zero_zero_nat,bot_bot_set_nat) ).

% atMost_0
tff(fact_6832_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: extended_enat] : bot_bo7653980558646680370d_enat != set_or8332593352340944941d_enat(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6833_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: real] : bot_bot_set_real != set_ord_atMost_real(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6834_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: nat] : bot_bot_set_nat != aa_nat_set_nat(set_ord_atMost_nat,H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6835_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: int] : bot_bot_set_int != set_ord_atMost_int(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6836_infinite__Iic,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_atMost_int(A))) ).

% infinite_Iic
tff(fact_6837_not__Iic__eq__Icc,axiom,
    ! [H2: int,L: int,H: int] : set_ord_atMost_int(H2) != set_or1266510415728281911st_int(L,H) ).

% not_Iic_eq_Icc
tff(fact_6838_not__Iic__eq__Icc,axiom,
    ! [H2: real,L: real,H: real] : set_ord_atMost_real(H2) != set_or1222579329274155063t_real(L,H) ).

% not_Iic_eq_Icc
tff(fact_6839_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_tj,U)) ).

% atMost_def
tff(fact_6840_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_av,U)) ).

% atMost_def
tff(fact_6841_atMost__def,axiom,
    ! [U: set_int] : set_or58775011639299419et_int(U) = collect_set_int(aTP_Lamm_aw(U)) ).

% atMost_def
tff(fact_6842_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ay,U)) ).

% atMost_def
tff(fact_6843_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_tk,U)) ).

% atMost_def
tff(fact_6844_atMost__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N) ).

% atMost_atLeast0
tff(fact_6845_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_6846_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_6847_not__Iic__le__Icc,axiom,
    ! [H: int,L2: int,H2: int] : ~ pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(H)),set_or1266510415728281911st_int(L2,H2))) ).

% not_Iic_le_Icc
tff(fact_6848_not__Iic__le__Icc,axiom,
    ! [H: real,L2: real,H2: real] : ~ pp(aa_set_real_bool(ord_less_eq_set_real(set_ord_atMost_real(H)),set_or1222579329274155063t_real(L2,H2))) ).

% not_Iic_le_Icc
tff(fact_6849_finite__nat__iff__bounded__le,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ? [K2: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_atMost_nat,K2))) ) ).

% finite_nat_iff_bounded_le
tff(fact_6850_frac__ge__0,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X))) ).

% frac_ge_0
tff(fact_6851_frac__lt__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(archim2898591450579166408c_real,X)),one_one_real)) ).

% frac_lt_1
tff(fact_6852_frac__1__eq,axiom,
    ! [X: real] : aa_real_real(archim2898591450579166408c_real,aa_real_real(plus_plus_real(X),one_one_real)) = aa_real_real(archim2898591450579166408c_real,X) ).

% frac_1_eq
tff(fact_6853_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_6854_Iic__subset__Iio__iff,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(set_or8332593352340944941d_enat(A)),set_or8419480210114673929d_enat(B2)))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,A),B2)) ) ).

% Iic_subset_Iio_iff
tff(fact_6855_Iic__subset__Iio__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_ord_atMost_real(A)),set_or5984915006950818249n_real(B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2)) ) ).

% Iic_subset_Iio_iff
tff(fact_6856_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,A)),aa_nat_set_nat(set_ord_lessThan_nat,B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2)) ) ).

% Iic_subset_Iio_iff
tff(fact_6857_Iic__subset__Iio__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(A)),set_ord_lessThan_int(B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2)) ) ).

% Iic_subset_Iio_iff
tff(fact_6858_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tl,M)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_6859_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lh(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_6860_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,zero_zero_nat)),groups7108830773950497114d_enat(aTP_Lamm_li(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_6861_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_6862_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kz(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_6863_sum__telescope,axiom,
    ! [F: fun_nat_int,I: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_my(F)),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_6864_sum__telescope,axiom,
    ! [F: fun_nat_real,I: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mz(F)),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_real_real(minus_minus_real(aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_6865_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_complex,N: nat,D: fun_nat_complex] :
      ( ! [X5: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(C),X5),aa_nat_set_nat(set_ord_atMost_nat,N)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(D),X5),aa_nat_set_nat(set_ord_atMost_nat,N))
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
         => ( aa_nat_complex(C,I3) = aa_nat_complex(D,I3) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_6866_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_real,N: nat,D: fun_nat_real] :
      ( ! [X5: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(C),X5)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(D),X5)),aa_nat_set_nat(set_ord_atMost_nat,N))
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
         => ( aa_nat_real(C,I3) = aa_nat_real(D,I3) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_6867_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(times_times_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_kz(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_6868_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,zero_zero_nat)),groups6464643781859351333omplex(aTP_Lamm_qw(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_6869_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,zero_zero_nat)),groups7961826882256487087d_enat(aTP_Lamm_li(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_6870_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_lh(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_6871_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_6872_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_to(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tq(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nested_swap'
tff(fact_6873_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_tr(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_tt(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nested_swap'
tff(fact_6874_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_tu(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_tw(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nested_swap'
tff(fact_6875_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_tx(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ty(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nested_swap'
tff(fact_6876_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tz,R2)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),N))),N) ).

% sum_choose_lower
tff(fact_6877_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ua,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_6878_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ua,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_6879_frac__eq,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = X )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% frac_eq
tff(fact_6880_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_complex,N: nat,K: nat] :
      ( ! [W: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(C),W),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( aa_nat_complex(C,K) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_6881_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_real,N: nat,K: nat] :
      ( ! [W: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(C),W)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( aa_nat_real(C,K) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_6882_polyfun__eq__0,axiom,
    ! [C: fun_nat_complex,N: nat] :
      ( ! [X5: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(C),X5),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
         => ( aa_nat_complex(C,I3) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_6883_polyfun__eq__0,axiom,
    ! [C: fun_nat_real,N: nat] :
      ( ! [X5: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(C),X5)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
         => ( aa_nat_real(C,I3) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_6884_frac__add,axiom,
    ! [X: real,Y: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(plus_plus_real(X),Y)) = aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(plus_plus_real(X),Y)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real) ) ) ) ).

% frac_add
tff(fact_6885_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lh(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_6886_sum_OatMost__shift,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,zero_zero_nat)),groups7108830773950497114d_enat(aTP_Lamm_li(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_6887_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_6888_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kz(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_6889_sum__up__index__split,axiom,
    ! [F: fun_nat_int,M: nat,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))) ).

% sum_up_index_split
tff(fact_6890_sum__up__index__split,axiom,
    ! [F: fun_na8797796813480665199d_enat,M: nat,N: nat] : groups7108830773950497114d_enat(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(F,aa_nat_set_nat(set_ord_atMost_nat,M))),groups7108830773950497114d_enat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))) ).

% sum_up_index_split
tff(fact_6891_sum__up__index__split,axiom,
    ! [F: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))) ).

% sum_up_index_split
tff(fact_6892_sum__up__index__split,axiom,
    ! [F: fun_nat_real,M: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))) ).

% sum_up_index_split
tff(fact_6893_prod_OatMost__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(times_times_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_kz(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_6894_prod_OatMost__shift,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,zero_zero_nat)),groups6464643781859351333omplex(aTP_Lamm_qw(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_6895_prod_OatMost__shift,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,zero_zero_nat)),groups7961826882256487087d_enat(aTP_Lamm_li(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_6896_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_lh(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_6897_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_6898_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_6899_gbinomial__parallel__sum,axiom,
    ! [A: complex,N: nat] : groups2073611262835488442omplex(aTP_Lamm_ub(A),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,N))),one_one_complex)),N) ).

% gbinomial_parallel_sum
tff(fact_6900_gbinomial__parallel__sum,axiom,
    ! [A: real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uc(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,N))),one_one_real)),N) ).

% gbinomial_parallel_sum
tff(fact_6901_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ud(N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_uf(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.triangle_reindex_eq
tff(fact_6902_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_real,N: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ud(N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uh(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.triangle_reindex_eq
tff(fact_6903_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_int,N: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ud(N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_uj(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.triangle_reindex_eq
tff(fact_6904_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ud(N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_uk(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.triangle_reindex_eq
tff(fact_6905_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ul(M),N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_6906_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_um(M),N),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),R2) ).

% vandermonde
tff(fact_6907_sum__gp__basic,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_6908_sum__gp__basic,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_6909_sum__gp__basic,axiom,
    ! [X: real,N: nat] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_6910_polyfun__roots__finite,axiom,
    ! [C: fun_nat_complex,K: nat,N: nat] :
      ( ( aa_nat_complex(C,K) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_un(C),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_6911_polyfun__roots__finite,axiom,
    ! [C: fun_nat_real,K: nat,N: nat] :
      ( ( aa_nat_real(C,K) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_uo(C),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_6912_polyfun__finite__roots,axiom,
    ! [C: fun_nat_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_un(C),N))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
          & ( aa_nat_complex(C,I3) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_6913_polyfun__finite__roots,axiom,
    ! [C: fun_nat_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_uo(C),N))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
          & ( aa_nat_real(C,I3) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_6914_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_int,A: int,N: nat] :
      ( ( aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_up(C),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int )
     => ~ ! [B3: fun_nat_int] :
            ~ ! [Z3: int] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_up(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z3),A)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_up(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_6915_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_complex,A: complex,N: nat] :
      ( ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(C),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex )
     => ~ ! [B3: fun_nat_complex] :
            ~ ! [Z3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z3),A)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_6916_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_real,A: real,N: nat] :
      ( ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(C),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real )
     => ~ ! [B3: fun_nat_real] :
            ~ ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_6917_polyfun__linear__factor,axiom,
    ! [C: fun_nat_int,N: nat,A: int] :
    ? [B3: fun_nat_int] :
    ! [Z3: int] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_up(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z3),A)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_up(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N)))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_up(C),A)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_6918_polyfun__linear__factor,axiom,
    ! [C: fun_nat_complex,N: nat,A: complex] :
    ? [B3: fun_nat_complex] :
    ! [Z3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z3),A)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(C),A),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_6919_polyfun__linear__factor,axiom,
    ! [C: fun_nat_real,N: nat,A: real] :
    ? [B3: fun_nat_real] :
    ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N)))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(C),A)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_6920_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),M)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_6921_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_6922_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(X),M)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_6923_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uq(N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_uf(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.triangle_reindex
tff(fact_6924_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_real,N: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uq(N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uh(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.triangle_reindex
tff(fact_6925_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_int,N: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uq(N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_uj(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.triangle_reindex
tff(fact_6926_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uq(N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_uk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.triangle_reindex
tff(fact_6927_binomial,axiom,
    ! [A: nat,B2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ur(A),B2),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial
tff(fact_6928_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lq(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_6929_sum_Oin__pairs__0,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = groups7108830773950497114d_enat(aTP_Lamm_lr(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_6930_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ls(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_6931_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lt(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_6932_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_int,N: nat,B2: fun_nat_int,X: int] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I4))
         => ( aa_nat_int(A,I4) = zero_zero_int ) )
     => ( ! [J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J3))
           => ( aa_nat_int(B2,J3) = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_up(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_up(B2),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_ut(A),B2),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_6933_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_complex,N: nat,B2: fun_nat_complex,X: complex] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I4))
         => ( aa_nat_complex(A,I4) = zero_zero_complex ) )
     => ( ! [J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J3))
           => ( aa_nat_complex(B2,J3) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(A),X),aa_nat_set_nat(set_ord_atMost_nat,M))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(B2),X),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_uv(A),B2),X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_6934_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_real,N: nat,B2: fun_nat_real,X: real] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I4))
         => ( aa_nat_real(A,I4) = zero_zero_real ) )
     => ( ! [J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J3))
           => ( aa_nat_real(B2,J3) = zero_zero_real ) )
       => ( aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(B2),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_ux(A),B2),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_6935_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_rh(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_6936_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = groups6464643781859351333omplex(aTP_Lamm_ri(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_6937_prod_Oin__pairs__0,axiom,
    ! [G: fun_na8797796813480665199d_enat,N: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = groups7961826882256487087d_enat(aTP_Lamm_rj(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_6938_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_rk(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_6939_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_rl(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_6940_polyfun__eq__const,axiom,
    ! [C: fun_nat_complex,N: nat,K: complex] :
      ( ! [X5: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(C),X5),aa_nat_set_nat(set_ord_atMost_nat,N)) = K
    <=> ( ( aa_nat_complex(C,zero_zero_nat) = K )
        & ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)))
           => ( aa_nat_complex(C,X5) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_6941_polyfun__eq__const,axiom,
    ! [C: fun_nat_real,N: nat,K: real] :
      ( ! [X5: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(C),X5)),aa_nat_set_nat(set_ord_atMost_nat,N)) = K
    <=> ( ( aa_nat_real(C,zero_zero_nat) = K )
        & ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)))
           => ( aa_nat_real(C,X5) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_6942_binomial__ring,axiom,
    ! [A: complex,B2: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(A),B2)),N) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_uy(A),B2),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_6943_binomial__ring,axiom,
    ! [A: extended_enat,B2: extended_enat,N: nat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2)),N) = groups7108830773950497114d_enat(aa_nat2025370463774033377d_enat(aa_Ext8984936454148280526d_enat(aTP_Lamm_uz(A),B2),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_6944_binomial__ring,axiom,
    ! [A: int,B2: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),N) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_va(A),B2),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_6945_binomial__ring,axiom,
    ! [A: nat,B2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ur(A),B2),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_6946_binomial__ring,axiom,
    ! [A: real,B2: real,N: nat] : aa_nat_real(power_power_real(aa_real_real(plus_plus_real(A),B2)),N) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_vb(A),B2),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_6947_pochhammer__binomial__sum,axiom,
    ! [A: complex,B2: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),B2)),N) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_vc(A),B2),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_6948_pochhammer__binomial__sum,axiom,
    ! [A: int,B2: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),N) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_vd(A),B2),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_6949_pochhammer__binomial__sum,axiom,
    ! [A: real,B2: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(A),B2)),N) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ve(A),B2),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_6950_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,N: nat,B2: fun_nat_nat,X: nat] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I4))
         => ( aa_nat_nat(A,I4) = zero_zero_nat ) )
     => ( ! [J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J3))
           => ( aa_nat_nat(B2,J3) = 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_vf(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vf(B2),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_vh(A),B2),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_6951_floor__add,axiom,
    ! [X: real,Y: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))),one_one_int) ) ) ) ).

% floor_add
tff(fact_6952_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_vi(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_vj(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6953_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_vk(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_vl(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6954_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_na8797796813480665199d_enat,H: fun_na8797796813480665199d_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups7108830773950497114d_enat(aa_fun2512298231148344956d_enat(aa_fun1703792696854984886d_enat(aTP_Lamm_vm(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups7108830773950497114d_enat(aa_fun2512298231148344956d_enat(aa_fun1703792696854984886d_enat(aTP_Lamm_vn(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6955_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_vo(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_vp(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6956_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_vq(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_vr(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6957_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_vs(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_vl(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6958_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_vt(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_vr(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6959_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_vu(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_vj(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6960_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_vv(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_vp(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6961_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: complex,X: complex,Y: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_vw(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_vx(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_6962_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: real,X: real,Y: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_vy(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_vz(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_6963_root__polyfun,axiom,
    ! [N: nat,Z2: complex,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_complex(power_power_complex(Z2),N) = A )
      <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_wa(N),Z2),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_6964_root__polyfun,axiom,
    ! [N: nat,Z2: int,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_int(power_power_int(Z2),N) = A )
      <=> ( aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_wb(N),Z2),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_6965_root__polyfun,axiom,
    ! [N: nat,Z2: real,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_real(power_power_real(Z2),N) = A )
      <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_wc(N),Z2),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_6966_sum__gp0,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N)))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp0
tff(fact_6967_sum__gp0,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X),aa_nat_nat(suc,N)))),aa_real_real(minus_minus_real(one_one_real),X)) ) ) ) ).

% sum_gp0
tff(fact_6968_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups2073611262835488442omplex(aTP_Lamm_wd(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_6969_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_we,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_6970_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_wf,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_6971_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : groups2073611262835488442omplex(aTP_Lamm_wg(M),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(bit0(one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_6972_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_wh,M)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_real(power_power_real(numeral_numeral_real(bit0(one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_6973_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: complex,X: complex,Y: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_vw(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_wi(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_6974_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: real,X: real,Y: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_vy(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_wj(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_6975_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_int,X: int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_up(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_up(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_wl(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_6976_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_complex,X: complex,Y: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(A),X),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_wn(N),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_6977_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_real,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_wp(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_6978_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) ).

% binomial_r_part_sum
tff(fact_6979_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups2073611262835488442omplex(aTP_Lamm_wq(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ).

% choose_alternating_sum
tff(fact_6980_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_wr,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_6981_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ws,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_6982_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_complex,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ? [M8: real] :
        ! [Z3: complex] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M8),aa_complex_real(real_V1022390504157884413omplex,Z3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N)))),aa_real_real(times_times_real(E2),aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,Z3)),aa_nat_nat(suc,N))))) ) ) ).

% polyfun_extremal_lemma
tff(fact_6983_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ? [M8: real] :
        ! [Z3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M8),aa_real_real(real_V7735802525324610683m_real,Z3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)))),aa_real_real(times_times_real(E2),aa_nat_real(power_power_real(aa_real_real(real_V7735802525324610683m_real,Z3)),aa_nat_nat(suc,N))))) ) ) ).

% polyfun_extremal_lemma
tff(fact_6984_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_int,X: int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_up(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_up(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_wu(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_6985_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_complex,X: complex,Y: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(A),X),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_ww(N),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_6986_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_real,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_wy(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_6987_sin__cos__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),pi)),numeral_numeral_real(bit0(one)))) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% sin_cos_npi
tff(fact_6988_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))))),numeral_numeral_real(bit0(one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_6989_gbinomial__code,axiom,
    ! [K: nat,A: complex] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_complex(gbinomial_complex(A),K) = one_one_complex ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_wz(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_6990_gbinomial__code,axiom,
    ! [K: nat,A: real] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_real(gbinomial_real(A),K) = one_one_real ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_xa(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_6991_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)),one_one_nat))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F))) ) ) ).

% sum_pos_lt_pair
tff(fact_6992_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_6993_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)) ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_6994_sgn__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(sgn_sgn_int,A)) = aa_int_int(sgn_sgn_int,A) ).

% sgn_sgn
tff(fact_6995_sgn__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(sgn_sgn_real,A)) = aa_real_real(sgn_sgn_real,A) ).

% sgn_sgn
tff(fact_6996_of__nat__id,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = N ).

% of_nat_id
tff(fact_6997_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_6998_sin__zero,axiom,
    aa_complex_complex(sin_complex,zero_zero_complex) = zero_zero_complex ).

% sin_zero
tff(fact_6999_sgn__0,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_0
tff(fact_7000_sgn__0,axiom,
    aa_int_int(sgn_sgn_int,zero_zero_int) = zero_zero_int ).

% sgn_0
tff(fact_7001_sgn__0,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_0
tff(fact_7002_sgn__zero,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_zero
tff(fact_7003_sgn__zero,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_zero
tff(fact_7004_sgn__1,axiom,
    aa_complex_complex(sgn_sgn_complex,one_one_complex) = one_one_complex ).

% sgn_1
tff(fact_7005_sgn__1,axiom,
    aa_int_int(sgn_sgn_int,one_one_int) = one_one_int ).

% sgn_1
tff(fact_7006_sgn__1,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_1
tff(fact_7007_sgn__divide,axiom,
    ! [A: real,B2: real] : aa_real_real(sgn_sgn_real,aa_real_real(divide_divide_real(A),B2)) = aa_real_real(divide_divide_real(aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B2)) ).

% sgn_divide
tff(fact_7008_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_7009_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_7010_summable__zero,axiom,
    summable_nat(aTP_Lamm_by) ).

% summable_zero
tff(fact_7011_summable__zero,axiom,
    summable_real(aTP_Lamm_go) ).

% summable_zero
tff(fact_7012_summable__zero,axiom,
    summable_int(aTP_Lamm_rs) ).

% summable_zero
tff(fact_7013_summable__zero,axiom,
    summable_complex(aTP_Lamm_rt) ).

% summable_zero
tff(fact_7014_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sc(I),F)) ).

% summable_single
tff(fact_7015_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_sd(I),F)) ).

% summable_single
tff(fact_7016_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_gy(I),F)) ).

% summable_single
tff(fact_7017_summable__single,axiom,
    ! [I: nat,F: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_se(I),F)) ).

% summable_single
tff(fact_7018_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_lb(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_7019_sgn__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sgn_sgn_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% sgn_less
tff(fact_7020_sgn__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(sgn_sgn_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% sgn_less
tff(fact_7021_sgn__greater,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sgn_sgn_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% sgn_greater
tff(fact_7022_sgn__greater,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(sgn_sgn_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% sgn_greater
tff(fact_7023_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_7024_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_7025_divide__sgn,axiom,
    ! [A: real,B2: real] : aa_real_real(divide_divide_real(A),aa_real_real(sgn_sgn_real,B2)) = aa_real_real(times_times_real(A),aa_real_real(sgn_sgn_real,B2)) ).

% divide_sgn
tff(fact_7026_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_7027_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_7028_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_7029_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_7030_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_sg(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_7031_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sh(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_7032_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_xb(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_7033_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_xc(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_7034_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ss(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7035_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_st(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7036_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_sq(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7037_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sr(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7038_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_so(P),F)) ) ).

% summable_If_finite
tff(fact_7039_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_sp(P),F)) ) ).

% summable_If_finite
tff(fact_7040_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_sm(P),F)) ) ).

% summable_If_finite
tff(fact_7041_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sn(P),F)) ) ).

% summable_If_finite
tff(fact_7042_sgn__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) ) ).

% sgn_pos
tff(fact_7043_sgn__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( aa_int_int(sgn_sgn_int,A) = one_one_int ) ) ).

% sgn_pos
tff(fact_7044_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_7045_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_7046_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_7047_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_7048_fact__Suc,axiom,
    ! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N))),aa_nat_complex(semiri5044797733671781792omplex,N)) ).

% fact_Suc
tff(fact_7049_fact__Suc,axiom,
    ! [N: nat] : aa_nat_Extended_enat(semiri4449623510593786356d_enat,aa_nat_nat(suc,N)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(suc,N))),aa_nat_Extended_enat(semiri4449623510593786356d_enat,N)) ).

% fact_Suc
tff(fact_7050_fact__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))),aa_nat_int(semiri1406184849735516958ct_int,N)) ).

% fact_Suc
tff(fact_7051_fact__Suc,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N))),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).

% fact_Suc
tff(fact_7052_fact__Suc,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,N)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))),aa_nat_real(semiri2265585572941072030t_real,N)) ).

% fact_Suc
tff(fact_7053_sgn__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,A)) = zero_n3304061248610475627l_real(aa_bool_bool(fNot,aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,A),zero_zero_real))) ).

% sgn_mult_self_eq
tff(fact_7054_sgn__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,A)) = zero_n2684676970156552555ol_int(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,A),zero_zero_int))) ).

% sgn_mult_self_eq
tff(fact_7055_sin__of__real__pi,axiom,
    aa_real_real(sin_real,aa_real_real(real_V1803761363581548252l_real,pi)) = zero_zero_real ).

% sin_of_real_pi
tff(fact_7056_sin__of__real__pi,axiom,
    aa_complex_complex(sin_complex,aa_real_complex(real_V4546457046886955230omplex,pi)) = zero_zero_complex ).

% sin_of_real_pi
tff(fact_7057_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,R2))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_7058_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),K)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_7059_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R2))),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_7060_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),L)),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_7061_sgn__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% sgn_neg
tff(fact_7062_sgn__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% sgn_neg
tff(fact_7063_sin__cos__squared__add3,axiom,
    ! [X: real] : aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(cos_real,X))),aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(sin_real,X))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_7064_sin__cos__squared__add3,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,X))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,X))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_7065_sgn__of__nat,axiom,
    ! [N: nat] : aa_real_real(sgn_sgn_real,aa_nat_real(semiri5074537144036343181t_real,N)) = zero_n3304061248610475627l_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_7066_sgn__of__nat,axiom,
    ! [N: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(semiri1314217659103216013at_int,N)) = zero_n2684676970156552555ol_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_7067_sin__cos__squared__add2,axiom,
    ! [X: real] : aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_7068_sin__cos__squared__add2,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_7069_sin__cos__squared__add,axiom,
    ! [X: real] : aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_7070_sin__cos__squared__add,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_7071_cos__of__real__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(bit0(one)))) = zero_zero_real ).

% cos_of_real_pi_half
tff(fact_7072_cos__of__real__pi__half,axiom,
    aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(bit0(one)))) = zero_zero_complex ).

% cos_of_real_pi_half
tff(fact_7073_sin__add,axiom,
    ! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X),Y)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(cos_real,Y))),aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(sin_real,Y))) ).

% sin_add
tff(fact_7074_sin__add,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(cos_complex,Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(sin_complex,Y))) ).

% sin_add
tff(fact_7075_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
     => ( aa_real_real(sin_real,X) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_7076_cos__one__sin__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) = one_one_complex )
     => ( aa_complex_complex(sin_complex,X) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_7077_fact__ge__self,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_self
tff(fact_7078_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% fact_mono_nat
tff(fact_7079_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
tff(fact_7080_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
tff(fact_7081_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
tff(fact_7082_sgn__0__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_0_0
tff(fact_7083_sgn__0__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_0_0
tff(fact_7084_sgn__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sgn_sgn_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% sgn_zero_iff
tff(fact_7085_sgn__zero__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,X) = zero_zero_complex )
    <=> ( X = zero_zero_complex ) ) ).

% sgn_zero_iff
tff(fact_7086_sgn__mult,axiom,
    ! [A: int,B2: int] : aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B2)) ).

% sgn_mult
tff(fact_7087_sgn__mult,axiom,
    ! [A: real,B2: real] : aa_real_real(sgn_sgn_real,aa_real_real(times_times_real(A),B2)) = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B2)) ).

% sgn_mult
tff(fact_7088_sgn__mult,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(times_times_complex(A),B2)) = aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B2)) ).

% sgn_mult
tff(fact_7089_same__sgn__sgn__add,axiom,
    ! [B2: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B2) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(sgn_sgn_int,A) ) ) ).

% same_sgn_sgn_add
tff(fact_7090_same__sgn__sgn__add,axiom,
    ! [B2: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B2) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(sgn_sgn_real,aa_real_real(plus_plus_real(A),B2)) = aa_real_real(sgn_sgn_real,A) ) ) ).

% same_sgn_sgn_add
tff(fact_7091_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) != zero_zero_int ).

% fact_nonzero
tff(fact_7092_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,N) != zero_zero_complex ).

% fact_nonzero
tff(fact_7093_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_Extended_enat(semiri4449623510593786356d_enat,N) != zero_z5237406670263579293d_enat ).

% fact_nonzero
tff(fact_7094_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) != zero_zero_nat ).

% fact_nonzero
tff(fact_7095_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,N) != zero_zero_real ).

% fact_nonzero
tff(fact_7096_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aTP_Lamm_xd(C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_7097_summable__const__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aTP_Lamm_xe(C))
    <=> ( C = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_7098_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_real(F) ) ) ).

% summable_comparison_test
tff(fact_7099_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_complex(F) ) ) ).

% summable_comparison_test
tff(fact_7100_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N5: nat,F: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
       => summable_real(F) ) ) ).

% summable_comparison_test'
tff(fact_7101_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N5: nat,F: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
       => summable_complex(F) ) ) ).

% summable_comparison_test'
tff(fact_7102_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ho(F),G)) ) ) ).

% summable_add
tff(fact_7103_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_sf(F),G)) ) ) ).

% summable_add
tff(fact_7104_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_hq(F),G)) ) ) ).

% summable_add
tff(fact_7105_cos__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(X),Y)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_diff
tff(fact_7106_cos__diff,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,Y))) ).

% cos_diff
tff(fact_7107_cos__add,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X),Y)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_add
tff(fact_7108_cos__add,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,Y))) ).

% cos_add
tff(fact_7109_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_kz(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_7110_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_lb(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_7111_sin__zero__norm__cos__one,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_7112_sin__zero__norm__cos__one,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sin_complex,X) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_7113_suminf__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2)))
     => ( summable_real(F)
       => ( summable_real(G)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),suminf_real(G))) ) ) ) ).

% suminf_le
tff(fact_7114_suminf__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G,N2)))
     => ( summable_nat(F)
       => ( summable_nat(G)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G))) ) ) ) ).

% suminf_le
tff(fact_7115_suminf__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G,N2)))
     => ( summable_int(F)
       => ( summable_int(G)
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),suminf_int(G))) ) ) ) ).

% suminf_le
tff(fact_7116_summable__finite,axiom,
    ! [N5: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N5))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => summable_nat(F) ) ) ).

% summable_finite
tff(fact_7117_summable__finite,axiom,
    ! [N5: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N5))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => summable_real(F) ) ) ).

% summable_finite
tff(fact_7118_summable__finite,axiom,
    ! [N5: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N5))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => summable_int(F) ) ) ).

% summable_finite
tff(fact_7119_summable__finite,axiom,
    ! [N5: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N5))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => summable_complex(F) ) ) ).

% summable_finite
tff(fact_7120_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_7121_sgn__not__eq__imp,axiom,
    ! [B2: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B2) != aa_int_int(sgn_sgn_int,A) )
     => ( ( aa_int_int(sgn_sgn_int,A) != zero_zero_int )
       => ( ( aa_int_int(sgn_sgn_int,B2) != zero_zero_int )
         => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,B2)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_7122_sgn__not__eq__imp,axiom,
    ! [B2: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B2) != aa_real_real(sgn_sgn_real,A) )
     => ( ( aa_real_real(sgn_sgn_real,A) != zero_zero_real )
       => ( ( aa_real_real(sgn_sgn_real,B2) != zero_zero_real )
         => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,B2)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_7123_sgn__minus__1,axiom,
    aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% sgn_minus_1
tff(fact_7124_sgn__minus__1,axiom,
    aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% sgn_minus_1
tff(fact_7125_sgn__minus__1,axiom,
    aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sgn_minus_1
tff(fact_7126_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_ge_zero
tff(fact_7127_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_zero
tff(fact_7128_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_ge_zero
tff(fact_7129_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_gt_zero
tff(fact_7130_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_gt_zero
tff(fact_7131_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_gt_zero
tff(fact_7132_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,N)),zero_zero_int)) ).

% fact_not_neg
tff(fact_7133_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),zero_zero_nat)) ).

% fact_not_neg
tff(fact_7134_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,N)),zero_zero_real)) ).

% fact_not_neg
tff(fact_7135_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L3: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_sgnE
tff(fact_7136_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_ge_1
tff(fact_7137_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_1
tff(fact_7138_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_ge_1
tff(fact_7139_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ).

% fact_mono
tff(fact_7140_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% fact_mono
tff(fact_7141_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,N))) ) ).

% fact_mono
tff(fact_7142_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_sg(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_7143_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sh(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_7144_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1406184849735516958ct_int,M))) ) ).

% fact_dvd
tff(fact_7145_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,M))) ) ).

% fact_dvd
tff(fact_7146_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,M))) ) ).

% fact_dvd
tff(fact_7147_summable__zero__power,axiom,
    summable_real(power_power_real(zero_zero_real)) ).

% summable_zero_power
tff(fact_7148_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_7149_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_7150_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_ho(F),G)) ) ) ) ).

% suminf_add
tff(fact_7151_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_sf(F),G)) ) ) ) ).

% suminf_add
tff(fact_7152_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(plus_plus_real(suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_hq(F),G)) ) ) ) ).

% suminf_add
tff(fact_7153_suminf__nonneg,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_nonneg
tff(fact_7154_suminf__nonneg,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_nonneg
tff(fact_7155_suminf__nonneg,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_nonneg
tff(fact_7156_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( ( suminf_real(F) = zero_zero_real )
        <=> ! [N4: nat] : aa_nat_real(F,N4) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7157_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( ( suminf_nat(F) = zero_zero_nat )
        <=> ! [N4: nat] : aa_nat_nat(F,N4) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7158_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( ( suminf_int(F) = zero_zero_int )
        <=> ! [N4: nat] : aa_nat_int(F,N4) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7159_suminf__pos,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_pos
tff(fact_7160_suminf__pos,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,N2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_pos
tff(fact_7161_suminf__pos,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,N2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_pos
tff(fact_7162_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_7163_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = one_one_real )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% sgn_1_pos
tff(fact_7164_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = one_one_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% sgn_1_pos
tff(fact_7165_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% dvd_fact
tff(fact_7166_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_kq(F)) ).

% summable_0_powser
tff(fact_7167_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_kp(F)) ).

% summable_0_powser
tff(fact_7168_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_xf(F)) ).

% summable_zero_power'
tff(fact_7169_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_kq(F)) ).

% summable_zero_power'
tff(fact_7170_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_kp(F)) ).

% summable_zero_power'
tff(fact_7171_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_xg(F),Z2))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_tn(F),Z2)) ) ).

% summable_powser_split_head
tff(fact_7172_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_xh(F),Z2))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tm(F),Z2)) ) ).

% summable_powser_split_head
tff(fact_7173_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_tn(F),Z2))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_xg(F),Z2)) ) ).

% powser_split_head(3)
tff(fact_7174_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tm(F),Z2))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_xh(F),Z2)) ) ).

% powser_split_head(3)
tff(fact_7175_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,M: nat,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xi(F),M),Z2))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_tn(F),Z2)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7176_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,M: nat,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_xj(F),M),Z2))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tm(F),Z2)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7177_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ) ).

% fact_less_mono
tff(fact_7178_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono
tff(fact_7179_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,N))) ) ) ).

% fact_less_mono
tff(fact_7180_sin__times__sin,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W2)),aa_complex_complex(sin_complex,Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W2),Z2))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W2),Z2)))),numera6690914467698888265omplex(bit0(one))) ).

% sin_times_sin
tff(fact_7181_sin__times__sin,axiom,
    ! [W2: real,Z2: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z2)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W2),Z2))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W2),Z2)))),numeral_numeral_real(bit0(one))) ).

% sin_times_sin
tff(fact_7182_sin__times__cos,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W2)),aa_complex_complex(cos_complex,Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W2),Z2))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W2),Z2)))),numera6690914467698888265omplex(bit0(one))) ).

% sin_times_cos
tff(fact_7183_sin__times__cos,axiom,
    ! [W2: real,Z2: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W2)),aa_real_real(cos_real,Z2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W2),Z2))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W2),Z2)))),numeral_numeral_real(bit0(one))) ).

% sin_times_cos
tff(fact_7184_cos__times__sin,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W2)),aa_complex_complex(sin_complex,Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W2),Z2))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W2),Z2)))),numera6690914467698888265omplex(bit0(one))) ).

% cos_times_sin
tff(fact_7185_cos__times__sin,axiom,
    ! [W2: real,Z2: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W2)),aa_real_real(sin_real,Z2)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W2),Z2))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W2),Z2)))),numeral_numeral_real(bit0(one))) ).

% cos_times_sin
tff(fact_7186_sin__plus__sin,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,W2)),aa_complex_complex(sin_complex,Z2)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z2)),numera6690914467698888265omplex(bit0(one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W2),Z2)),numera6690914467698888265omplex(bit0(one))))) ).

% sin_plus_sin
tff(fact_7187_sin__plus__sin,axiom,
    ! [W2: real,Z2: real] : aa_real_real(plus_plus_real(aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z2)) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W2),Z2)),numeral_numeral_real(bit0(one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W2),Z2)),numeral_numeral_real(bit0(one))))) ).

% sin_plus_sin
tff(fact_7188_sin__diff__sin,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,W2)),aa_complex_complex(sin_complex,Z2)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W2),Z2)),numera6690914467698888265omplex(bit0(one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z2)),numera6690914467698888265omplex(bit0(one))))) ).

% sin_diff_sin
tff(fact_7189_sin__diff__sin,axiom,
    ! [W2: real,Z2: real] : aa_real_real(minus_minus_real(aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z2)) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W2),Z2)),numeral_numeral_real(bit0(one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W2),Z2)),numeral_numeral_real(bit0(one))))) ).

% sin_diff_sin
tff(fact_7190_cos__diff__cos,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,W2)),aa_complex_complex(cos_complex,Z2)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z2)),numera6690914467698888265omplex(bit0(one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Z2),W2)),numera6690914467698888265omplex(bit0(one))))) ).

% cos_diff_cos
tff(fact_7191_cos__diff__cos,axiom,
    ! [W2: real,Z2: real] : aa_real_real(minus_minus_real(aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z2)) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W2),Z2)),numeral_numeral_real(bit0(one)))))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Z2),W2)),numeral_numeral_real(bit0(one))))) ).

% cos_diff_cos
tff(fact_7192_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(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,N))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_7193_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,N))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_7194_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_7195_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(modulo_modulo_int(aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1406184849735516958ct_int,M)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_7196_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_7197_fact__le__power,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_7198_fact__le__power,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_7199_fact__le__power,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_7200_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_xk(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_7201_suminf__pos2,axiom,
    ! [F: fun_nat_real,I: nat] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ) ).

% suminf_pos2
tff(fact_7202_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I: nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ) ).

% suminf_pos2
tff(fact_7203_suminf__pos2,axiom,
    ! [F: fun_nat_int,I: nat] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ) ).

% suminf_pos2
tff(fact_7204_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F)))
        <=> ? [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I3))) ) ) ) ).

% suminf_pos_iff
tff(fact_7205_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F)))
        <=> ? [I3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I3))) ) ) ) ).

% suminf_pos_iff
tff(fact_7206_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F)))
        <=> ? [I3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I3))) ) ) ) ).

% suminf_pos_iff
tff(fact_7207_minus__sin__cos__eq,axiom,
    ! [X: real] : aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) = aa_real_real(cos_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(bit0(one))))) ).

% minus_sin_cos_eq
tff(fact_7208_minus__sin__cos__eq,axiom,
    ! [X: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sin_complex,X)) = aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(bit0(one))))) ).

% minus_sin_cos_eq
tff(fact_7209_suminf__le__const,axiom,
    ! [F: fun_nat_int,X: int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),X)) ) ) ).

% suminf_le_const
tff(fact_7210_suminf__le__const,axiom,
    ! [F: fun_nat_nat,X: nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),X)) ) ) ).

% suminf_le_const
tff(fact_7211_suminf__le__const,axiom,
    ! [F: fun_nat_real,X: real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),X)) ) ) ).

% suminf_le_const
tff(fact_7212_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% fact_diff_Suc
tff(fact_7213_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% sgn_1_neg
tff(fact_7214_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% sgn_1_neg
tff(fact_7215_sgn__if,axiom,
    ! [X: int] :
      ( ( ( X = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,X) = zero_zero_int ) )
      & ( ( X != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
           => ( aa_int_int(sgn_sgn_int,X) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
           => ( aa_int_int(sgn_sgn_int,X) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% sgn_if
tff(fact_7216_sgn__if,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(sgn_sgn_real,X) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
           => ( aa_real_real(sgn_sgn_real,X) = one_one_real ) )
          & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
           => ( aa_real_real(sgn_sgn_real,X) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

% sgn_if
tff(fact_7217_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,I) = zero_zero_int ) )
      & ( ( I != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% zsgn_def
tff(fact_7218_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R2))) ) ).

% fact_div_fact_le_pow
tff(fact_7219_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_7220_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,X: int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7221_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,X: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7222_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,X: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7223_norm__sgn,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,X)) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,X)) = one_one_real ) ) ) ).

% norm_sgn
tff(fact_7224_norm__sgn,axiom,
    ! [X: complex] :
      ( ( ( X = zero_zero_complex )
       => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,X)) = zero_zero_real ) )
      & ( ( X != zero_zero_complex )
       => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,X)) = one_one_real ) ) ) ).

% norm_sgn
tff(fact_7225_bounded__imp__summable,axiom,
    ! [A: fun_nat_int,B: int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(A,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(A),aa_nat_set_nat(set_ord_atMost_nat,N2))),B))
       => summable_int(A) ) ) ).

% bounded_imp_summable
tff(fact_7226_bounded__imp__summable,axiom,
    ! [A: fun_nat_nat,B: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(A,N2)))
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_ord_atMost_nat,N2))),B))
       => summable_nat(A) ) ) ).

% bounded_imp_summable
tff(fact_7227_bounded__imp__summable,axiom,
    ! [A: fun_nat_real,B: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(A),aa_nat_set_nat(set_ord_atMost_nat,N2))),B))
       => summable_real(A) ) ) ).

% bounded_imp_summable
tff(fact_7228_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_kz(F)) = aa_real_real(minus_minus_real(suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_7229_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(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,N),K)))),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ).

% choose_dvd
tff(fact_7230_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% choose_dvd
tff(fact_7231_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_real(semiri2265585572941072030t_real,N))) ) ).

% choose_dvd
tff(fact_7232_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cd),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_7233_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(bit0(one))))) = aa_real_real(cos_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(bit0(one))))) ).

% sin_expansion_lemma
tff(fact_7234_sum__le__suminf,axiom,
    ! [F: fun_nat_int,I6: set_nat] :
      ( summable_int(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I6)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),I6)),suminf_int(F))) ) ) ) ).

% sum_le_suminf
tff(fact_7235_sum__le__suminf,axiom,
    ! [F: fun_nat_nat,I6: set_nat] :
      ( summable_nat(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I6)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),I6)),suminf_nat(F))) ) ) ) ).

% sum_le_suminf
tff(fact_7236_sum__le__suminf,axiom,
    ! [F: fun_nat_real,I6: set_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I6)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),I6)),suminf_real(F))) ) ) ) ).

% sum_le_suminf
tff(fact_7237_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(bit0(one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(bit0(one)))))) ).

% cos_expansion_lemma
tff(fact_7238_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_7239_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(F) = aa_real_real(plus_plus_real(suminf_real(aa_nat_fun_nat_real(aTP_Lamm_lb(F),K))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_split_initial_segment
tff(fact_7240_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_lb(F),K)) = aa_real_real(minus_minus_real(suminf_real(F)),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_minus_initial_segment
tff(fact_7241_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cd),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_7242_sum__less__suminf,axiom,
    ! [F: fun_nat_int,N: nat] :
      ( summable_int(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,M3))) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_int(F))) ) ) ).

% sum_less_suminf
tff(fact_7243_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( summable_nat(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M3))) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_nat(F))) ) ) ).

% sum_less_suminf
tff(fact_7244_sum__less__suminf,axiom,
    ! [F: fun_nat_real,N: nat] :
      ( summable_real(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,M3))) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_real(F))) ) ) ).

% sum_less_suminf
tff(fact_7245_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_tn(F),Z2))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_tn(F),Z2)) = aa_real_real(plus_plus_real(aa_nat_real(F,zero_zero_nat)),aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_xg(F),Z2))),Z2)) ) ) ).

% powser_split_head(1)
tff(fact_7246_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tm(F),Z2))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_tm(F),Z2)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(F,zero_zero_nat)),aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_xh(F),Z2))),Z2)) ) ) ).

% powser_split_head(1)
tff(fact_7247_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_tn(F),Z2))
     => ( aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_xg(F),Z2))),Z2) = aa_real_real(minus_minus_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_tn(F),Z2))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7248_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tm(F),Z2))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_xh(F),Z2))),Z2) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_tm(F),Z2))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7249_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E2: real] :
      ( summable_complex(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ~ ! [N8: nat] :
              ~ ! [M5: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M5))
                 => ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M5),N6)))),E2)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_7250_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E2: real] :
      ( summable_real(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ~ ! [N8: nat] :
              ~ ! [M5: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M5))
                 => ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M5),N6)))),E2)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_7251_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_real(F)
       => ? [N8: nat] :
          ! [N6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_lb(F),N6)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_7252_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_complex(F)
       => ? [N8: nat] :
          ! [N6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_xl(F),N6)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_7253_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_7254_sin__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aTP_Lamm_xm(X)),aa_real_real(sin_real,X))) ).

% sin_paired
tff(fact_7255_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(semiri5044797733671781792omplex,M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(semiri5044797733671781792omplex,M) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7256_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_Extended_enat(semiri4449623510593786356d_enat,M) = one_on7984719198319812577d_enat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_Extended_enat(semiri4449623510593786356d_enat,M) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,M)),aa_nat_Extended_enat(semiri4449623510593786356d_enat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7257_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(semiri1406184849735516958ct_int,M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(semiri1406184849735516958ct_int,M) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7258_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7259_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_real(semiri2265585572941072030t_real,M) = one_one_real ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_real(semiri2265585572941072030t_real,M) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7260_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(semiri5044797733671781792omplex,N) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7261_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_Extended_enat(semiri4449623510593786356d_enat,N) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)),aa_nat_Extended_enat(semiri4449623510593786356d_enat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7262_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7263_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7264_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(semiri2265585572941072030t_real,N) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7265_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,N)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_binomial
tff(fact_7266_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_binomial
tff(fact_7267_binomial__fact,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,N)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_fact
tff(fact_7268_binomial__fact,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,N)),aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_fact
tff(fact_7269_cos__times__cos,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W2)),aa_complex_complex(cos_complex,Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W2),Z2))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W2),Z2)))),numera6690914467698888265omplex(bit0(one))) ).

% cos_times_cos
tff(fact_7270_cos__times__cos,axiom,
    ! [W2: real,Z2: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W2),Z2))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W2),Z2)))),numeral_numeral_real(bit0(one))) ).

% cos_times_cos
tff(fact_7271_cos__plus__cos,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,W2)),aa_complex_complex(cos_complex,Z2)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z2)),numera6690914467698888265omplex(bit0(one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W2),Z2)),numera6690914467698888265omplex(bit0(one))))) ).

% cos_plus_cos
tff(fact_7272_cos__plus__cos,axiom,
    ! [W2: real,Z2: real] : aa_real_real(plus_plus_real(aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z2)) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W2),Z2)),numeral_numeral_real(bit0(one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W2),Z2)),numeral_numeral_real(bit0(one))))) ).

% cos_plus_cos
tff(fact_7273_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N2)))),aa_real_real(times_times_real(C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_7274_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N2)))),aa_real_real(times_times_real(C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_7275_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,N: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M3))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_int(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7276_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,N: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M3))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_nat(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7277_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,N: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M3))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_real(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7278_gbinomial__pochhammer_H,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer'
tff(fact_7279_gbinomial__pochhammer_H,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),K) = aa_real_real(divide_divide_real(aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer'
tff(fact_7280_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_nat_fun_nat_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_xn(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_7281_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_xo(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_7282_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_nat_fun_int_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_xp(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_7283_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_na5723442673449079972x_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_xq(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).

% Maclaurin_zero
tff(fact_7284_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_na2978037997607195054t_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun2748096013305118955t_real(aTP_Lamm_xr(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_Ext2541572767299617110t_real(aa_nat6736956760766371991t_real(Diff,zero_zero_nat),zero_z5237406670263579293d_enat) ) ) ) ).

% Maclaurin_zero
tff(fact_7285_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X))
            & ( aa_real_real(cos_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xs(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T6),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_7286_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),zero_zero_real))
            & ( aa_real_real(cos_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xs(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T6),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_7287_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_7288_gbinomial__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)) = aa_real_real(divide_divide_real(aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_xt(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_7289_gbinomial__Suc,axiom,
    ! [A: int,K: nat] : aa_nat_int(gbinomial_int(A),aa_nat_nat(suc,K)) = aa_int_int(divide_divide_int(aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_xu(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_7290_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xv,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_7291_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_int ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_7292_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X))
            & ( aa_real_real(sin_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xw(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T6),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_7293_listrel1p__def,axiom,
    ! [R2: fun_nat_fun_nat_bool,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(listrel1p_nat(R2,Xs),Ys))
    <=> member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(R2)))) ) ).

% listrel1p_def
tff(fact_7294_listrel1p__def,axiom,
    ! [R2: fun_int_fun_int_bool,Xs: list_int,Ys: list_int] :
      ( pp(aa_list_int_bool(listrel1p_int(R2,Xs),Ys))
    <=> member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(collec213857154873943460nt_int(produc8560705615207772305t_bool(R2)))) ) ).

% listrel1p_def
tff(fact_7295_sin__x__sin__y,axiom,
    ! [X: complex,Y: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_xy(X),Y)),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,Y)))) ).

% sin_x_sin_y
tff(fact_7296_sin__x__sin__y,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_ya(X),Y)),aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))) ).

% sin_x_sin_y
tff(fact_7297_sums__cos__x__plus__y,axiom,
    ! [X: complex,Y: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_yc(X),Y)),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y)))) ).

% sums_cos_x_plus_y
tff(fact_7298_sums__cos__x__plus__y,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_ye(X),Y)),aa_real_real(cos_real,aa_real_real(plus_plus_real(X),Y)))) ).

% sums_cos_x_plus_y
tff(fact_7299_cos__x__cos__y,axiom,
    ! [X: complex,Y: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_yg(X),Y)),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y)))) ).

% cos_x_cos_y
tff(fact_7300_cos__x__cos__y,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_yi(X),Y)),aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))) ).

% cos_x_cos_y
tff(fact_7301_sin__coeff__def,axiom,
    ! [X2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),X2))
       => ( aa_nat_real(sin_coeff,X2) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),X2))
       => ( aa_nat_real(sin_coeff,X2) = aa_real_real(divide_divide_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_real(semiri2265585572941072030t_real,X2)) ) ) ) ).

% sin_coeff_def
tff(fact_7302_scaleR__cancel__right,axiom,
    ! [A: real,X: real,B2: real] :
      ( ( aa_real_real(real_V1485227260804924795R_real(A),X) = aa_real_real(real_V1485227260804924795R_real(B2),X) )
    <=> ( ( A = B2 )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_cancel_right
tff(fact_7303_scaleR__cancel__right,axiom,
    ! [A: real,X: complex,B2: real] :
      ( ( aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),X) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,B2),X) )
    <=> ( ( A = B2 )
        | ( X = zero_zero_complex ) ) ) ).

% scaleR_cancel_right
tff(fact_7304_scaleR__zero__right,axiom,
    ! [A: real] : aa_real_real(real_V1485227260804924795R_real(A),zero_zero_real) = zero_zero_real ).

% scaleR_zero_right
tff(fact_7305_scaleR__zero__right,axiom,
    ! [A: real] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),zero_zero_complex) = zero_zero_complex ).

% scaleR_zero_right
tff(fact_7306_scaleR__eq__0__iff,axiom,
    ! [A: real,X: real] :
      ( ( aa_real_real(real_V1485227260804924795R_real(A),X) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_eq_0_iff
tff(fact_7307_scaleR__eq__0__iff,axiom,
    ! [A: real,X: complex] :
      ( ( aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),X) = zero_zero_complex )
    <=> ( ( A = zero_zero_real )
        | ( X = zero_zero_complex ) ) ) ).

% scaleR_eq_0_iff
tff(fact_7308_scaleR__zero__left,axiom,
    ! [X: real] : aa_real_real(real_V1485227260804924795R_real(zero_zero_real),X) = zero_zero_real ).

% scaleR_zero_left
tff(fact_7309_scaleR__zero__left,axiom,
    ! [X: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,zero_zero_real),X) = zero_zero_complex ).

% scaleR_zero_left
tff(fact_7310_scaleR__eq__iff,axiom,
    ! [B2: real,U: real,A: real] :
      ( ( aa_real_real(plus_plus_real(B2),aa_real_real(real_V1485227260804924795R_real(U),A)) = aa_real_real(plus_plus_real(A),aa_real_real(real_V1485227260804924795R_real(U),B2)) )
    <=> ( ( A = B2 )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
tff(fact_7311_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_7312_scaleR__collapse,axiom,
    ! [U: real,A: real] : aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(minus_minus_real(one_one_real),U)),A)),aa_real_real(real_V1485227260804924795R_real(U),A)) = A ).

% scaleR_collapse
tff(fact_7313_scaleR__half__double,axiom,
    ! [A: real] : aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_real_real(plus_plus_real(A),A)) = A ).

% scaleR_half_double
tff(fact_7314_scaleR__right__imp__eq,axiom,
    ! [X: real,A: real,B2: real] :
      ( ( X != zero_zero_real )
     => ( ( aa_real_real(real_V1485227260804924795R_real(A),X) = aa_real_real(real_V1485227260804924795R_real(B2),X) )
       => ( A = B2 ) ) ) ).

% scaleR_right_imp_eq
tff(fact_7315_scaleR__right__imp__eq,axiom,
    ! [X: complex,A: real,B2: real] :
      ( ( X != zero_zero_complex )
     => ( ( aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),X) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,B2),X) )
       => ( A = B2 ) ) ) ).

% scaleR_right_imp_eq
tff(fact_7316_scaleR__right__distrib,axiom,
    ! [A: real,X: real,Y: real] : aa_real_real(real_V1485227260804924795R_real(A),aa_real_real(plus_plus_real(X),Y)) = aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(A),X)),aa_real_real(real_V1485227260804924795R_real(A),Y)) ).

% scaleR_right_distrib
tff(fact_7317_scaleR__left_Oadd,axiom,
    ! [X: real,Y: real,Xa2: real] : aa_real_real(real_V1485227260804924795R_real(aa_real_real(plus_plus_real(X),Y)),Xa2) = aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(X),Xa2)),aa_real_real(real_V1485227260804924795R_real(Y),Xa2)) ).

% scaleR_left.add
tff(fact_7318_scaleR__left__distrib,axiom,
    ! [A: real,B2: real,X: real] : aa_real_real(real_V1485227260804924795R_real(aa_real_real(plus_plus_real(A),B2)),X) = aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(A),X)),aa_real_real(real_V1485227260804924795R_real(B2),X)) ).

% scaleR_left_distrib
tff(fact_7319_scaleR__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),C)),aa_real_real(real_V1485227260804924795R_real(B2),C))) ) ) ).

% scaleR_right_mono_neg
tff(fact_7320_scaleR__right__mono,axiom,
    ! [A: real,B2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),aa_real_real(real_V1485227260804924795R_real(B2),X))) ) ) ).

% scaleR_right_mono
tff(fact_7321_scaleR__le__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(real_V1485227260804924795R_real(C),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_7322_scaleR__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(real_V1485227260804924795R_real(C),B2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_7323_scaleR__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(real_V1485227260804924795R_real(C),B2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_7324_scaleR__left__mono,axiom,
    ! [X: real,Y: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),aa_real_real(real_V1485227260804924795R_real(A),Y))) ) ) ).

% scaleR_left_mono
tff(fact_7325_scaleR__left__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(real_V1485227260804924795R_real(C),B2))) ) ) ).

% scaleR_left_mono_neg
tff(fact_7326_eq__vector__fraction__iff,axiom,
    ! [X: real,U: real,V: real,A: real] :
      ( ( X = aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(U),V)),A) )
    <=> ( ( ( V = zero_zero_real )
         => ( X = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( aa_real_real(real_V1485227260804924795R_real(V),X) = aa_real_real(real_V1485227260804924795R_real(U),A) ) ) ) ) ).

% eq_vector_fraction_iff
tff(fact_7327_eq__vector__fraction__iff,axiom,
    ! [X: complex,U: real,V: real,A: complex] :
      ( ( X = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(divide_divide_real(U),V)),A) )
    <=> ( ( ( V = zero_zero_real )
         => ( X = zero_zero_complex ) )
        & ( ( V != zero_zero_real )
         => ( aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,V),X) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,U),A) ) ) ) ) ).

% eq_vector_fraction_iff
tff(fact_7328_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: real,X: real] :
      ( ( aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(U),V)),A) = X )
    <=> ( ( ( V = zero_zero_real )
         => ( X = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( aa_real_real(real_V1485227260804924795R_real(U),A) = aa_real_real(real_V1485227260804924795R_real(V),X) ) ) ) ) ).

% vector_fraction_eq_iff
tff(fact_7329_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: complex,X: complex] :
      ( ( aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(divide_divide_real(U),V)),A) = X )
    <=> ( ( ( V = zero_zero_real )
         => ( X = zero_zero_complex ) )
        & ( ( V != zero_zero_real )
         => ( aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,U),A) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,V),X) ) ) ) ) ).

% vector_fraction_eq_iff
tff(fact_7330_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(B2),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(minus_minus_real(B2),A)),E2)),D))) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_7331_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(B2),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(minus_minus_real(A),B2)),E2)),C)),D)) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_7332_zero__le__scaleR__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(real_V1485227260804924795R_real(A),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) )
        | ( A = zero_zero_real ) ) ) ).

% zero_le_scaleR_iff
tff(fact_7333_scaleR__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),B2)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) )
        | ( A = zero_zero_real ) ) ) ).

% scaleR_le_0_iff
tff(fact_7334_scaleR__nonpos__nonpos,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(real_V1485227260804924795R_real(A),B2))) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_7335_scaleR__nonpos__nonneg,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),zero_zero_real)) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_7336_scaleR__nonneg__nonpos,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),zero_zero_real)) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_7337_scaleR__nonneg__nonneg,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(real_V1485227260804924795R_real(A),X))) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_7338_split__scaleR__pos__le,axiom,
    ! [A: real,B2: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(real_V1485227260804924795R_real(A),B2))) ) ).

% split_scaleR_pos_le
tff(fact_7339_split__scaleR__neg__le,axiom,
    ! [A: real,X: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),zero_zero_real)) ) ).

% split_scaleR_neg_le
tff(fact_7340_scaleR__mono_H,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),C)),aa_real_real(real_V1485227260804924795R_real(B2),D))) ) ) ) ) ).

% scaleR_mono'
tff(fact_7341_scaleR__mono,axiom,
    ! [A: real,B2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),aa_real_real(real_V1485227260804924795R_real(B2),Y))) ) ) ) ) ).

% scaleR_mono
tff(fact_7342_scaleR__left__le__one__le,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),X)) ) ) ).

% scaleR_left_le_one_le
tff(fact_7343_scaleR__2,axiom,
    ! [X: real] : aa_real_real(real_V1485227260804924795R_real(numeral_numeral_real(bit0(one))),X) = aa_real_real(plus_plus_real(X),X) ).

% scaleR_2
tff(fact_7344_sin__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,N)) = aa_real_real(divide_divide_real(aa_nat_real(cos_coeff,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% sin_coeff_Suc
tff(fact_7345_cos__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,N)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,N))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% cos_coeff_Suc
tff(fact_7346_summable__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => summable_real(aTP_Lamm_yj(X)) ) ).

% summable_arctan_series
tff(fact_7347_diffs__equiv,axiom,
    ! [C: fun_nat_complex,X: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_yk(C),X))
     => pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_yl(C),X)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_yk(C),X)))) ) ).

% diffs_equiv
tff(fact_7348_diffs__equiv,axiom,
    ! [C: fun_nat_real,X: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ym(C),X))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_yn(C),X)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_ym(C),X)))) ) ).

% diffs_equiv
tff(fact_7349_tan__double,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_complex_complex(tan_complex,X))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ) ) ).

% tan_double
tff(fact_7350_tan__double,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),X)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),X)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(tan_real,X))),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ) ) ).

% tan_double
tff(fact_7351_tan__half,axiom,
    ! [X: complex] : aa_complex_complex(tan_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X))),one_one_complex)) ).

% tan_half
tff(fact_7352_tan__half,axiom,
    ! [X: real] : aa_real_real(tan_real,X) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),X))),aa_real_real(plus_plus_real(aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),X))),one_one_real)) ).

% tan_half
tff(fact_7353_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_yo(D))) ).

% int_ge_less_than2_def
tff(fact_7354_abs__idempotent,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_idempotent
tff(fact_7355_abs__idempotent,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_idempotent
tff(fact_7356_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_7357_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_7358_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_7359_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_7360_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_7361_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_7362_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_7363_abs__eq__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0
tff(fact_7364_abs__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_7365_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(abs_abs_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% abs_0_eq
tff(fact_7366_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_7367_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_7368_abs__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(times_times_real(A),A) ).

% abs_mult_self_eq
tff(fact_7369_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_7370_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_7371_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_7372_abs__add__abs,axiom,
    ! [A: int,B2: 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,B2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)) ).

% abs_add_abs
tff(fact_7373_abs__add__abs,axiom,
    ! [A: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2))) = aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ).

% abs_add_abs
tff(fact_7374_abs__divide,axiom,
    ! [A: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(A),B2)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ).

% abs_divide
tff(fact_7375_abs__minus__cancel,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus_cancel
tff(fact_7376_abs__minus__cancel,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus_cancel
tff(fact_7377_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_7378_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_7379_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(abs_abs_real,M)),K))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).

% abs_dvd_iff
tff(fact_7380_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(abs_abs_int,M)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).

% abs_dvd_iff
tff(fact_7381_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),aa_real_real(abs_abs_real,K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).

% dvd_abs_iff
tff(fact_7382_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),aa_int_int(abs_abs_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).

% dvd_abs_iff
tff(fact_7383_abs__of__nat,axiom,
    ! [N: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% abs_of_nat
tff(fact_7384_abs__of__nat,axiom,
    ! [N: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% abs_of_nat
tff(fact_7385_of__int__abs,axiom,
    ! [X: int] : aa_int_int(ring_1_of_int_int,aa_int_int(abs_abs_int,X)) = aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,X)) ).

% of_int_abs
tff(fact_7386_of__int__abs,axiom,
    ! [X: int] : aa_int_real(ring_1_of_int_real,aa_int_int(abs_abs_int,X)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,X)) ).

% of_int_abs
tff(fact_7387_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_7388_tan__zero,axiom,
    aa_complex_complex(tan_complex,zero_zero_complex) = zero_zero_complex ).

% tan_zero
tff(fact_7389_abs__bool__eq,axiom,
    ! [P: bool] : aa_real_real(abs_abs_real,zero_n3304061248610475627l_real(P)) = zero_n3304061248610475627l_real(P) ).

% abs_bool_eq
tff(fact_7390_abs__bool__eq,axiom,
    ! [P: bool] : aa_int_int(abs_abs_int,zero_n2684676970156552555ol_int(P)) = zero_n2684676970156552555ol_int(P) ).

% abs_bool_eq
tff(fact_7391_abs__of__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_7392_abs__of__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_7393_abs__le__self__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% abs_le_self_iff
tff(fact_7394_abs__le__self__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% abs_le_self_iff
tff(fact_7395_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_7396_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int))
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_7397_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A)))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_7398_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A)))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_7399_sum__abs,axiom,
    ! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A2))),groups4538972089207619220nt_int(aTP_Lamm_yp(F),A2))) ).

% sum_abs
tff(fact_7400_sum__abs,axiom,
    ! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yq(F)),A2))) ).

% sum_abs
tff(fact_7401_divide__le__0__abs__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(A),aa_real_real(abs_abs_real,B2))),zero_zero_real))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
        | ( B2 = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_7402_zero__le__divide__abs__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_real_real(abs_abs_real,B2))))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        | ( B2 = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_7403_abs__of__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_nonpos
tff(fact_7404_abs__of__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_7405_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ).

% abs_sgn_eq_1
tff(fact_7406_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ).

% abs_sgn_eq_1
tff(fact_7407_sgn__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = zero_n3304061248610475627l_real(aa_bool_bool(fNot,aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,A),zero_zero_real))) ).

% sgn_abs
tff(fact_7408_sgn__abs,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(sgn_sgn_complex,A)) = zero_n1201886186963655149omplex(aa_bool_bool(fNot,aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,A),zero_zero_complex))) ).

% sgn_abs
tff(fact_7409_sgn__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = zero_n2684676970156552555ol_int(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,A),zero_zero_int))) ).

% sgn_abs
tff(fact_7410_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(abs_abs_real,A)) = zero_n3304061248610475627l_real(aa_bool_bool(fNot,aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,A),zero_zero_real))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7411_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(abs_abs_complex,A)) = zero_n1201886186963655149omplex(aa_bool_bool(fNot,aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,A),zero_zero_complex))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7412_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(abs_abs_int,A)) = zero_n2684676970156552555ol_int(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,A),zero_zero_int))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7413_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_yp(F),A2))) ).

% sum_abs_ge_zero
tff(fact_7414_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yq(F)),A2))) ).

% sum_abs_ge_zero
tff(fact_7415_zero__less__power__abs__iff,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N)))
    <=> ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7416_zero__less__power__abs__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N)))
    <=> ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7417_norm__of__real__add1,axiom,
    ! [X: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,X)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(X),one_one_real)) ).

% norm_of_real_add1
tff(fact_7418_norm__of__real__add1,axiom,
    ! [X: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(X),one_one_real)) ).

% norm_of_real_add1
tff(fact_7419_norm__of__real__addn,axiom,
    ! [X: real,B2: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,X)),numeral_numeral_real(B2))) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(X),numeral_numeral_real(B2))) ).

% norm_of_real_addn
tff(fact_7420_norm__of__real__addn,axiom,
    ! [X: real,B2: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),numera6690914467698888265omplex(B2))) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(X),numeral_numeral_real(B2))) ).

% norm_of_real_addn
tff(fact_7421_abs__minus__commute,axiom,
    ! [A: int,B2: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A)) ).

% abs_minus_commute
tff(fact_7422_abs__minus__commute,axiom,
    ! [A: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B2)) = aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B2),A)) ).

% abs_minus_commute
tff(fact_7423_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,X) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( X = Y )
        | ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_7424_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,X) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( X = Y )
        | ( X = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_7425_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_7426_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_7427_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_7428_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_7429_abs__mult,axiom,
    ! [A: int,B2: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)) ).

% abs_mult
tff(fact_7430_abs__mult,axiom,
    ! [A: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(times_times_real(A),B2)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ).

% abs_mult
tff(fact_7431_abs__mult,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(times_times_complex(A),B2)) = aa_complex_complex(times_times_complex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B2)) ).

% abs_mult
tff(fact_7432_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_7433_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_7434_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_7435_abs__le__D1,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2)) ) ).

% abs_le_D1
tff(fact_7436_abs__le__D1,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2)) ) ).

% abs_le_D1
tff(fact_7437_abs__ge__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(abs_abs_real,A))) ).

% abs_ge_self
tff(fact_7438_abs__ge__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(abs_abs_int,A))) ).

% abs_ge_self
tff(fact_7439_abs__ge__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A))) ).

% abs_ge_zero
tff(fact_7440_abs__ge__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A))) ).

% abs_ge_zero
tff(fact_7441_abs__of__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_7442_abs__of__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_7443_abs__not__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real)) ).

% abs_not_less_zero
tff(fact_7444_abs__not__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int)) ).

% abs_not_less_zero
tff(fact_7445_abs__triangle__ineq,axiom,
    ! [A: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(A),B2))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)))) ).

% abs_triangle_ineq
tff(fact_7446_abs__triangle__ineq,axiom,
    ! [A: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)))) ).

% abs_triangle_ineq
tff(fact_7447_abs__mult__less,axiom,
    ! [A: real,C: real,B2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,B2)),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2))),aa_real_real(times_times_real(C),D))) ) ) ).

% abs_mult_less
tff(fact_7448_abs__mult__less,axiom,
    ! [A: int,C: int,B2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,B2)),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))) ) ) ).

% abs_mult_less
tff(fact_7449_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B2),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_7450_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_7451_abs__triangle__ineq3,axiom,
    ! [A: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B2)))) ).

% abs_triangle_ineq3
tff(fact_7452_abs__triangle__ineq3,axiom,
    ! [A: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)))) ).

% abs_triangle_ineq3
tff(fact_7453_abs__triangle__ineq2,axiom,
    ! [A: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B2)))) ).

% abs_triangle_ineq2
tff(fact_7454_abs__triangle__ineq2,axiom,
    ! [A: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)))) ).

% abs_triangle_ineq2
tff(fact_7455_nonzero__abs__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(A),B2)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ) ) ).

% nonzero_abs_divide
tff(fact_7456_abs__ge__minus__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A))) ).

% abs_ge_minus_self
tff(fact_7457_abs__ge__minus__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A))) ).

% abs_ge_minus_self
tff(fact_7458_abs__le__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B2)) ) ) ).

% abs_le_iff
tff(fact_7459_abs__le__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B2)) ) ) ).

% abs_le_iff
tff(fact_7460_abs__le__D2,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B2)) ) ).

% abs_le_D2
tff(fact_7461_abs__le__D2,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B2)) ) ).

% abs_le_D2
tff(fact_7462_abs__leI,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B2)) ) ) ).

% abs_leI
tff(fact_7463_abs__leI,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B2)) ) ) ).

% abs_leI
tff(fact_7464_abs__less__iff,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B2)) ) ) ).

% abs_less_iff
tff(fact_7465_abs__less__iff,axiom,
    ! [A: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),B2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B2)) ) ) ).

% abs_less_iff
tff(fact_7466_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: int] : aa_int_int(abs_abs_int,K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_7467_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: real] : aa_real_real(abs_abs_real,K) = aa_real_real(times_times_real(K),aa_real_real(sgn_sgn_real,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_7468_abs__mult__sgn,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(sgn_sgn_int,A)) = A ).

% abs_mult_sgn
tff(fact_7469_abs__mult__sgn,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(sgn_sgn_real,A)) = A ).

% abs_mult_sgn
tff(fact_7470_abs__mult__sgn,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(sgn_sgn_complex,A)) = A ).

% abs_mult_sgn
tff(fact_7471_sgn__mult__abs,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(abs_abs_int,A)) = A ).

% sgn_mult_abs
tff(fact_7472_sgn__mult__abs,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_real_real(abs_abs_real,A)) = A ).

% sgn_mult_abs
tff(fact_7473_sgn__mult__abs,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(abs_abs_complex,A)) = A ).

% sgn_mult_abs
tff(fact_7474_mult__sgn__abs,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,X)),aa_int_int(abs_abs_int,X)) = X ).

% mult_sgn_abs
tff(fact_7475_mult__sgn__abs,axiom,
    ! [X: real] : aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,X)),aa_real_real(abs_abs_real,X)) = X ).

% mult_sgn_abs
tff(fact_7476_same__sgn__abs__add,axiom,
    ! [B2: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B2) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)) ) ) ).

% same_sgn_abs_add
tff(fact_7477_same__sgn__abs__add,axiom,
    ! [B2: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B2) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(A),B2)) = aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ) ) ).

% same_sgn_abs_add
tff(fact_7478_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),E)) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_7479_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(times_times_real(aa_real_real(abs_abs_real,Y)),X) = aa_real_real(abs_abs_real,aa_real_real(times_times_real(Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7480_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7481_abs__eq__mult,axiom,
    ! [A: real,B2: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(times_times_real(A),B2)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ) ) ).

% abs_eq_mult
tff(fact_7482_abs__eq__mult,axiom,
    ! [A: int,B2: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int)) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)) ) ) ).

% abs_eq_mult
tff(fact_7483_abs__eq__iff_H,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(abs_abs_real,A) = B2 )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
        & ( ( A = B2 )
          | ( A = aa_real_real(uminus_uminus_real,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7484_abs__eq__iff_H,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(abs_abs_int,A) = B2 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
        & ( ( A = B2 )
          | ( A = aa_int_int(uminus_uminus_int,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7485_eq__abs__iff_H,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(abs_abs_real,B2) )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        & ( ( B2 = A )
          | ( B2 = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7486_eq__abs__iff_H,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(abs_abs_int,B2) )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
        & ( ( B2 = A )
          | ( B2 = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7487_abs__minus__le__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real)) ).

% abs_minus_le_zero
tff(fact_7488_abs__minus__le__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int)) ).

% abs_minus_le_zero
tff(fact_7489_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,X)),Y) = aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(X),Y)) ) ) ).

% abs_div_pos
tff(fact_7490_zero__le__power__abs,axiom,
    ! [A: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N))) ).

% zero_le_power_abs
tff(fact_7491_zero__le__power__abs,axiom,
    ! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N))) ).

% zero_le_power_abs
tff(fact_7492_abs__if__raw,axiom,
    ! [X2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X2) = aa_int_int(uminus_uminus_int,X2) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X2) = X2 ) ) ) ).

% abs_if_raw
tff(fact_7493_abs__if__raw,axiom,
    ! [X2: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X2) = aa_real_real(uminus_uminus_real,X2) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X2) = X2 ) ) ) ).

% abs_if_raw
tff(fact_7494_abs__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A) = A ) ) ) ).

% abs_if
tff(fact_7495_abs__if,axiom,
    ! [A: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = A ) ) ) ).

% abs_if
tff(fact_7496_abs__of__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_7497_abs__of__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_neg
tff(fact_7498_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),R2)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(plus_plus_real(A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_7499_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_7500_abs__triangle__ineq4,axiom,
    ! [A: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B2))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)))) ).

% abs_triangle_ineq4
tff(fact_7501_abs__triangle__ineq4,axiom,
    ! [A: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)))) ).

% abs_triangle_ineq4
tff(fact_7502_abs__diff__triangle__ineq,axiom,
    ! [A: real,B2: real,C: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B2)),aa_real_real(plus_plus_real(C),D)))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),C))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B2),D))))) ).

% abs_diff_triangle_ineq
tff(fact_7503_abs__diff__triangle__ineq,axiom,
    ! [A: int,B2: int,C: int,D: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),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,B2),D))))) ).

% abs_diff_triangle_ineq
tff(fact_7504_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),R2)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(plus_plus_real(A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_7505_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_7506_abs__sgn__eq,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ) ).

% abs_sgn_eq
tff(fact_7507_abs__sgn__eq,axiom,
    ! [A: int] :
      ( ( ( A = zero_zero_int )
       => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = zero_zero_int ) )
      & ( ( A != zero_zero_int )
       => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ) ).

% abs_sgn_eq
tff(fact_7508_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_yq(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_7509_abs__add__one__gt__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),aa_real_real(abs_abs_real,X)))) ).

% abs_add_one_gt_zero
tff(fact_7510_abs__add__one__gt__zero,axiom,
    ! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,X)))) ).

% abs_add_one_gt_zero
tff(fact_7511_of__int__leD,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% of_int_leD
tff(fact_7512_of__int__leD,axiom,
    ! [N: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),X)) ) ) ).

% of_int_leD
tff(fact_7513_of__int__lessD,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% of_int_lessD
tff(fact_7514_of__int__lessD,axiom,
    ! [N: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),X)) ) ) ).

% of_int_lessD
tff(fact_7515_sgn__power__injE,axiom,
    ! [A: real,N: nat,X: real,B2: real] :
      ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N)) = X )
     => ( ( X = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,B2)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,B2)),N)) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( A = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_7516_round__diff__minimal,axiom,
    ! [Z2: real,M: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z2),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z2))))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z2),aa_int_real(ring_1_of_int_real,M))))) ).

% round_diff_minimal
tff(fact_7517_diffs__def,axiom,
    ! [C: fun_nat_complex,X2: nat] : aa_nat_complex(diffs_complex(C),X2) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,X2))),aa_nat_complex(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_7518_diffs__def,axiom,
    ! [C: fun_nat_real,X2: nat] : aa_nat_real(diffs_real(C),X2) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X2))),aa_nat_real(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_7519_diffs__def,axiom,
    ! [C: fun_nat_int,X2: nat] : aa_nat_int(diffs_int(C),X2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X2))),aa_nat_int(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_7520_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% abs_le_square_iff
tff(fact_7521_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% abs_le_square_iff
tff(fact_7522_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_7523_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_7524_abs__sqrt__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,X: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,X4),aa_nat_real(power_power_real(X4),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,aa_real_real(abs_abs_real,X)),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% abs_sqrt_wlog
tff(fact_7525_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_bool,X: int] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X4))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,X4),aa_nat_int(power_power_int(X4),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(abs_abs_int,X)),aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% abs_sqrt_wlog
tff(fact_7526_abs__square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).

% abs_square_le_1
tff(fact_7527_abs__square__le__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).

% abs_square_le_1
tff(fact_7528_abs__square__less__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).

% abs_square_less_1
tff(fact_7529_abs__square__less__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).

% abs_square_less_1
tff(fact_7530_power__mono__even,axiom,
    ! [N: nat,A: real,B2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B2),N))) ) ) ).

% power_mono_even
tff(fact_7531_power__mono__even,axiom,
    ! [N: nat,A: int,B2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B2),N))) ) ) ).

% power_mono_even
tff(fact_7532_convex__sum__bound__le,axiom,
    ! [I6: set_Extended_enat,X: fun_Ex1451620480498224165t_real,A: fun_Ex1451620480498224165t_real,B2: real,Delta: real] :
      ( ! [I4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(X,I4))) )
     => ( ( groups4148127829035722712t_real(X,I6) = one_one_real )
       => ( ! [I4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_Ext2541572767299617110t_real(A,I4)),B2))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_yr(X),A),I6)),B2))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7533_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: fun_real_real,A: fun_real_real,B2: real,Delta: real] :
      ( ! [I4: real] :
          ( pp(aa_set_real_bool(member_real(I4),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(X,I4))) )
     => ( ( groups8097168146408367636l_real(X,I6) = one_one_real )
       => ( ! [I4: real] :
              ( pp(aa_set_real_bool(member_real(I4),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(A,I4)),B2))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ys(X),A),I6)),B2))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7534_convex__sum__bound__le,axiom,
    ! [I6: set_set_nat,X: fun_set_nat_real,A: fun_set_nat_real,B2: real,Delta: real] :
      ( ! [I4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(I4),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_set_nat_real(X,I4))) )
     => ( ( groups5107569545109728110t_real(X,I6) = one_one_real )
       => ( ! [I4: set_nat] :
              ( pp(aa_set_set_nat_bool(member_set_nat(I4),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_set_nat_real(A,I4)),B2))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups5107569545109728110t_real(aa_fun3569254276315353500t_real(aTP_Lamm_yt(X),A),I6)),B2))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7535_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: fun_int_real,A: fun_int_real,B2: real,Delta: real] :
      ( ! [I4: int] :
          ( pp(aa_set_int_bool(member_int(I4),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(X,I4))) )
     => ( ( groups8778361861064173332t_real(X,I6) = one_one_real )
       => ( ! [I4: int] :
              ( pp(aa_set_int_bool(member_int(I4),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_int_real(A,I4)),B2))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_yu(X),A),I6)),B2))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7536_convex__sum__bound__le,axiom,
    ! [I6: set_Extended_enat,X: fun_Ex3482447378695272101at_int,A: fun_Ex3482447378695272101at_int,B2: int,Delta: int] :
      ( ! [I4: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_Extended_enat_int(X,I4))) )
     => ( ( groups2025484359314973016at_int(X,I6) = one_one_int )
       => ( ! [I4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I4),I6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Extended_enat_int(A,I4)),B2))),Delta)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups2025484359314973016at_int(aa_fun1695976788104708604at_int(aTP_Lamm_yv(X),A),I6)),B2))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7537_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: fun_real_int,A: fun_real_int,B2: int,Delta: int] :
      ( ! [I4: real] :
          ( pp(aa_set_real_bool(member_real(I4),I6))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(X,I4))) )
     => ( ( groups1932886352136224148al_int(X,I6) = one_one_int )
       => ( ! [I4: real] :
              ( pp(aa_set_real_bool(member_real(I4),I6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(A,I4)),B2))),Delta)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aTP_Lamm_yw(X),A),I6)),B2))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7538_convex__sum__bound__le,axiom,
    ! [I6: set_set_nat,X: fun_set_nat_int,A: fun_set_nat_int,B2: int,Delta: int] :
      ( ! [I4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(I4),I6))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_set_nat_int(X,I4))) )
     => ( ( groups8292507037921071086at_int(X,I6) = one_one_int )
       => ( ! [I4: set_nat] :
              ( pp(aa_set_set_nat_bool(member_set_nat(I4),I6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(A,I4)),B2))),Delta)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups8292507037921071086at_int(aa_fun3600621996379952540at_int(aTP_Lamm_yx(X),A),I6)),B2))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7539_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: fun_nat_int,A: fun_nat_int,B2: int,Delta: int] :
      ( ! [I4: nat] :
          ( pp(aa_set_nat_bool(member_nat(I4),I6))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(X,I4))) )
     => ( ( aa_set_nat_int(groups3539618377306564664at_int(X),I6) = one_one_int )
       => ( ! [I4: nat] :
              ( pp(aa_set_nat_bool(member_nat(I4),I6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(A,I4)),B2))),Delta)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_yy(X),A)),I6)),B2))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7540_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: fun_int_int,A: fun_int_int,B2: int,Delta: int] :
      ( ! [I4: int] :
          ( pp(aa_set_int_bool(member_int(I4),I6))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(X,I4))) )
     => ( ( groups4538972089207619220nt_int(X,I6) = one_one_int )
       => ( ! [I4: int] :
              ( pp(aa_set_int_bool(member_int(I4),I6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(A,I4)),B2))),Delta)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_yz(X),A),I6)),B2))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7541_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: fun_nat_real,A: fun_nat_real,B2: real,Delta: real] :
      ( ! [I4: nat] :
          ( pp(aa_set_nat_bool(member_nat(I4),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(X,I4))) )
     => ( ( aa_set_nat_real(groups6591440286371151544t_real(X),I6) = one_one_real )
       => ( ! [I4: nat] :
              ( pp(aa_set_nat_bool(member_nat(I4),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(A,I4)),B2))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_za(X),A)),I6)),B2))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7542_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => topolo6980174941875973593q_real(aTP_Lamm_zb(X)) ) ).

% monoseq_arctan_series
tff(fact_7543_arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(arctan,X) = suminf_real(aTP_Lamm_yj(X)) ) ) ).

% arctan_series
tff(fact_7544_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_zc(D))) ).

% int_ge_less_than_def
tff(fact_7545_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T6)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(exp_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zd(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_7546_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ).

% xor_Suc_0_eq
tff(fact_7547_zdvd1__eq,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),one_one_int))
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_7548_zabs__less__one__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,Z2)),one_one_int))
    <=> ( Z2 = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_7549_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,bit0(X)) ).

% xor_nat_numerals(4)
tff(fact_7550_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,bit0(X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_7551_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,bit0(Y)) ).

% xor_nat_numerals(2)
tff(fact_7552_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,bit0(Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_7553_zdvd__antisym__abs,axiom,
    ! [A: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A))
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B2) ) ) ) ).

% zdvd_antisym_abs
tff(fact_7554_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_7555_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ! [M2: int] :
        ? [N4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M2),aa_int_int(abs_abs_int,N4)))
          & pp(aa_set_int_bool(member_int(N4),S2)) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_7556_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ! [M2: int] :
        ? [N4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M2),aa_int_int(abs_abs_int,N4)))
          & pp(aa_set_int_bool(member_int(N4),S2)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_7557_zabs__def,axiom,
    ! [I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = aa_int_int(uminus_uminus_int,I) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = I ) ) ) ).

% zabs_def
tff(fact_7558_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),I))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I))) ) ) ).

% dvd_imp_le_int
tff(fact_7559_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)),M))
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_7560_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: fun_nat_int,K: int] :
      ( ! [I4: nat] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I4))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I4))),aa_nat_int(F,I4)))),one_one_int)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,M)),K))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
           => ? [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I4))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
                & ( aa_nat_int(F,I4) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_7561_nat__ivt__aux,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I4))),aa_nat_int(F,I4)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
         => ? [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
              & ( aa_nat_int(F,I4) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_7562_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_7563_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I4),one_one_nat))),aa_nat_int(F,I4)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
         => ? [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
              & ( aa_nat_int(F,I4) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_7564_xor__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% xor_nat_rec
tff(fact_7565_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,X))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_7566_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(minus_minus_real(one_one_real),aa_real_real(divide_divide_real(X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_7567_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ).

% Suc_0_xor_eq
tff(fact_7568_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(K),L) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_7569_nat__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,N)) = N ).

% nat_int
tff(fact_7570_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,numeral_numeral_int(K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_numeral
tff(fact_7571_nat__of__bool,axiom,
    ! [P: bool] : aa_int_nat(nat2,zero_n2684676970156552555ol_int(P)) = zero_n2687167440665602831ol_nat(P) ).

% nat_of_bool
tff(fact_7572_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_7573_nat__le__0,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),zero_zero_int))
     => ( aa_int_nat(nat2,Z2) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_7574_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int)) ) ).

% nat_0_iff
tff(fact_7575_zless__nat__conj,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z2)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ) ).

% zless_nat_conj
tff(fact_7576_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,numeral_numeral_int(K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_7577_nat__zminus__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_7578_int__nat__eq,axiom,
    ! [Z2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z2)) = Z2 ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z2)) = zero_zero_int ) ) ) ).

% int_nat_eq
tff(fact_7579_zero__less__nat__eq,axiom,
    ! [Z2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2)) ) ).

% zero_less_nat_eq
tff(fact_7580_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(V)),numeral_numeral_int(V3))) ).

% diff_nat_numeral
tff(fact_7581_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(numeral_numeral_int(X)),N) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_7582_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(numeral_numeral_int(X)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_7583_dvd__nat__abs__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ).

% dvd_nat_abs_iff
tff(fact_7584_nat__abs__dvd__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_abs_dvd_iff
tff(fact_7585_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,A))) ) ).

% nat_ceiling_le_eq
tff(fact_7586_one__less__nat__eq,axiom,
    ! [Z2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z2)) ) ).

% one_less_nat_eq
tff(fact_7587_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_7588_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(numeral_numeral_int(X)),N)),A)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_7589_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(numeral_numeral_int(X)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_7590_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(numeral_numeral_int(X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_7591_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(X)),N)),A)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_7592_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_7593_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y))) ) ).

% nat_mono
tff(fact_7594_ex__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ? [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
          & pp(aa_nat_bool(P,aa_int_nat(nat2,X5))) ) ) ).

% ex_nat
tff(fact_7595_all__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ! [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ! [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
         => pp(aa_nat_bool(P,aa_int_nat(nat2,X5))) ) ) ).

% all_nat
tff(fact_7596_eq__nat__nat__iff,axiom,
    ! [Z2: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
       => ( ( aa_int_nat(nat2,Z2) = aa_int_nat(nat2,Z5) )
        <=> ( Z2 = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_7597_nat__mono__iff,axiom,
    ! [Z2: int,W2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ) ).

% nat_mono_iff
tff(fact_7598_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_int_nat(nat2,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z2)) ) ).

% zless_nat_eq_int_zless
tff(fact_7599_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_le_iff
tff(fact_7600_int__eq__iff,axiom,
    ! [M: nat,Z2: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z2 )
    <=> ( ( M = aa_int_nat(nat2,Z2) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2)) ) ) ).

% int_eq_iff
tff(fact_7601_nat__0__le,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z2)) = Z2 ) ) ).

% nat_0_le
tff(fact_7602_nat__int__add,axiom,
    ! [A: nat,B2: 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,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) ).

% nat_int_add
tff(fact_7603_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z2: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z2))) ).

% nat_abs_mult_distrib
tff(fact_7604_nat__plus__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_7605_nat__less__eq__zless,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ) ).

% nat_less_eq_zless
tff(fact_7606_nat__le__eq__zle,axiom,
    ! [W2: int,Z2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),W2))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z2)) ) ) ).

% nat_le_eq_zle
tff(fact_7607_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa_int_nat(nat2,W2) = M )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
tff(fact_7608_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa_int_nat(nat2,W2) )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
tff(fact_7609_split__nat,axiom,
    ! [P: fun_nat_bool,I: int] :
      ( pp(aa_nat_bool(P,aa_int_nat(nat2,I)))
    <=> ( ! [N4: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N4) )
           => pp(aa_nat_bool(P,N4)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
         => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_nat
tff(fact_7610_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ) ).

% le_nat_iff
tff(fact_7611_nat__add__distrib,axiom,
    ! [Z2: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_7612_nat__mult__distrib,axiom,
    ! [Z2: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z2),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_7613_Suc__as__int,axiom,
    ! [X2: nat] : aa_nat_nat(suc,X2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X2)),one_one_int)) ).

% Suc_as_int
tff(fact_7614_forall__pos__mono__1,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D4: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D4),E))
         => ( pp(aa_real_bool(P,D4))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N2: nat] : pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono_1
tff(fact_7615_real__arch__inverse,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
    <=> ? [N4: nat] :
          ( ( N4 != zero_zero_nat )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4))),E2)) ) ) ).

% real_arch_inverse
tff(fact_7616_forall__pos__mono,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D4: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D4),E))
         => ( pp(aa_real_bool(P,D4))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2)))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono
tff(fact_7617_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_7618_nat__diff__distrib,axiom,
    ! [Z5: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z5),Z2))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z2),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_7619_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ).

% nat_abs_triangle_ineq
tff(fact_7620_nat__floor__neg,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_7621_nat__power__eq,axiom,
    ! [Z2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z2)),N) ) ) ).

% nat_power_eq
tff(fact_7622_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq3
tff(fact_7623_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A)))) ) ).

% le_nat_floor
tff(fact_7624_nat__2,axiom,
    aa_int_nat(nat2,numeral_numeral_int(bit0(one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_7625_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z2)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z2)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_7626_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),M))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,M))) ) ) ).

% nat_less_iff
tff(fact_7627_nat__mult__distrib__neg,axiom,
    ! [Z2: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),zero_zero_int))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z2),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z2))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_7628_nat__abs__int__diff,axiom,
    ! [A: nat,B2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
       => ( 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,B2)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
       => ( 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,B2)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2) ) ) ) ).

% nat_abs_int_diff
tff(fact_7629_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq4
tff(fact_7630_diff__nat__eq__if,axiom,
    ! [Z5: int,Z2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z5)) = aa_int_nat(nat2,Z2) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z2),Z5)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z2),Z5))) ) ) ) ).

% diff_nat_eq_if
tff(fact_7631_nat__dvd__iff,axiom,
    ! [Z2: int,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,Z2)),M))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z2),aa_nat_int(semiri1314217659103216013at_int,M))) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
tff(fact_7632_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_7633_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cd),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% Sum_Ico_nat
tff(fact_7634_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_7635_Cauchy__iff2,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [J2: nat] :
        ? [M9: nat] :
        ! [M2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),M2))
         => ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),N4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(X6,M2)),aa_nat_real(X6,N4)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J2))))) ) ) ) ).

% Cauchy_iff2
tff(fact_7636_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),U))) ).

% finite_atLeastLessThan
tff(fact_7637_atLeastLessThan__singleton,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_7638_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_7639_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(4)
tff(fact_7640_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,bit0(X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(3)
tff(fact_7641_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,bit0(Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_7642_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
          & pp(aa_nat_bool(P,M2)) )
    <=> ? [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
          & pp(aa_nat_bool(P,X5)) ) ) ).

% ex_nat_less_eq
tff(fact_7643_all__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
         => pp(aa_nat_bool(P,M2)) )
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
         => pp(aa_nat_bool(P,X5)) ) ) ).

% all_nat_less_eq
tff(fact_7644_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(L),aa_nat_nat(suc,U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_7645_lessThan__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ).

% lessThan_atLeast0
tff(fact_7646_atLeastLessThan0,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_7647_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_7648_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N5)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_7649_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
tff(fact_7650_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_Suc_fact
tff(fact_7651_prod__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_fact
tff(fact_7652_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(pred_numeral,K))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_7653_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_7654_Suc__0__or__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% Suc_0_or_eq
tff(fact_7655_or__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% or_Suc_0_eq
tff(fact_7656_or__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(fdisj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% or_nat_rec
tff(fact_7657_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K)),one_one_nat) ).

% sum_power2
tff(fact_7658_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: fun_nat_nat,B2: fun_nat_nat] :
      ( ! [I4: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(A,I4)),aa_nat_nat(A,J3))) ) )
     => ( ! [I4: nat,J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),J3))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(B2,J3)),aa_nat_nat(B2,I4))) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ze(A),B2)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_set_nat_nat(groups3542108847815614940at_nat(B2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_7659_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_7660_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or4662586982721622107an_int(L,U))) ).

% finite_atLeastLessThan_int
tff(fact_7661_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : pp(aa_set_int_bool(finite_finite_int,set_or4662586982721622107an_int(zero_zero_int,U))) ).

% finite_atLeastZeroLessThan_int
tff(fact_7662_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_7663_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf(X21,X222)) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_7664_card__lessThan,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_ord_lessThan_nat,U)) = U ).

% card_lessThan
tff(fact_7665_card__Collect__less__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,N))) = N ).

% card_Collect_less_nat
tff(fact_7666_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_7667_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_atLeastLessThan
tff(fact_7668_card__Collect__le__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ay,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_7669_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_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_7670_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_7671_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_7672_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) ).

% push_bit_of_Suc_0
tff(fact_7673_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_card_int(set_or4662586982721622107an_int(zero_zero_int,U)) = aa_int_nat(nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_7674_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_bg),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_7675_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_zf),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_7676_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),K)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_7677_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M7))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zg(M7),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zh(M7),I))) ) ) ).

% card_less_Suc2
tff(fact_7678_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M7))
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zg(M7),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zh(M7),I))) ) ) ).

% card_less_Suc
tff(fact_7679_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M7))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zh(M7),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_7680_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q3)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_7681_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2)))))
     => ( A2 = aa_nat_set_nat(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_7682_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aa_nat_fun_nat_bool(ord_less_eq_nat,M)),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_7683_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_zi,N)),M) ).

% max_Suc2
tff(fact_7684_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_zj,N)),M) ).

% max_Suc1
tff(fact_7685_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N5)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_7686_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cd),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_set_nat_nat(finite_card_nat,S2)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cd),S2))) ).

% card_sum_le_nat_sum
tff(fact_7687_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_zk(C),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_7688_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_cw,N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_7689_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_cd),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ).

% diff_Suc
tff(fact_7690_binomial__def,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = finite_card_set_nat(collect_set_nat(aa_nat8857836526276733660t_bool(aTP_Lamm_zl(N),K))) ).

% binomial_def
tff(fact_7691_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_cd),Nat) ).

% pred_def
tff(fact_7692_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ).

% bezw_0
tff(fact_7693_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,K))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_7694_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa2) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
         => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
         => ( Y = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_7695_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ? [R3: fun_nat_nat] :
          ( pp(aa_set_nat_bool(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S2))))
          & ! [N6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N6),aa_set_nat_nat(finite_card_nat,S2)))
             => pp(aa_set_nat_bool(member_nat(aa_nat_nat(R3,N6)),S2)) ) ) ) ).

% finite_enumerate
tff(fact_7696_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(powr_real(X),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% root_powr_inverse
tff(fact_7697_real__root__Suc__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),X) = X ).

% real_root_Suc_0
tff(fact_7698_root__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),X) = zero_zero_real ).

% root_0
tff(fact_7699_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(aa_nat_fun_real_real(root,N),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_7700_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% drop_bit_of_Suc_0
tff(fact_7701_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_int_int(uminus_uminus_int,numeral_numeral_int(K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_7702_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_7703_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% real_root_less_iff
tff(fact_7704_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% real_root_le_iff
tff(fact_7705_real__root__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_7706_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = one_one_real )
      <=> ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_7707_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ) ).

% real_root_gt_0_iff
tff(fact_7708_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ) ).

% real_root_lt_0_iff
tff(fact_7709_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ) ).

% real_root_ge_0_iff
tff(fact_7710_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ) ).

% real_root_le_0_iff
tff(fact_7711_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ) ).

% real_root_gt_1_iff
tff(fact_7712_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% real_root_lt_1_iff
tff(fact_7713_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ) ).

% real_root_ge_1_iff
tff(fact_7714_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% real_root_le_1_iff
tff(fact_7715_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,numeral_numeral_int(aa_num_num(bit1,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_7716_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_7717_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_less_mono
tff(fact_7718_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_le_mono
tff(fact_7719_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(power_power_real(X),K)) = aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),K) ) ) ).

% real_root_power
tff(fact_7720_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(abs_abs_real,X)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ) ) ).

% real_root_abs
tff(fact_7721_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) = aa_real_real(sgn_sgn_real,X) ) ) ).

% sgn_root
tff(fact_7722_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).

% real_root_gt_zero
tff(fact_7723_real__root__strict__decreasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N5),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_7724_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(power_power_real(Y),N))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_7725_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).

% real_root_pos_pos
tff(fact_7726_real__root__strict__increasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N5),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_7727_real__root__decreasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N5),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_7728_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos
tff(fact_7729_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(power_power_real(Y),N) = X )
         => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_7730_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(power_power_real(X),N)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_7731_real__root__increasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N5),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_7732_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_7733_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),N)) = X ) ) ).

% sgn_power_root
tff(fact_7734_ln__root,axiom,
    ! [N: nat,B2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,N),B2)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B2)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% ln_root
tff(fact_7735_log__root,axiom,
    ! [N: nat,A: real,B2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( aa_real_real(log(B2),aa_real_real(aa_nat_fun_real_real(root,N),A)) = aa_real_real(divide_divide_real(aa_real_real(log(B2),A)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% log_root
tff(fact_7736_log__base__root,axiom,
    ! [N: nat,B2: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
       => ( aa_real_real(log(aa_real_real(aa_nat_fun_real_real(root,N),B2)),X) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(B2),X)) ) ) ) ).

% log_base_root
tff(fact_7737_split__root,axiom,
    ! [P: fun_real_bool,N: nat,X: real] :
      ( pp(aa_real_bool(P,aa_real_real(aa_nat_fun_real_real(root,N),X)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_real_bool(P,zero_zero_real)) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ! [Y5: real] :
              ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y5)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Y5)),N)) = X )
             => pp(aa_real_bool(P,Y5)) ) ) ) ) ).

% split_root
tff(fact_7738_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_7739_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(complete_Inf_Inf_nat,K5)),K5)) ) ).

% Inf_nat_def1
tff(fact_7740_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_7741_Suc__funpow,axiom,
    ! [N: nat] : compow_fun_nat_nat(N,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N) ).

% Suc_funpow
tff(fact_7742_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or5832277885323065728an_int(L,U))) ).

% finite_greaterThanLessThan_int
tff(fact_7743_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_7744_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U))) ).

% finite_greaterThanLessThan
tff(fact_7745_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_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_7746_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or5834768355832116004an_nat(L),U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_7747_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_ay,aTP_Lamm_ax) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_7748_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_zn),Xa2),X)) ).

% times_int.abs_eq
tff(fact_7749_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M7))
     => ( aa_set_nat_nat(gcd_Gcd_nat,M7) = aa_set_nat_nat(gcd_Gcd_nat,minus_minus_set_nat(M7,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_7750_int_Oabs__induct,axiom,
    ! [P: fun_int_bool,X: int] :
      ( ! [Y3: product_prod_nat_nat] : pp(aa_int_bool(P,aa_Pro4240683299535347697at_int(abs_Integ,Y3)))
     => pp(aa_int_bool(P,X)) ) ).

% int.abs_induct
tff(fact_7751_eq__Abs__Integ,axiom,
    ! [Z2: int] :
      ~ ! [X4: nat,Y3: nat] : Z2 != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3)) ).

% eq_Abs_Integ
tff(fact_7752_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),X) ).

% nat.abs_eq
tff(fact_7753_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_7754_int__def,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),zero_zero_nat)) ).

% int_def
tff(fact_7755_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_zo),X)) ).

% uminus_int.abs_eq
tff(fact_7756_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_7757_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_zq),Xa2),X)) ) ).

% less_int.abs_eq
tff(fact_7758_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_zs),Xa2),X)) ) ).

% less_eq_int.abs_eq
tff(fact_7759_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_zu),Xa2),X)) ).

% plus_int.abs_eq
tff(fact_7760_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_zw),Xa2),X)) ).

% minus_int.abs_eq
tff(fact_7761_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( num_of_nat(aa_nat_nat(suc,N)) = inc(num_of_nat(N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( num_of_nat(aa_nat_nat(suc,N)) = one ) ) ) ).

% num_of_nat.simps(2)
tff(fact_7762_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_7763_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_num_nat(numeral_numeral_nat,num_of_nat(N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_7764_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),one_one_nat))
     => ( num_of_nat(N) = one ) ) ).

% num_of_nat_One
tff(fact_7765_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( num_of_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)) = bit0(num_of_nat(N)) ) ) ).

% num_of_nat_double
tff(fact_7766_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( num_of_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = plus_plus_num(num_of_nat(M),num_of_nat(N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_7767_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I))))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_7768_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_zs),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).

% less_eq_int.rep_eq
tff(fact_7769_less__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_zq),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).

% less_int.rep_eq
tff(fact_7770_nat_Orep__eq,axiom,
    ! [X: int] : aa_int_nat(nat2,X) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,X)) ).

% nat.rep_eq
tff(fact_7771_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_7772_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_zo)) ).

% uminus_int_def
tff(fact_7773_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_zx) ).

% prod_encode_def
tff(fact_7774_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or6659071591806873216st_nat(L),U))) ).

% finite_greaterThanAtMost
tff(fact_7775_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_greaterThanAtMost
tff(fact_7776_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or6659071591806873216st_nat(L),U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_7777_le__prod__encode__1,axiom,
    ! [A: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B2)))) ).

% le_prod_encode_1
tff(fact_7778_le__prod__encode__2,axiom,
    ! [B2: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B2)))) ).

% le_prod_encode_2
tff(fact_7779_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),M) ).

% prod_encode_prod_decode_aux
tff(fact_7780_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_zn)) ).

% times_int_def
tff(fact_7781_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_zw)) ).

% minus_int_def
tff(fact_7782_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_zu)) ).

% plus_int_def
tff(fact_7783_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zy,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zy,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = insert_nat(zero_zero_nat,bot_bot_set_nat) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zy,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_7784_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or6656581121297822940st_int(L,U))) ).

% finite_greaterThanAtMost_int
tff(fact_7785_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_7786_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(I),J)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_7787_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_7788_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),image_nat_nat(suc,A2))) ).

% zero_notin_Suc_image
tff(fact_7789_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_7790_image__Suc__lessThan,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N) ).

% image_Suc_lessThan
tff(fact_7791_image__Suc__atMost,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,N)) ).

% image_Suc_atMost
tff(fact_7792_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_7793_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_7794_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_7795_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_7796_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_7797_less__int__def,axiom,
    ord_less_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_zq)) ).

% less_int_def
tff(fact_7798_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_zs)) ).

% less_eq_int_def
tff(fact_7799_finite__int__iff__bounded__le,axiom,
    ! [S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ? [K2: int] : pp(aa_set_int_bool(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_atMost_int(K2))) ) ).

% finite_int_iff_bounded_le
tff(fact_7800_finite__int__iff__bounded,axiom,
    ! [S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ? [K2: int] : pp(aa_set_int_bool(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_lessThan_int(K2))) ) ).

% finite_int_iff_bounded
tff(fact_7801_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_7802_image__int__atLeastAtMost,axiom,
    ! [A: nat,B2: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% image_int_atLeastAtMost
tff(fact_7803_image__int__atLeastLessThan,axiom,
    ! [A: nat,B2: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B2)) = set_or4662586982721622107an_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% image_int_atLeastLessThan
tff(fact_7804_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_zz(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_7805_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_7806_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_7807_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( image_real_real(times_times_real(A),top_top_set_real) = insert_real(zero_zero_real,bot_bot_set_real) ) )
      & ( ( A != zero_zero_real )
       => ( image_real_real(times_times_real(A),top_top_set_real) = top_top_set_real ) ) ) ).

% range_mult
tff(fact_7808_nat__not__finite,axiom,
    ~ pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ).

% nat_not_finite
tff(fact_7809_infinite__UNIV__nat,axiom,
    ~ pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ).

% infinite_UNIV_nat
tff(fact_7810_infinite__UNIV__int,axiom,
    ~ pp(aa_set_int_bool(finite_finite_int,top_top_set_int)) ).

% infinite_UNIV_int
tff(fact_7811_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_7812_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_7813_int__in__range__abs,axiom,
    ! [N: nat] : pp(aa_set_int_bool(member_int(aa_nat_int(semiri1314217659103216013at_int,N)),image_int_int(abs_abs_int,top_top_set_int))) ).

% int_in_range_abs
tff(fact_7814_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_7815_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aaa,N),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ) ) ).

% range_mod
tff(fact_7816_root__def,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_aab,N)),X) ) ) ) ).

% root_def
tff(fact_7817_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D5: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( X != zero_zero_real )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
               => ( D5 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
               => ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),D5,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_7818_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_7819_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R)))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_aac(F),X4)) )
     => ( pp(aa_set_real_bool(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
         => has_fi5821293074295781190e_real(aTP_Lamm_aae(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_aac(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_7820_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_7821_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aaf(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_7822_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M3: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X4),topolo2177554685111907308n_real(X4,top_top_set_real)) )
     => ? [T6: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
          & ( aa_real_real(F,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aaf(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_7823_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( ( X != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_7824_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),zero_zero_real))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M3: nat,T6: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),zero_zero_real)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
           => ? [T6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),zero_zero_real))
                & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_xo(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_7825_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T6: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),H)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ? [T6: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),H))
              & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_xo(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_7826_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M3: nat,T6: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),H)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
           => ? [T6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),H))
                & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_xo(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_7827_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( X != zero_zero_real )
         => ( ! [M3: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
           => ? [T6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T6)))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
                & ( aa_real_real(F,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aaf(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_7828_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,T6: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X))) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aaf(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_7829_Taylor,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B2: real,C: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T6: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),B2)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B2))
                 => ( ( X != C )
                   => ? [T6: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T6))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),C)) ) )
                        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T6))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X)) ) )
                        & ( aa_real_real(F,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aag(Diff),C),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(X),C)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_7830_Taylor__up,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B2: real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T6: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),B2)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B2))
             => ? [T6: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),B2))
                  & ( aa_real_real(F,B2) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aah(Diff),B2),C)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(B2),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_7831_Taylor__down,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B2: real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T6: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),B2)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B2))
             => ? [T6: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),C))
                  & ( aa_real_real(F,A) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aah(Diff),A),C)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(A),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_7832_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B: real] :
      ( ! [M3: nat,T6: real] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),H)) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
     => ( ( N = aa_nat_nat(suc,K) )
       => ! [M5: nat,T7: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T7))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),H)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_aaj(N),Diff),B),M5),aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T7)),aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_aak(Diff),M5),T7)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M5))))),aa_real_real(times_times_real(B),aa_real_real(divide_divide_real(aa_nat_real(power_power_real(T7),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M5)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M5))))))),topolo2177554685111907308n_real(T7,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_7833_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => has_fi5821293074295781190e_real(aTP_Lamm_aal,suminf_real(aTP_Lamm_aam(X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_7834_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_aan,N),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S)) ).

% DERIV_pow
tff(fact_7835_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M7))
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M7))
         => ( aa_set_nat_nat(gcd_Gcd_nat,M7) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_aao,M7))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_7836_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ba,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_7837_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S2)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S2)))) ) ).

% card_le_Suc_Max
tff(fact_7838_Sup__nat__def,axiom,
    ! [X6: set_nat] :
      ( ( ( X6 = bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = zero_zero_nat ) )
      & ( ( X6 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = aa_set_nat_nat(lattic8265883725875713057ax_nat,X6) ) ) ) ).

% Sup_nat_def
tff(fact_7839_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aap(N),M))) ) ) ) ).

% divide_nat_def
tff(fact_7840_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real))
         => ! [N6: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_aaq(A))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aaq(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N6)),one_one_nat))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aaq(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N6)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_7841_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat)))
         => ! [N6: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_aaq(A))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aaq(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N6))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aaq(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N6)),one_one_nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_7842_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(aTP_Lamm_aaq(A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aaq(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),one_one_nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_7843_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => filterlim_nat_real(aTP_Lamm_aar(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_aaq(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_7844_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_7845_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_7846_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aas,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_7847_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_7848_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N2))),aa_nat_real(G,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2)))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_aat(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N6)),L3))
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),aa_nat_real(G,N6)))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_7849_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun_nat_real] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R3),aa_nat_real(X6,N2))) )
     => filterlim_nat_real(aTP_Lamm_aau(X6),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_7850_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_aav,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_7851_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_aaw(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_7852_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),L))
       => ( ! [E: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
             => ? [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),aa_real_real(plus_plus_real(aa_nat_real(F,N6)),E))) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_7853_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_aax(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_7854_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_aay(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_7855_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => summable_real(aTP_Lamm_aaq(A)) ) ) ) ).

% summable
tff(fact_7856_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => filterlim_nat_real(aTP_Lamm_zb(X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_7857_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aaq(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),suminf_real(aTP_Lamm_aaq(A)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_7858_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => filterlim_nat_real(aTP_Lamm_aaz(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_aaq(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_7859_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L3: real] :
              ( ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aaq(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N6)))),L3))
              & filterlim_nat_real(aTP_Lamm_aaz(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aaq(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N6)),one_one_nat)))))
              & filterlim_nat_real(aTP_Lamm_aar(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_7860_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_aar(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_aaq(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_7861_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(aTP_Lamm_aba(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_7862_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_abb(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_7863_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X4))
         => pp(aa_nat_bool(P,X4)) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_7864_eventually__sequentially,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N9: nat] :
        ! [N4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N4))
         => pp(aa_nat_bool(P,N4)) ) ) ).

% eventually_sequentially
tff(fact_7865_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N9: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),F3) ) ).

% le_sequentially
tff(fact_7866_sequentially__offset,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_abb(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_7867_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: fun_real_real,F3: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abc(N),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_7868_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: fun_real_real,F3: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abc(N),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_7869_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_7870_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ~ pp(aa_set_nat_bool(finite_finite_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),S2))) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_7871_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_7872_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N)) ) ).

% mono_times_nat
tff(fact_7873_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_7874_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_7875_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abd,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_7876_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_7877_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_7878_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_7879_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_7880_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_7881_range__abs__Nats,axiom,
    image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_7882_inj__sgn__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_real_bool(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_aab,N)),top_top_set_real)) ) ).

% inj_sgn_power
tff(fact_7883_measure__function__int,axiom,
    fun_is_measure_int(comp_int_nat_int(nat2,abs_abs_int)) ).

% measure_function_int
tff(fact_7884_inj__Suc,axiom,
    ! [N5: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N5)) ).

% inj_Suc
tff(fact_7885_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( pp(aa_set_nat_bool(member_nat(N2),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2)) )
     => pp(aa_set_nat_bool(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zy,K)),N5)) ) ).

% inj_on_diff_nat
tff(fact_7886_inj__on__set__encode,axiom,
    pp(aa_set_set_nat_bool(inj_on_set_nat_nat(nat_set_encode),collect_set_nat(finite_finite_nat))) ).

% inj_on_set_encode
tff(fact_7887_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_abe)) ).

% pred_nat_def
tff(fact_7888_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( ( Deg2 = Xa2 )
                & ! [X4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                   => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
                & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))))
                & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
                & case_o7912701867834613557at_nat(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abf(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_7889_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X2: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(vEBT_VEBT_valid(X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
                  & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))))
                  & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
                  & case_o7912701867834613557at_nat(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abf(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_7890_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( pp(Y)
          <=> ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( pp(Y)
              <=> ~ ( ( Deg2 = Xa2 )
                    & ! [X5: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X5),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))))
                    & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
                    & case_o7912701867834613557at_nat(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abf(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_7891_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList,Summary)),Deg3))
    <=> ( ( Deg = Deg3 )
        & ! [X5: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
           => pp(aa_nat_bool(vEBT_VEBT_valid(X5),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
        & pp(aa_nat_bool(vEBT_VEBT_valid(Summary),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))
        & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
        & case_o7912701867834613557at_nat(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abf(Deg),TreeList),Summary)),Mima2) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_7892_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( pp(Y)
                <=> ( Xa2 = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( ( pp(Y)
                  <=> ( ( Deg2 = Xa2 )
                      & ! [X5: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X5),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))))
                      & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
                      & case_o7912701867834613557at_nat(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abf(Deg2),TreeList2),Summary2)),Mima) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa2)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_7893_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2))
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa2))
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X2: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))))
                      & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
                      & case_o7912701867834613557at_nat(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abf(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_7894_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2))
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa2))
                 => ( ( Deg2 = Xa2 )
                    & ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))))
                    & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
                    & case_o7912701867834613557at_nat(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abf(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_7895_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),J),aa_nat_set_nat(set_or4665077453230672383an_nat(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_7896_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_7897_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_7898_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_abg) ).

% Rats_eq_int_div_nat
tff(fact_7899_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N),transi6264000038957366511cl_nat(pred_nat))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% less_eq
tff(fact_7900_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N),transi2905341329935302413cl_nat(pred_nat))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% pred_nat_trancl_eq_le
tff(fact_7901_bdd__above__nat,axiom,
    ! [X6: set_nat] :
      ( pp(aa_set_nat_bool(condit2214826472909112428ve_nat,X6))
    <=> pp(aa_set_nat_bool(finite_finite_nat,X6)) ) ).

% bdd_above_nat
tff(fact_7902_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero_nat,M) = none_num ).

% take_bit_num_simps(1)
tff(fact_7903_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : bit_take_bit_num(aa_nat_nat(suc,N),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_7904_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),bit0(M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_abh,bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_7905_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(N,M))) ).

% take_bit_num_simps(4)
tff(fact_7906_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat )
       => ( bit_take_bit_num(N,M) = none_num ) )
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) != zero_zero_nat )
       => ( bit_take_bit_num(N,M) = some_num(num_of_nat(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)))) ) ) ) ).

% take_bit_num_def
tff(fact_7907_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)) ).

% min_Suc_Suc
tff(fact_7908_min__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).

% min_0L
tff(fact_7909_min__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_7910_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),N)) ).

% min_numeral_Suc
tff(fact_7911_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_7912_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).

% min_diff
tff(fact_7913_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ).

% nat_mult_min_left
tff(fact_7914_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ).

% nat_mult_min_right
tff(fact_7915_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_abi,N)),M) ).

% min_Suc2
tff(fact_7916_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_abj,N)),M) ).

% min_Suc1
tff(fact_7917_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_7918_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
tff(fact_7919_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_complex_bool(bij_betw_nat_complex(aTP_Lamm_abk(N),aa_nat_set_nat(set_ord_lessThan_nat,N)),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_cw,N)))) ) ).

% bij_betw_roots_unity
tff(fact_7920_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_7921_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_7922_bij__betw__Suc,axiom,
    ! [M7: set_nat,N5: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(bij_betw_nat_nat(suc),M7),N5))
    <=> ( image_nat_nat(suc,M7) = N5 ) ) ).

% bij_betw_Suc
tff(fact_7923_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_set_complex_bool(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,N),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(divide_divide_real(aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,N))))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_cw,N))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_zk(C),N)))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_7924_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_7925_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_7926_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X4: nat,Xs2: list_nat] :
              ( ( X = cons_nat(X4,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_7927_list__encode_Osimps_I1_J,axiom,
    nat_list_encode(nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_7928_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] : nat_list_encode(cons_nat(X,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),nat_list_encode(Xs)))) ).

% list_encode.simps(2)
tff(fact_7929_upto__aux__rec,axiom,
    ! [J: int,I: int,Js: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = Js ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js)) ) ) ) ).

% upto_aux_rec
tff(fact_7930_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I,J))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = nil_int ) ) ) ) ).

% upto.psimps
tff(fact_7931_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(X,Xa2))
       => ~ ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
               => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa2)) ) )
              & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
               => ( Y = nil_int ) ) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(X,Xa2)) ) ) ) ).

% upto.pelims
tff(fact_7932_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil
tff(fact_7933_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil2
tff(fact_7934_upto__empty,axiom,
    ! [J: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_7935_upto__single,axiom,
    ! [I: int] : upto(I,I) = cons_int(I,nil_int) ).

% upto_single
tff(fact_7936_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J))
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_7937_length__upto,axiom,
    ! [I: int,J: int] : aa_list_int_nat(size_size_list_int,upto(I,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I)),one_one_int)) ).

% length_upto
tff(fact_7938_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(M)),numeral_numeral_int(N)))
       => ( upto(numeral_numeral_int(M),numeral_numeral_int(N)) = cons_int(numeral_numeral_int(M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(M)),one_one_int),numeral_numeral_int(N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(M)),numeral_numeral_int(N)))
       => ( upto(numeral_numeral_int(M),numeral_numeral_int(N)) = nil_int ) ) ) ).

% upto_rec_numeral(1)
tff(fact_7939_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(M)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))))
       => ( upto(numeral_numeral_int(M),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = cons_int(numeral_numeral_int(M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(M)),one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(M)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))))
       => ( upto(numeral_numeral_int(M),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = nil_int ) ) ) ).

% upto_rec_numeral(2)
tff(fact_7940_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),numeral_numeral_int(N)))
       => ( upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(M)),numeral_numeral_int(N)) = cons_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),one_one_int),numeral_numeral_int(N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),numeral_numeral_int(N)))
       => ( upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(M)),numeral_numeral_int(N)) = nil_int ) ) ) ).

% upto_rec_numeral(3)
tff(fact_7941_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))))
       => ( upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(M)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = cons_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M))),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))))
       => ( upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(M)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = nil_int ) ) ) ).

% upto_rec_numeral(4)
tff(fact_7942_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil_int) ).

% upto_code
tff(fact_7943_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list_int] : upto_aux(I,J,Js) = append_int(upto(I,J),Js) ).

% upto_aux_def
tff(fact_7944_distinct__upto,axiom,
    ! [I: int,J: int] : pp(aa_list_int_bool(distinct_int,upto(I,J))) ).

% distinct_upto
tff(fact_7945_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1266510415728281911st_int(I,J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_7946_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_7947_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_7948_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_7949_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or6656581121297822940st_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_7950_upto_Osimps,axiom,
    ! [I: int,J: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = nil_int ) ) ) ).

% upto.simps
tff(fact_7951_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
         => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa2)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
tff(fact_7952_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_7953_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_7954_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_7955_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_7956_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),X))
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),nil_nat)) ) )
         => ~ ! [X4: nat,Xs2: list_nat] :
                ( ( X = cons_nat(X4,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),nat_list_encode(Xs2)))) )
                 => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),cons_nat(X4,Xs2))) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_7957_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = cons_nat(aa_num_nat(numeral_numeral_nat,M),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = nil_nat ) ) ) ).

% upt_rec_numeral
tff(fact_7958_remdups__upt,axiom,
    ! [M: nat,N: nat] : remdups_nat(upt(M,N)) = upt(M,N) ).

% remdups_upt
tff(fact_7959_tl__upt,axiom,
    ! [M: nat,N: nat] : tl_nat(upt(M,N)) = upt(aa_nat_nat(suc,M),N) ).

% tl_upt
tff(fact_7960_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_7961_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] : drop_nat(M,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M),J) ).

% drop_upt
tff(fact_7962_length__upt,axiom,
    ! [I: nat,J: nat] : aa_list_nat_nat(size_size_list_nat,upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_7963_take__upt,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)),N))
     => ( take_nat(M,upt(I,N)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).

% take_upt
tff(fact_7964_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_7965_sorted__list__of__set__range,axiom,
    ! [M: nat,N: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = upt(M,N) ).

% sorted_list_of_set_range
tff(fact_7966_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_7967_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J))
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_7968_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_7969_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q3: nat] :
      ( ( cons_nat(M,cons_nat(N,Ns)) = upt(M,Q3) )
    <=> ( cons_nat(N,Ns) = upt(aa_nat_nat(suc,M),Q3) ) ) ).

% upt_conv_Cons_Cons
tff(fact_7970_greaterThanAtMost__upt,axiom,
    ! [N: nat,M: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(N),M) = set_nat2(upt(aa_nat_nat(suc,N),aa_nat_nat(suc,M))) ).

% greaterThanAtMost_upt
tff(fact_7971_distinct__upt,axiom,
    ! [I: nat,J: nat] : pp(aa_list_nat_bool(distinct_nat,upt(I,J))) ).

% distinct_upt
tff(fact_7972_atLeastAtMost__upt,axiom,
    ! [N: nat,M: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(N),M) = set_nat2(upt(N,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_7973_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(I),J) = set_nat2(upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_7974_atLeast__upt,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_nat2(upt(zero_zero_nat,N)) ).

% atLeast_upt
tff(fact_7975_greaterThanLessThan__upt,axiom,
    ! [N: nat,M: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(N),M) = set_nat2(upt(aa_nat_nat(suc,N),M)) ).

% greaterThanLessThan_upt
tff(fact_7976_atMost__upto,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_7977_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_7978_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_7979_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(X,Xs) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( I = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_7980_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = nil_nat ) ) ) ).

% upt_rec
tff(fact_7981_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = nil_nat ) ) ) ).

% upt_Suc
tff(fact_7982_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_7983_map__Suc__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(suc,upt(M,N)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N)) ).

% map_Suc_upt
tff(fact_7984_map__add__upt,axiom,
    ! [N: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abl,N),upt(zero_zero_nat,M)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% map_add_upt
tff(fact_7985_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_abm,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_7986_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups4561878855575611511st_nat(upt(M,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cd),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) ) ).

% sum_list_upt
tff(fact_7987_card__length__sum__list__rec,axiom,
    ! [M: nat,N5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_abn(M),N5))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_abo(M),N5)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_abp(M),N5)))) ) ) ).

% card_length_sum_list_rec
tff(fact_7988_card__length__sum__list,axiom,
    ! [M: nat,N5: nat] : finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_abn(M),N5))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),M)),one_one_nat)),N5) ).

% card_length_sum_list
tff(fact_7989_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),upt(M,N))) ).

% sorted_wrt_upt
tff(fact_7990_sorted__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),upt(M,N))) ).

% sorted_upt
tff(fact_7991_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ns))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Ns)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I))) ) ) ).

% sorted_wrt_less_idx
tff(fact_7992_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),upto(I,J))) ).

% sorted_wrt_upto
tff(fact_7993_sorted__upto,axiom,
    ! [M: int,N: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),upto(M,N))) ).

% sorted_upto
tff(fact_7994_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ud(M))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,M),aTP_Lamm_abq(M)) ).

% pairs_le_eq_Sigma
tff(fact_7995_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)) ).

% natLess_def
tff(fact_7996_Restr__natLeq,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,N)),aTP_Lamm_abr(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_abs(N))) ).

% Restr_natLeq
tff(fact_7997_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_7998_Restr__natLeq2,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N),aTP_Lamm_abt(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_abs(N))) ).

% Restr_natLeq2
tff(fact_7999_natLeq__underS__less,axiom,
    ! [N: nat] : aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,N)) ).

% natLeq_underS_less
tff(fact_8000_sort__upt,axiom,
    ! [M: nat,N: nat] : linord738340561235409698at_nat(aTP_Lamm_cd,upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_8001_sort__upto,axiom,
    ! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_ce,upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_8002_Field__natLeq__on,axiom,
    ! [N: nat] : field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_abs(N)))) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,N)) ).

% Field_natLeq_on
tff(fact_8003_wf__less,axiom,
    wf_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat))) ).

% wf_less
tff(fact_8004_wf__int__ge__less__than2,axiom,
    ! [D: int] : wf_int(int_ge_less_than2(D)) ).

% wf_int_ge_less_than2
tff(fact_8005_wf__int__ge__less__than,axiom,
    ! [D: int] : wf_int(int_ge_less_than(D)) ).

% wf_int_ge_less_than
tff(fact_8006_infinite__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ? [R3: fun_nat_nat] :
          ( order_5726023648592871131at_nat(R3)
          & ! [N6: nat] : pp(aa_set_nat_bool(member_nat(aa_nat_nat(R3,N6)),S2)) ) ) ).

% infinite_enumerate
tff(fact_8007_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( order_5726023648592871131at_nat(F)
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(F,N))) ) ).

% strict_mono_imp_increasing
tff(fact_8008_enumerate__Ex,axiom,
    ! [S2: set_nat,S: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(member_nat(S),S2))
       => ? [N2: nat] : aa_nat_nat(infini8530281810654367211te_nat(S2),N2) = S ) ) ).

% enumerate_Ex
tff(fact_8009_le__enumerate,axiom,
    ! [S2: set_nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S2),N))) ) ).

% le_enumerate
tff(fact_8010_strict__mono__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => order_5726023648592871131at_nat(infini8530281810654367211te_nat(S2)) ) ).

% strict_mono_enumerate
tff(fact_8011_range__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( image_nat_nat(infini8530281810654367211te_nat(S2),top_top_set_nat) = S2 ) ) ).

% range_enumerate
tff(fact_8012_finite__le__enumerate,axiom,
    ! [S2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_set_nat_nat(finite_card_nat,S2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S2),N))) ) ) ).

% finite_le_enumerate
tff(fact_8013_bij__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(bij_betw_nat_nat(infini8530281810654367211te_nat(S2)),top_top_set_nat),S2)) ) ).

% bij_enumerate
tff(fact_8014_Least__eq__0,axiom,
    ! [P: fun_nat_bool] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ord_Least_nat(P) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_8015_Least__Suc2,axiom,
    ! [P: fun_nat_bool,N: nat,Q: fun_nat_bool,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(Q,M))
       => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
         => ( ! [K3: nat] :
                ( pp(aa_nat_bool(P,aa_nat_nat(suc,K3)))
              <=> pp(aa_nat_bool(Q,K3)) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_8016_Least__Suc,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_aba(P))) ) ) ) ).

% Least_Suc
tff(fact_8017_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( last_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_8018_vimage__Suc__insert__0,axiom,
    ! [A2: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A2)) = vimage_nat_nat(suc,A2) ).

% vimage_Suc_insert_0
tff(fact_8019_finite__vimage__Suc__iff,axiom,
    ! [F3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,vimage_nat_nat(suc,F3)))
    <=> pp(aa_set_nat_bool(finite_finite_nat,F3)) ) ).

% finite_vimage_Suc_iff
tff(fact_8020_vimage__Suc__insert__Suc,axiom,
    ! [N: nat,A2: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,N),A2)) = insert_nat(N,vimage_nat_nat(suc,A2)) ).

% vimage_Suc_insert_Suc
tff(fact_8021_set__decode__div__2,axiom,
    ! [X: nat] : aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))) = vimage_nat_nat(suc,aa_nat_set_nat(nat_set_decode,X)) ).

% set_decode_div_2
tff(fact_8022_set__encode__vimage__Suc,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A2)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% set_encode_vimage_Suc
tff(fact_8023_GreatestI__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B2: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B2)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_nat
tff(fact_8024_Greatest__le__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B2: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B2)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),order_Greatest_nat(P))) ) ) ).

% Greatest_le_nat
tff(fact_8025_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_bool,B2: nat] :
      ( ? [X_1: nat] : pp(aa_nat_bool(P,X_1))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B2)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_ex_nat
tff(fact_8026_natLeq__on__wo__rel,axiom,
    ! [N: nat] : bNF_We3818239936649020644el_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_abs(N)))) ).

% natLeq_on_wo_rel
tff(fact_8027_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_8028_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_8029_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Y))
     => ( bezw(X,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).

% bezw_non_0
tff(fact_8030_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = product_Pair_int_int(one_one_int,zero_zero_int) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y = product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) ) ) ).

% bezw.elims
tff(fact_8031_bezw_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( bezw(X,Y) = product_Pair_int_int(one_one_int,zero_zero_int) ) )
      & ( ( Y != zero_zero_nat )
       => ( bezw(X,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ) ).

% bezw.simps
tff(fact_8032_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = product_Pair_int_int(one_one_int,zero_zero_int) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y = product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% bezw.pelims
tff(fact_8033_pair__lessI2,axiom,
    ! [A: nat,B2: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),T)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_8034_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z2)),fun_pair_less)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z2)) ) ).

% pair_less_iff1
tff(fact_8035_pair__lessI1,axiom,
    ! [A: nat,B2: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),T)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_8036_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
               => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
               => ( Y = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_8037_pair__leqI2,axiom,
    ! [A: nat,B2: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),T)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_8038_pair__leqI1,axiom,
    ! [A: nat,B2: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B2))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),T)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_8039_gcd__nat_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_abu),zero_zero_nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_8040_bot__nat__0_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(aTP_Lamm_ay,aTP_Lamm_ax),zero_zero_nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_8041_less__eq__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,M),N))
    <=> pp(aa_Ext2772979874139974381t_bool(extend8126323780198025762t_bool(aa_Ext6309394871002001526t_bool(aTP_Lamm_abv,M),fTrue),N)) ) ).

% less_eq_enat_def
tff(fact_8042_less__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,M),N))
    <=> pp(aa_Ext2772979874139974381t_bool(extend8126323780198025762t_bool(aa_Ext6309394871002001526t_bool(aTP_Lamm_abw,N),fFalse),M)) ) ).

% less_enat_def
tff(fact_8043_prod__decode__triangle__add,axiom,
    ! [K: nat,M: nat] : aa_nat7603221163087113075at_nat(nat_prod_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),M)) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) ).

% prod_decode_triangle_add
tff(fact_8044_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero_nat) ).

% prod_decode_def
tff(fact_8045_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),A0))
     => ( ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2)))
             => ( ! [X2: nat,Y4: nat] :
                    ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,N2) )
                   => pp(aa_nat_bool(P,Y4)) )
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,A0)) ) ) ) ).

% list_decode.pinduct
tff(fact_8046_list__decode_Oelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( nat_list_decode(X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N2: nat] :
              ( ( X = aa_nat_nat(suc,N2) )
             => ( Y != produc2761476792215241774st_nat(aTP_Lamm_abx,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) ) ) ) ) ).

% list_decode.elims
tff(fact_8047_list__decode_Opsimps_I1_J,axiom,
    ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat))
   => ( nat_list_decode(zero_zero_nat) = nil_nat ) ) ).

% list_decode.psimps(1)
tff(fact_8048_list__decode_Osimps_I1_J,axiom,
    nat_list_decode(zero_zero_nat) = nil_nat ).

% list_decode.simps(1)
tff(fact_8049_list__decode_Opsimps_I2_J,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N)))
     => ( nat_list_decode(aa_nat_nat(suc,N)) = produc2761476792215241774st_nat(aTP_Lamm_abx,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ) ) ).

% list_decode.psimps(2)
tff(fact_8050_list__decode_Osimps_I2_J,axiom,
    ! [N: nat] : nat_list_decode(aa_nat_nat(suc,N)) = produc2761476792215241774st_nat(aTP_Lamm_abx,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ).

% list_decode.simps(2)
tff(fact_8051_list__decode_Opelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( nat_list_decode(X) = Y )
     => ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat)) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa_nat_nat(suc,N2) )
               => ( ( Y = produc2761476792215241774st_nat(aTP_Lamm_abx,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) )
                 => ~ pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2))) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_8052_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))) ).

% UNIV_char_of_nat
tff(fact_8053_inj__on__char__of__nat,axiom,
    pp(aa_set_nat_bool(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one)))))))))))) ).

% inj_on_char_of_nat
tff(fact_8054_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one)))))))))) ).

% range_nat_of_char
tff(fact_8055_one__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int)) ).

% one_int.transfer
tff(fact_8056_zero__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int)) ).

% zero_int.transfer
tff(fact_8057_nat__of__char__less__256,axiom,
    ! [C: char] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))) ).

% nat_of_char_less_256
tff(fact_8058_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( pp(aa_set_real_bool(member_real(X),field_5140801741446780682s_real))
     => ~ ! [M3: nat,N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( ( aa_real_real(abs_abs_real,X) = aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,M3)),aa_nat_real(semiri5074537144036343181t_real,N2)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,M3),N2)) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_8059_coprime__Suc__left__nat,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,N)),N)) ).

% coprime_Suc_left_nat
tff(fact_8060_coprime__Suc__right__nat,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(suc,N))) ).

% coprime_Suc_right_nat
tff(fact_8061_coprime__Suc__0__left,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ).

% coprime_Suc_0_left
tff(fact_8062_coprime__Suc__0__right,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(suc,zero_zero_nat))) ).

% coprime_Suc_0_right
tff(fact_8063_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N9: nat] :
        ! [M2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),M2))
         => ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N4))
             => pp(aa_Pro2232559814647311250t_bool(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N4),M2))) ) ) ) ).

% eventually_prod_sequentially
tff(fact_8064_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N)) ) ).

% coprime_diff_one_left_nat
tff(fact_8065_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_8066_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_zw),minus_minus_int) ).

% minus_int.transfer
tff(fact_8067_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_zu),plus_plus_int) ).

% plus_int.transfer
tff(fact_8068_less__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_zq),ord_less_int) ).

% less_int.transfer
tff(fact_8069_less__eq__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_zs),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_8070_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_zn),times_times_int) ).

% times_int.transfer
tff(fact_8071_uminus__int_Otransfer,axiom,
    pp(aa_fun_int_int_bool(aa_fun7527181275634263258t_bool(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_zo)),uminus_uminus_int)) ).

% uminus_int.transfer
tff(fact_8072_nat_Otransfer,axiom,
    bNF_re4555766996558763186at_nat(pcr_int,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),nat2) ).

% nat.transfer
tff(fact_8073_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_aby,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_8074_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_zn),produc4528391901023682385at_nat(aTP_Lamm_zn)) ).

% times_int.rsp
tff(fact_8075_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V)))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),V) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).

% intrel_iff
tff(fact_8076_nat_Orsp,axiom,
    bNF_re8246922863344978751at_nat(intrel,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat.rsp
tff(fact_8077_uminus__int_Orsp,axiom,
    pp(aa_fun4519828053337421206t_bool(aa_fun2399787666232550060t_bool(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_zo)),produc2626176000494625587at_nat(aTP_Lamm_zo))) ).

% uminus_int.rsp
tff(fact_8078_int_Oabs__eq__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( aa_Pro4240683299535347697at_int(abs_Integ,X) = aa_Pro4240683299535347697at_int(abs_Integ,Y) )
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,X),Y)) ) ).

% int.abs_eq_iff
tff(fact_8079_zero__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat))) ).

% zero_int.rsp
tff(fact_8080_one__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat))) ).

% one_int.rsp
tff(fact_8081_intrel__def,axiom,
    intrel = produc5023667602222594023t_bool(aTP_Lamm_aca) ).

% intrel_def
tff(fact_8082_less__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_zq),produc5023667602222594023t_bool(aTP_Lamm_zq)) ).

% less_int.rsp
tff(fact_8083_less__eq__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_zs),produc5023667602222594023t_bool(aTP_Lamm_zs)) ).

% less_eq_int.rsp
tff(fact_8084_int_Orel__eq__transfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),intrel,fequal_int) ).

% int.rel_eq_transfer
tff(fact_8085_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_zw),produc4528391901023682385at_nat(aTP_Lamm_zw)) ).

% minus_int.rsp
tff(fact_8086_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_zu),produc4528391901023682385at_nat(aTP_Lamm_zu)) ).

% plus_int.rsp
tff(fact_8087_int_Obi__total,axiom,
    bi_tot896582865486249351at_int(pcr_int) ).

% int.bi_total
tff(fact_8088_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),less_than)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% less_than_iff
tff(fact_8089_elimnum,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = vEBT_Node(Info,Deg,TreeList,Summary) ) ) ).

% elimnum
tff(fact_8090_idiff__enat__0__right,axiom,
    ! [N: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(N),aa_nat_Extended_enat(extended_enat2,zero_zero_nat)) = N ).

% idiff_enat_0_right
tff(fact_8091_idiff__enat__0,axiom,
    ! [N: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(aa_nat_Extended_enat(extended_enat2,zero_zero_nat)),N) = aa_nat_Extended_enat(extended_enat2,zero_zero_nat) ).

% idiff_enat_0
tff(fact_8092_plus__enat__simps_I1_J,axiom,
    ! [M: nat,N: nat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(extended_enat2,M)),aa_nat_Extended_enat(extended_enat2,N)) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% plus_enat_simps(1)
tff(fact_8093_enat__ord__simps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(extended_enat2,M)),aa_nat_Extended_enat(extended_enat2,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% enat_ord_simps(2)
tff(fact_8094_enat__ord__simps_I1_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(extended_enat2,M)),aa_nat_Extended_enat(extended_enat2,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% enat_ord_simps(1)
tff(fact_8095_numeral__less__enat__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(M)),aa_nat_Extended_enat(extended_enat2,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),N)) ) ).

% numeral_less_enat_iff
tff(fact_8096_numeral__le__enat__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(M)),aa_nat_Extended_enat(extended_enat2,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),N)) ) ).

% numeral_le_enat_iff
tff(fact_8097_Suc__ile__eq,axiom,
    ! [M: nat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(extended_enat2,aa_nat_nat(suc,M))),N))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_nat_Extended_enat(extended_enat2,M)),N)) ) ).

% Suc_ile_eq
tff(fact_8098_finite__enat__bounded,axiom,
    ! [A2: set_Extended_enat,N: nat] :
      ( ! [Y3: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y3),A2))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Y3),aa_nat_Extended_enat(extended_enat2,N))) )
     => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2)) ) ).

% finite_enat_bounded
tff(fact_8099_iadd__le__enat__iff,axiom,
    ! [X: extended_enat,Y: extended_enat,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),Y)),aa_nat_Extended_enat(extended_enat2,N)))
    <=> ? [Y6: nat,X7: nat] :
          ( ( X = aa_nat_Extended_enat(extended_enat2,X7) )
          & ( Y = aa_nat_Extended_enat(extended_enat2,Y6) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X7),Y6)),N)) ) ) ).

% iadd_le_enat_iff
tff(fact_8100_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_z5237406670263579293d_enat = aa_nat_Extended_enat(extended_enat2,X) )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(2)
tff(fact_8101_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( aa_nat_Extended_enat(extended_enat2,X) = zero_z5237406670263579293d_enat )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(1)
tff(fact_8102_zero__enat__def,axiom,
    zero_z5237406670263579293d_enat = aa_nat_Extended_enat(extended_enat2,zero_zero_nat) ).

% zero_enat_def
tff(fact_8103_less__enatE,axiom,
    ! [N: extended_enat,M: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,N),aa_nat_Extended_enat(extended_enat2,M)))
     => ~ ! [K3: nat] :
            ( ( N = aa_nat_Extended_enat(extended_enat2,K3) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),M)) ) ) ).

% less_enatE
tff(fact_8104_elimcomplete,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),extend5688581933313929465d_enat) = vEBT_Node(Info,Deg,TreeList,Summary) ) ) ).

% elimcomplete
tff(fact_8105_times__enat__simps_I4_J,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(extended_enat2,M)),extend5688581933313929465d_enat) = zero_z5237406670263579293d_enat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(extended_enat2,M)),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(4)
tff(fact_8106_times__enat__simps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),aa_nat_Extended_enat(extended_enat2,N)) = zero_z5237406670263579293d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),aa_nat_Extended_enat(extended_enat2,N)) = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(3)
tff(fact_8107_Inf__enat__def,axiom,
    ! [A2: set_Extended_enat] :
      ( ( ( A2 = bot_bo7653980558646680370d_enat )
       => ( comple2295165028678016749d_enat(A2) = extend5688581933313929465d_enat ) )
      & ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( comple2295165028678016749d_enat(A2) = ord_Le1955565732374568822d_enat(aTP_Lamm_a(A2)) ) ) ) ).

% Inf_enat_def
tff(fact_8108_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_8109_Sup__enat__def,axiom,
    ! [A2: set_Extended_enat] :
      ( ( ( A2 = bot_bo7653980558646680370d_enat )
       => ( comple4398354569131411667d_enat(A2) = zero_z5237406670263579293d_enat ) )
      & ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
           => ( comple4398354569131411667d_enat(A2) = lattic921264341876707157d_enat(A2) ) )
          & ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
           => ( comple4398354569131411667d_enat(A2) = extend5688581933313929465d_enat ) ) ) ) ) ).

% Sup_enat_def
tff(fact_8110_times__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(M),N) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_acc(N),aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,N),zero_z5237406670263579293d_enat),zero_z5237406670263579293d_enat),extend5688581933313929465d_enat)),M) ).

% times_enat_def
tff(fact_8111_plus__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(M),N) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_ace(N),extend5688581933313929465d_enat),M) ).

% plus_enat_def
tff(fact_8112_eSuc__Max,axiom,
    ! [A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( aa_Ext7969882492699152700d_enat(extended_eSuc,lattic921264341876707157d_enat(A2)) = lattic921264341876707157d_enat(image_80655429650038917d_enat(extended_eSuc,A2)) ) ) ) ).

% eSuc_Max
tff(fact_8113_eSuc__def,axiom,
    ! [I: extended_enat] : aa_Ext7969882492699152700d_enat(extended_eSuc,I) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_acf,extend5688581933313929465d_enat),I) ).

% eSuc_def
tff(fact_8114_enat__eSuc__iff,axiom,
    ! [Y: nat,X: extended_enat] :
      ( ( aa_nat_Extended_enat(extended_enat2,Y) = aa_Ext7969882492699152700d_enat(extended_eSuc,X) )
    <=> ? [N4: nat] :
          ( ( Y = aa_nat_nat(suc,N4) )
          & ( aa_nat_Extended_enat(extended_enat2,N4) = X ) ) ) ).

% enat_eSuc_iff
tff(fact_8115_eSuc__enat__iff,axiom,
    ! [X: extended_enat,Y: nat] :
      ( ( aa_Ext7969882492699152700d_enat(extended_eSuc,X) = aa_nat_Extended_enat(extended_enat2,Y) )
    <=> ? [N4: nat] :
          ( ( Y = aa_nat_nat(suc,N4) )
          & ( X = aa_nat_Extended_enat(extended_enat2,N4) ) ) ) ).

% eSuc_enat_iff
tff(fact_8116_eSuc__enat,axiom,
    ! [N: nat] : aa_Ext7969882492699152700d_enat(extended_eSuc,aa_nat_Extended_enat(extended_enat2,N)) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(suc,N)) ).

% eSuc_enat
tff(fact_8117_eSuc__Sup,axiom,
    ! [A2: set_Extended_enat] :
      ( ( A2 != bot_bo7653980558646680370d_enat )
     => ( aa_Ext7969882492699152700d_enat(extended_eSuc,comple4398354569131411667d_enat(A2)) = comple4398354569131411667d_enat(image_80655429650038917d_enat(extended_eSuc,A2)) ) ) ).

% eSuc_Sup
tff(fact_8118_natLeq__on__well__order__on,axiom,
    ! [N: nat] : order_2888998067076097458on_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,N)),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_abs(N)))) ).

% natLeq_on_well_order_on
tff(fact_8119_natLeq__on__Well__order,axiom,
    ! [N: nat] : order_2888998067076097458on_nat(field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_abs(N)))),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_abs(N)))) ).

% natLeq_on_Well_order
tff(fact_8120_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_tc,Uu) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_8121_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_rr,Uu) = aa_nat_real(power_power_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_8122_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ns,Uu) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uu)),aa_nat_real(power_power_real(zero_zero_real),Uu)) ).

% ATP.lambda_3
tff(fact_8123_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_abm,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_4
tff(fact_8124_ATP_Olambda__5,axiom,
    ! [Uu: extended_enat] : aa_Ext7969882492699152700d_enat(aTP_Lamm_mo,Uu) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Uu),one_on7984719198319812577d_enat) ).

% ATP.lambda_5
tff(fact_8125_ATP_Olambda__6,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_mn,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).

% ATP.lambda_6
tff(fact_8126_ATP_Olambda__7,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_mp,Uu) = aa_real_real(plus_plus_real(Uu),one_one_real) ).

% ATP.lambda_7
tff(fact_8127_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_mr,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_8
tff(fact_8128_ATP_Olambda__9,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_mq,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_9
tff(fact_8129_ATP_Olambda__10,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_aby,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).

% ATP.lambda_10
tff(fact_8130_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_aal,Uu) = suminf_real(aTP_Lamm_yj(Uu)) ).

% ATP.lambda_11
tff(fact_8131_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_aao,Uu) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ba,Uu)) ).

% ATP.lambda_12
tff(fact_8132_ATP_Olambda__13,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_aav,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_13
tff(fact_8133_ATP_Olambda__14,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_abh,Uu) = some_num(bit0(Uu)) ).

% ATP.lambda_14
tff(fact_8134_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_gn,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_15
tff(fact_8135_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa_nat_Extended_enat(aTP_Lamm_acf,Uu) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(suc,Uu)) ).

% ATP.lambda_16
tff(fact_8136_ATP_Olambda__17,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_abg,Uu))
    <=> ? [I3: int,N4: nat] :
          ( ( Uu = aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,I3)),aa_nat_real(semiri5074537144036343181t_real,N4)) )
          & ( N4 != zero_zero_nat ) ) ) ).

% ATP.lambda_17
tff(fact_8137_ATP_Olambda__18,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_td(Uu),Uua) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_complex) ).

% ATP.lambda_18
tff(fact_8138_ATP_Olambda__19,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_tf,Uu),Uua) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_real) ).

% ATP.lambda_19
tff(fact_8139_ATP_Olambda__20,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_te,Uu),Uua) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_int) ).

% ATP.lambda_20
tff(fact_8140_ATP_Olambda__21,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_mf,Uu),Uua) = aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uu))),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uu))),one_on7984719198319812577d_enat)) ).

% ATP.lambda_21
tff(fact_8141_ATP_Olambda__22,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_me,Uu),Uua) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_nat_complex(semiri8010041392384452111omplex,Uu))),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_nat_complex(semiri8010041392384452111omplex,Uu))),one_one_complex)) ).

% ATP.lambda_22
tff(fact_8142_ATP_Olambda__23,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mg,Uu),Uua) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Uu))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Uu))),one_one_real)) ).

% ATP.lambda_23
tff(fact_8143_ATP_Olambda__24,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mi,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(semiri1316708129612266289at_nat,Uu))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(semiri1316708129612266289at_nat,Uu))),one_one_nat)) ).

% ATP.lambda_24
tff(fact_8144_ATP_Olambda__25,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_mh,Uu),Uua) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_nat_int(semiri1314217659103216013at_int,Uu))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_nat_int(semiri1314217659103216013at_int,Uu))),one_one_int)) ).

% ATP.lambda_25
tff(fact_8145_ATP_Olambda__26,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_acc(Uu),Uua) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_acb(Uua),aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),zero_z5237406670263579293d_enat),extend5688581933313929465d_enat)),Uu) ).

% ATP.lambda_26
tff(fact_8146_ATP_Olambda__27,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_ace(Uu),Uua) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_acd(Uua),extend5688581933313929465d_enat),Uu) ).

% ATP.lambda_27
tff(fact_8147_ATP_Olambda__28,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( pp(aa_nat_bool(aa_Ext6309394871002001526t_bool(aTP_Lamm_abv,Uu),Uua))
    <=> pp(aa_Ext2772979874139974381t_bool(extend8126323780198025762t_bool(aa_nat_fun_nat_bool(aTP_Lamm_ay,Uua),fFalse),Uu)) ) ).

% ATP.lambda_28
tff(fact_8148_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_tg(Uu),Uua) = aa_complex_complex(if_complex(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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_8149_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ti,Uu),Uua) = aa_real_real(if_real(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_real) ).

% ATP.lambda_30
tff(fact_8150_ATP_Olambda__31,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_th,Uu),Uua) = aa_int_int(if_int(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_int) ).

% ATP.lambda_31
tff(fact_8151_ATP_Olambda__32,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( pp(aa_nat_bool(aa_Ext6309394871002001526t_bool(aTP_Lamm_abw,Uu),Uua))
    <=> pp(aa_Ext2772979874139974381t_bool(extend8126323780198025762t_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),fTrue),Uu)) ) ).

% ATP.lambda_32
tff(fact_8152_ATP_Olambda__33,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_uj(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ui(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_33
tff(fact_8153_ATP_Olambda__34,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_uk(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ue(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_34
tff(fact_8154_ATP_Olambda__35,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_uh(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ug(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_35
tff(fact_8155_ATP_Olambda__36,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_uf(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ue(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_36
tff(fact_8156_ATP_Olambda__37,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_sx(Uu),Uua) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_37
tff(fact_8157_ATP_Olambda__38,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_xm(Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),one_one_nat)))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),one_one_nat))) ).

% ATP.lambda_38
tff(fact_8158_ATP_Olambda__39,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_wd(Uu),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,Uua))),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_39
tff(fact_8159_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_wf,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_40
tff(fact_8160_ATP_Olambda__41,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_we,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,Uua))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_41
tff(fact_8161_ATP_Olambda__42,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_wg(Uu),Uua) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_complex(power_power_complex(numera6690914467698888265omplex(bit0(one))),Uua)) ).

% ATP.lambda_42
tff(fact_8162_ATP_Olambda__43,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_wh,Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_real(power_power_real(numeral_numeral_real(bit0(one))),Uua)) ).

% ATP.lambda_43
tff(fact_8163_ATP_Olambda__44,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_yj(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat)))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat)))) ).

% ATP.lambda_44
tff(fact_8164_ATP_Olambda__45,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_aam(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% ATP.lambda_45
tff(fact_8165_ATP_Olambda__46,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aaq(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_46
tff(fact_8166_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_wq(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_47
tff(fact_8167_ATP_Olambda__48,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ws,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_48
tff(fact_8168_ATP_Olambda__49,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_wr,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_49
tff(fact_8169_ATP_Olambda__50,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ub(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))),Uua) ).

% ATP.lambda_50
tff(fact_8170_ATP_Olambda__51,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_uc(Uu),Uua) = aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_51
tff(fact_8171_ATP_Olambda__52,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_sy(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu),numera6690914467698888265omplex(bit0(one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_52
tff(fact_8172_ATP_Olambda__53,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_sz(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(gbinomial_real(Uu),Uua)),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Uu),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_53
tff(fact_8173_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_abu,Uu),Uua))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_54
tff(fact_8174_ATP_Olambda__55,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_zd(Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(power_power_real(Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_55
tff(fact_8175_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abd,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).

% ATP.lambda_56
tff(fact_8176_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ua,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_57
tff(fact_8177_ATP_Olambda__58,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tz,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_58
tff(fact_8178_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_cw,Uu),Uua))
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_59
tff(fact_8179_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_cv(Uu),Uua))
    <=> ( aa_nat_real(power_power_real(Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_60
tff(fact_8180_ATP_Olambda__61,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_zb(Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat)))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat))) ).

% ATP.lambda_61
tff(fact_8181_ATP_Olambda__62,axiom,
    ! [Uu: fun_na8797796813480665199d_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_rj(Uu),Uua) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_Extended_enat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_62
tff(fact_8182_ATP_Olambda__63,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ri(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_complex(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_63
tff(fact_8183_ATP_Olambda__64,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rh(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_64
tff(fact_8184_ATP_Olambda__65,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_rl(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_65
tff(fact_8185_ATP_Olambda__66,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_rk(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_66
tff(fact_8186_ATP_Olambda__67,axiom,
    ! [Uu: fun_na8797796813480665199d_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_lr(Uu),Uua) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_Extended_enat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_67
tff(fact_8187_ATP_Olambda__68,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lt(Uu),Uua) = aa_real_real(plus_plus_real(aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_68
tff(fact_8188_ATP_Olambda__69,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ls(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_69
tff(fact_8189_ATP_Olambda__70,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lq(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_70
tff(fact_8190_ATP_Olambda__71,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lk(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_71
tff(fact_8191_ATP_Olambda__72,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lj(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_72
tff(fact_8192_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_kp(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_73
tff(fact_8193_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_kq(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(Uu,Uua)),aa_nat_real(power_power_real(zero_zero_real),Uua)) ).

% ATP.lambda_74
tff(fact_8194_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_xf(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_75
tff(fact_8195_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lp(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_76
tff(fact_8196_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lo(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_77
tff(fact_8197_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ll(Uu),Uua) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_78
tff(fact_8198_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ln(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_79
tff(fact_8199_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lm(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_80
tff(fact_8200_ATP_Olambda__81,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mz(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_81
tff(fact_8201_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_my(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_82
tff(fact_8202_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_tx(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_83
tff(fact_8203_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_tu(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_84
tff(fact_8204_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_tr(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_85
tff(fact_8205_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_to(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_86
tff(fact_8206_ATP_Olambda__87,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_kb(Uu),Uua))
    <=> ( aa_Ext2541572767299617110t_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_87
tff(fact_8207_ATP_Olambda__88,axiom,
    ! [Uu: fun_Ex7660298398204468809at_nat,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_jx(Uu),Uua))
    <=> ( aa_Extended_enat_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_88
tff(fact_8208_ATP_Olambda__89,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_jz(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_89
tff(fact_8209_ATP_Olambda__90,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_jv(Uu),Uua))
    <=> ( aa_complex_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_90
tff(fact_8210_ATP_Olambda__91,axiom,
    ! [Uu: fun_complex_int,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_kd(Uu),Uua))
    <=> ( aa_complex_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_91
tff(fact_8211_ATP_Olambda__92,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_jy(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_92
tff(fact_8212_ATP_Olambda__93,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ju(Uu),Uua))
    <=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_93
tff(fact_8213_ATP_Olambda__94,axiom,
    ! [Uu: fun_real_int,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_kc(Uu),Uua))
    <=> ( aa_real_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_94
tff(fact_8214_ATP_Olambda__95,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ka(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_95
tff(fact_8215_ATP_Olambda__96,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_jw(Uu),Uua))
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_96
tff(fact_8216_ATP_Olambda__97,axiom,
    ! [Uu: fun_Ex7660298398204468809at_nat,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_qn(Uu),Uua))
    <=> ( aa_Extended_enat_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_97
tff(fact_8217_ATP_Olambda__98,axiom,
    ! [Uu: fun_Ex3482447378695272101at_int,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_qq(Uu),Uua))
    <=> ( aa_Extended_enat_int(Uu,Uua) = one_one_int ) ) ).

% ATP.lambda_98
tff(fact_8218_ATP_Olambda__99,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_qs(Uu),Uua))
    <=> ( aa_complex_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_99
tff(fact_8219_ATP_Olambda__100,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_ql(Uu),Uua))
    <=> ( aa_complex_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_100
tff(fact_8220_ATP_Olambda__101,axiom,
    ! [Uu: fun_complex_int,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_qp(Uu),Uua))
    <=> ( aa_complex_int(Uu,Uua) = one_one_int ) ) ).

% ATP.lambda_101
tff(fact_8221_ATP_Olambda__102,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_qr(Uu),Uua))
    <=> ( aa_real_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_102
tff(fact_8222_ATP_Olambda__103,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_qk(Uu),Uua))
    <=> ( aa_real_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_103
tff(fact_8223_ATP_Olambda__104,axiom,
    ! [Uu: fun_real_int,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_qo(Uu),Uua))
    <=> ( aa_real_int(Uu,Uua) = one_one_int ) ) ).

% ATP.lambda_104
tff(fact_8224_ATP_Olambda__105,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_qt(Uu),Uua))
    <=> ( aa_int_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_105
tff(fact_8225_ATP_Olambda__106,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_qm(Uu),Uua))
    <=> ( aa_int_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_106
tff(fact_8226_ATP_Olambda__107,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aar(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aaq(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),one_one_nat))) ).

% ATP.lambda_107
tff(fact_8227_ATP_Olambda__108,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aaz(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aaq(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))) ).

% ATP.lambda_108
tff(fact_8228_ATP_Olambda__109,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zx,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_109
tff(fact_8229_ATP_Olambda__110,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ob(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_110
tff(fact_8230_ATP_Olambda__111,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_oa(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_111
tff(fact_8231_ATP_Olambda__112,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_aab,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_112
tff(fact_8232_ATP_Olambda__113,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_xw(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(sin_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_113
tff(fact_8233_ATP_Olambda__114,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_xs(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_114
tff(fact_8234_ATP_Olambda__115,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ta(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_115
tff(fact_8235_ATP_Olambda__116,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_tb,Uu),Uua) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_116
tff(fact_8236_ATP_Olambda__117,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ey,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,bit0(one))),Uua)),one_one_nat)) ).

% ATP.lambda_117
tff(fact_8237_ATP_Olambda__118,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ex,Uu),Uua) = product_Pair_int_int(Uu,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Uua)),one_one_int)) ).

% ATP.lambda_118
tff(fact_8238_ATP_Olambda__119,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_nt(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(bit0(one)))) ).

% ATP.lambda_119
tff(fact_8239_ATP_Olambda__120,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_nu(Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Uua)),numeral_numeral_real(bit0(one)))) ).

% ATP.lambda_120
tff(fact_8240_ATP_Olambda__121,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_aay(Uu),Uua) = aa_real_real(times_times_real(Uu),aa_real_real(plus_plus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_121
tff(fact_8241_ATP_Olambda__122,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_aax(Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_122
tff(fact_8242_ATP_Olambda__123,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_aaw(Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_123
tff(fact_8243_ATP_Olambda__124,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_xt(Uu),Uua) = aa_real_real(minus_minus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_124
tff(fact_8244_ATP_Olambda__125,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xv,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_125
tff(fact_8245_ATP_Olambda__126,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_xu(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_126
tff(fact_8246_ATP_Olambda__127,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_qz(Uu),Uua) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Uu),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uua)) ).

% ATP.lambda_127
tff(fact_8247_ATP_Olambda__128,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ra(Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_128
tff(fact_8248_ATP_Olambda__129,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rc,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_129
tff(fact_8249_ATP_Olambda__130,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_rb(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_130
tff(fact_8250_ATP_Olambda__131,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_abx,Uu),Uua) = cons_nat(Uu,nat_list_decode(Uua)) ).

% ATP.lambda_131
tff(fact_8251_ATP_Olambda__132,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_abe,Uu),Uua))
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_132
tff(fact_8252_ATP_Olambda__133,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_tj,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),Uu)) ) ).

% ATP.lambda_133
tff(fact_8253_ATP_Olambda__134,axiom,
    ! [Uu: set_Extended_enat,Uua: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(aTP_Lamm_au(Uu),Uua))
    <=> pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(Uua),Uu)) ) ).

% ATP.lambda_134
tff(fact_8254_ATP_Olambda__135,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( pp(aa_set_complex_bool(aTP_Lamm_at(Uu),Uua))
    <=> pp(aa_set_complex_bool(ord_le211207098394363844omplex(Uua),Uu)) ) ).

% ATP.lambda_135
tff(fact_8255_ATP_Olambda__136,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_av,Uu),Uua))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Uua),Uu)) ) ).

% ATP.lambda_136
tff(fact_8256_ATP_Olambda__137,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( pp(aa_set_int_bool(aTP_Lamm_aw(Uu),Uua))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Uua),Uu)) ) ).

% ATP.lambda_137
tff(fact_8257_ATP_Olambda__138,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ay,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).

% ATP.lambda_138
tff(fact_8258_ATP_Olambda__139,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_tk,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uua),Uu)) ) ).

% ATP.lambda_139
tff(fact_8259_ATP_Olambda__140,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aaa,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_140
tff(fact_8260_ATP_Olambda__141,axiom,
    ! [Uu: extended_enat,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(aTP_Lamm_mt,Uu),Uua))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Uua),Uu)) ) ).

% ATP.lambda_141
tff(fact_8261_ATP_Olambda__142,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_mu,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),Uu)) ) ).

% ATP.lambda_142
tff(fact_8262_ATP_Olambda__143,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_ms,Uu),Uua))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,Uua),Uu)) ) ).

% ATP.lambda_143
tff(fact_8263_ATP_Olambda__144,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ax,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).

% ATP.lambda_144
tff(fact_8264_ATP_Olambda__145,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_mv,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uu)) ) ).

% ATP.lambda_145
tff(fact_8265_ATP_Olambda__146,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aas,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_146
tff(fact_8266_ATP_Olambda__147,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zy,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_147
tff(fact_8267_ATP_Olambda__148,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_aan,Uu),Uua) = aa_nat_real(power_power_real(Uua),Uu) ).

% ATP.lambda_148
tff(fact_8268_ATP_Olambda__149,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abl,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_149
tff(fact_8269_ATP_Olambda__150,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_zz(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_150
tff(fact_8270_ATP_Olambda__151,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_az,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,Uua),Uu)) ) ).

% ATP.lambda_151
tff(fact_8271_ATP_Olambda__152,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ba,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).

% ATP.lambda_152
tff(fact_8272_ATP_Olambda__153,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_bb,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu)) ) ).

% ATP.lambda_153
tff(fact_8273_ATP_Olambda__154,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_zo,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_154
tff(fact_8274_ATP_Olambda__155,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tl,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_155
tff(fact_8275_ATP_Olambda__156,axiom,
    ! [Uu: set_Extended_enat,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_a(Uu),Uua))
    <=> pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uua),Uu)) ) ).

% ATP.lambda_156
tff(fact_8276_ATP_Olambda__157,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_aa(Uu),Uua))
    <=> pp(aa_set_real_bool(member_real(Uua),Uu)) ) ).

% ATP.lambda_157
tff(fact_8277_ATP_Olambda__158,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_ab(Uu),Uua))
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_158
tff(fact_8278_ATP_Olambda__159,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_ac(Uu),Uua))
    <=> pp(aa_set_set_nat_bool(member_set_nat(Uua),Uu)) ) ).

% ATP.lambda_159
tff(fact_8279_ATP_Olambda__160,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ad(Uu),Uua))
    <=> pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_160
tff(fact_8280_ATP_Olambda__161,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ae(Uu),Uua))
    <=> pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_161
tff(fact_8281_ATP_Olambda__162,axiom,
    ! [Uu: extended_enat,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(aTP_Lamm_es,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_162
tff(fact_8282_ATP_Olambda__163,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_et,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_163
tff(fact_8283_ATP_Olambda__164,axiom,
    ! [Uu: list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_eq(Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_164
tff(fact_8284_ATP_Olambda__165,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_er,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_165
tff(fact_8285_ATP_Olambda__166,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_eu,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_166
tff(fact_8286_ATP_Olambda__167,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ev,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_167
tff(fact_8287_ATP_Olambda__168,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_abt(Uu),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu) ).

% ATP.lambda_168
tff(fact_8288_ATP_Olambda__169,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nr(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),one_one_nat)) ).

% ATP.lambda_169
tff(fact_8289_ATP_Olambda__170,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nq(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)) ).

% ATP.lambda_170
tff(fact_8290_ATP_Olambda__171,axiom,
    ! [Uu: fun_na8797796813480665199d_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_li(Uu),Uua) = aa_nat_Extended_enat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_171
tff(fact_8291_ATP_Olambda__172,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_qw(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_172
tff(fact_8292_ATP_Olambda__173,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_kz(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_173
tff(fact_8293_ATP_Olambda__174,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_aba(Uu),Uua))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_174
tff(fact_8294_ATP_Olambda__175,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ky(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_175
tff(fact_8295_ATP_Olambda__176,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lh(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_176
tff(fact_8296_ATP_Olambda__177,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_aca,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_abz(Uu),Uua)) ).

% ATP.lambda_177
tff(fact_8297_ATP_Olambda__178,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_zw,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_zv(Uu),Uua)) ).

% ATP.lambda_178
tff(fact_8298_ATP_Olambda__179,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_zu,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_zt(Uu),Uua)) ).

% ATP.lambda_179
tff(fact_8299_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_zs,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_zr(Uu),Uua)) ).

% ATP.lambda_180
tff(fact_8300_ATP_Olambda__181,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_zq,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_zp(Uu),Uua)) ).

% ATP.lambda_181
tff(fact_8301_ATP_Olambda__182,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_zn,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_zm(Uu),Uua)) ).

% ATP.lambda_182
tff(fact_8302_ATP_Olambda__183,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_aae(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_aad(Uu),Uua)) ).

% ATP.lambda_183
tff(fact_8303_ATP_Olambda__184,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_abk(Uu),Uua) = aa_real_complex(cis,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_184
tff(fact_8304_ATP_Olambda__185,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_wz(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_185
tff(fact_8305_ATP_Olambda__186,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_xa(Uu),Uua) = times_times_real(aa_real_real(minus_minus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_186
tff(fact_8306_ATP_Olambda__187,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat9102221060835343569d_enat(aTP_Lamm_nw(Uu),Uua) = times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Uu),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uua))) ).

% ATP.lambda_187
tff(fact_8307_ATP_Olambda__188,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_nv(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_188
tff(fact_8308_ATP_Olambda__189,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_nx(Uu),Uua) = times_times_real(aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_189
tff(fact_8309_ATP_Olambda__190,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_nz(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_190
tff(fact_8310_ATP_Olambda__191,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_ny(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_191
tff(fact_8311_ATP_Olambda__192,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_abq(Uu),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).

% ATP.lambda_192
tff(fact_8312_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_acb(Uu),Uua) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uua)) ).

% ATP.lambda_193
tff(fact_8313_ATP_Olambda__194,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_acd(Uu),Uua) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)) ).

% ATP.lambda_194
tff(fact_8314_ATP_Olambda__195,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abj,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_195
tff(fact_8315_ATP_Olambda__196,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abi,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_196
tff(fact_8316_ATP_Olambda__197,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zj,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_197
tff(fact_8317_ATP_Olambda__198,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zi,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_198
tff(fact_8318_ATP_Olambda__199,axiom,
    ! [Uu: set_Extended_enat,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_fk(Uu),Uua))
    <=> ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uua),Uu)) ) ).

% ATP.lambda_199
tff(fact_8319_ATP_Olambda__200,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_fl(Uu),Uua))
    <=> ~ pp(aa_set_real_bool(member_real(Uua),Uu)) ) ).

% ATP.lambda_200
tff(fact_8320_ATP_Olambda__201,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_fm(Uu),Uua))
    <=> ~ member_list_nat(Uua,Uu) ) ).

% ATP.lambda_201
tff(fact_8321_ATP_Olambda__202,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_fn(Uu),Uua))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(Uua),Uu)) ) ).

% ATP.lambda_202
tff(fact_8322_ATP_Olambda__203,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_fo(Uu),Uua))
    <=> ~ pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_203
tff(fact_8323_ATP_Olambda__204,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_fp(Uu),Uua))
    <=> ~ pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_204
tff(fact_8324_ATP_Olambda__205,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_xk(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_205
tff(fact_8325_ATP_Olambda__206,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aau(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_206
tff(fact_8326_ATP_Olambda__207,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_lx(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_207
tff(fact_8327_ATP_Olambda__208,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_ma(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_208
tff(fact_8328_ATP_Olambda__209,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_lz(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_209
tff(fact_8329_ATP_Olambda__210,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_ly(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_210
tff(fact_8330_ATP_Olambda__211,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_lw(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_211
tff(fact_8331_ATP_Olambda__212,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: extended_enat] : aa_Ext2541572767299617110t_real(aTP_Lamm_sa(Uu),Uua) = aa_real_real(ln_ln_real,aa_Ext2541572767299617110t_real(Uu,Uua)) ).

% ATP.lambda_212
tff(fact_8332_ATP_Olambda__213,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_real(aTP_Lamm_ry(Uu),Uua) = aa_real_real(ln_ln_real,aa_complex_real(Uu,Uua)) ).

% ATP.lambda_213
tff(fact_8333_ATP_Olambda__214,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_rw(Uu),Uua) = aa_real_real(ln_ln_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_214
tff(fact_8334_ATP_Olambda__215,axiom,
    ! [Uu: fun_set_nat_real,Uua: set_nat] : aa_set_nat_real(aTP_Lamm_rx(Uu),Uua) = aa_real_real(ln_ln_real,aa_set_nat_real(Uu,Uua)) ).

% ATP.lambda_215
tff(fact_8335_ATP_Olambda__216,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sb(Uu),Uua) = aa_real_real(ln_ln_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_216
tff(fact_8336_ATP_Olambda__217,axiom,
    ! [Uu: fun_int_real,Uua: int] : aa_int_real(aTP_Lamm_rz(Uu),Uua) = aa_real_real(ln_ln_real,aa_int_real(Uu,Uua)) ).

% ATP.lambda_217
tff(fact_8337_ATP_Olambda__218,axiom,
    ! [Uu: fun_na8797796813480665199d_enat,Uua: nat] : aa_nat9102221060835343569d_enat(aTP_Lamm_pw(Uu),Uua) = times_7803423173614009249d_enat(aa_nat_Extended_enat(Uu,Uua)) ).

% ATP.lambda_218
tff(fact_8338_ATP_Olambda__219,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_pv(Uu),Uua) = times_times_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_219
tff(fact_8339_ATP_Olambda__220,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_pu(Uu),Uua) = times_times_real(aa_nat_real(Uu,Uua)) ).

% ATP.lambda_220
tff(fact_8340_ATP_Olambda__221,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_py(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_221
tff(fact_8341_ATP_Olambda__222,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_px(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_222
tff(fact_8342_ATP_Olambda__223,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_hj(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_223
tff(fact_8343_ATP_Olambda__224,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_hk(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_224
tff(fact_8344_ATP_Olambda__225,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_oc(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_225
tff(fact_8345_ATP_Olambda__226,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_hl(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_226
tff(fact_8346_ATP_Olambda__227,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_hm(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_227
tff(fact_8347_ATP_Olambda__228,axiom,
    ! [Uu: fun_na8797796813480665199d_enat,Uua: nat] : aa_nat9102221060835343569d_enat(aTP_Lamm_ro(Uu),Uua) = plus_p3455044024723400733d_enat(aa_nat_Extended_enat(Uu,Uua)) ).

% ATP.lambda_228
tff(fact_8348_ATP_Olambda__229,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_rn(Uu),Uua) = plus_plus_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_229
tff(fact_8349_ATP_Olambda__230,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_rq(Uu),Uua) = plus_plus_real(aa_nat_real(Uu,Uua)) ).

% ATP.lambda_230
tff(fact_8350_ATP_Olambda__231,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_rp(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_231
tff(fact_8351_ATP_Olambda__232,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_rm(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_232
tff(fact_8352_ATP_Olambda__233,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_yq(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_233
tff(fact_8353_ATP_Olambda__234,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_yp(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_234
tff(fact_8354_ATP_Olambda__235,axiom,
    ! [Uu: fun_real_bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ff(Uu),Uua))
    <=> ~ pp(aa_real_bool(Uu,Uua)) ) ).

% ATP.lambda_235
tff(fact_8355_ATP_Olambda__236,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_fg(Uu),Uua))
    <=> ~ pp(aa_list_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_236
tff(fact_8356_ATP_Olambda__237,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_fh(Uu),Uua))
    <=> ~ pp(aa_set_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_237
tff(fact_8357_ATP_Olambda__238,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_fi(Uu),Uua))
    <=> ~ pp(aa_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_238
tff(fact_8358_ATP_Olambda__239,axiom,
    ! [Uu: fun_int_bool,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_fj(Uu),Uua))
    <=> ~ pp(aa_int_bool(Uu,Uua)) ) ).

% ATP.lambda_239
tff(fact_8359_ATP_Olambda__240,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_abr(Uu),Uua) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,Uu)) ).

% ATP.lambda_240
tff(fact_8360_ATP_Olambda__241,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_np(Uu),Uua),Uub) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub),aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_241
tff(fact_8361_ATP_Olambda__242,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ep(Uu),Uua),Uub) = if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),Uua)),Uub)) ).

% ATP.lambda_242
tff(fact_8362_ATP_Olambda__243,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_eo(Uu),Uua),Uub) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(Uu)),Uub),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Uua)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),numeral_numeral_int(Uu))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Uua),Uub)) ).

% ATP.lambda_243
tff(fact_8363_ATP_Olambda__244,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sr(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_set_nat_bool(member_nat(Uub),Uu),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_244
tff(fact_8364_ATP_Olambda__245,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_st(Uu),Uua),Uub) = aa_real_real(if_real(aa_set_nat_bool(member_nat(Uub),Uu),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_245
tff(fact_8365_ATP_Olambda__246,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ss(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_set_nat_bool(member_nat(Uub),Uu)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_246
tff(fact_8366_ATP_Olambda__247,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_sq(Uu),Uua),Uub) = aa_int_int(if_int(aa_set_nat_bool(member_nat(Uub),Uu),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_247
tff(fact_8367_ATP_Olambda__248,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aTP_Lamm_hg(Uu),Uua),Uub) = aa_real_real(if_real(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uu),Uub),aa_Ext2541572767299617110t_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_248
tff(fact_8368_ATP_Olambda__249,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: extended_enat] : aa_Extended_enat_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_hc(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uu),Uub)),aa_Extended_enat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_249
tff(fact_8369_ATP_Olambda__250,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_he(Uu),Uua),Uub) = aa_real_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_250
tff(fact_8370_ATP_Olambda__251,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ha(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub)),aa_complex_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_251
tff(fact_8371_ATP_Olambda__252,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_hd(Uu),Uua),Uub) = aa_real_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_252
tff(fact_8372_ATP_Olambda__253,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_253
tff(fact_8373_ATP_Olambda__254,axiom,
    ! [Uu: real,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_hh(Uu),Uua),Uub) = aa_int_int(if_int(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_254
tff(fact_8374_ATP_Olambda__255,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_hi(Uu),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_255
tff(fact_8375_ATP_Olambda__256,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_hf(Uu),Uua),Uub) = aa_real_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_256
tff(fact_8376_ATP_Olambda__257,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_hb(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_257
tff(fact_8377_ATP_Olambda__258,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: extended_enat] : aa_Extended_enat_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_oq(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uu),Uub)),aa_Extended_enat_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_258
tff(fact_8378_ATP_Olambda__259,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex3482447378695272101at_int,Uub: extended_enat] : aa_Extended_enat_int(aa_fun1695976788104708604at_int(aTP_Lamm_ot(Uu),Uua),Uub) = aa_int_int(if_int(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uu),Uub),aa_Extended_enat_int(Uua,Uub)),one_one_int) ).

% ATP.lambda_259
tff(fact_8379_ATP_Olambda__260,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_ow(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_260
tff(fact_8380_ATP_Olambda__261,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_oo(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub)),aa_complex_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_261
tff(fact_8381_ATP_Olambda__262,axiom,
    ! [Uu: complex,Uua: fun_complex_int,Uub: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_os(Uu),Uua),Uub) = aa_int_int(if_int(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_int(Uua,Uub)),one_one_int) ).

% ATP.lambda_262
tff(fact_8382_ATP_Olambda__263,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_ou(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_263
tff(fact_8383_ATP_Olambda__264,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_on(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_264
tff(fact_8384_ATP_Olambda__265,axiom,
    ! [Uu: real,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_or(Uu),Uua),Uub) = aa_int_int(if_int(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_int(Uua,Uub)),one_one_int) ).

% ATP.lambda_265
tff(fact_8385_ATP_Olambda__266,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ov(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub),aa_nat_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_266
tff(fact_8386_ATP_Olambda__267,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_op(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_267
tff(fact_8387_ATP_Olambda__268,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aTP_Lamm_gw(Uu),Uua),Uub) = aa_real_real(if_real(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uub),Uu),aa_Ext2541572767299617110t_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_268
tff(fact_8388_ATP_Olambda__269,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: extended_enat] : aa_Extended_enat_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_gs(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uub),Uu)),aa_Extended_enat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_269
tff(fact_8389_ATP_Olambda__270,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_gu(Uu),Uua),Uub) = aa_real_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_270
tff(fact_8390_ATP_Olambda__271,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_gq(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu)),aa_complex_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_271
tff(fact_8391_ATP_Olambda__272,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gt(Uu),Uua),Uub) = aa_real_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_272
tff(fact_8392_ATP_Olambda__273,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gp(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_273
tff(fact_8393_ATP_Olambda__274,axiom,
    ! [Uu: real,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_gx(Uu),Uua),Uub) = aa_int_int(if_int(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_274
tff(fact_8394_ATP_Olambda__275,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_se(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_275
tff(fact_8395_ATP_Olambda__276,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sd(Uu),Uua),Uub) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_276
tff(fact_8396_ATP_Olambda__277,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sc(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_277
tff(fact_8397_ATP_Olambda__278,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_gy(Uu),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_278
tff(fact_8398_ATP_Olambda__279,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_gv(Uu),Uua),Uub) = aa_real_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_279
tff(fact_8399_ATP_Olambda__280,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_gr(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_280
tff(fact_8400_ATP_Olambda__281,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: extended_enat] : aa_Extended_enat_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_og(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uub),Uu)),aa_Extended_enat_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_281
tff(fact_8401_ATP_Olambda__282,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex3482447378695272101at_int,Uub: extended_enat] : aa_Extended_enat_int(aa_fun1695976788104708604at_int(aTP_Lamm_oj(Uu),Uua),Uub) = aa_int_int(if_int(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uub),Uu),aa_Extended_enat_int(Uua,Uub)),one_one_int) ).

% ATP.lambda_282
tff(fact_8402_ATP_Olambda__283,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_om(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_283
tff(fact_8403_ATP_Olambda__284,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_oe(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu)),aa_complex_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_284
tff(fact_8404_ATP_Olambda__285,axiom,
    ! [Uu: complex,Uua: fun_complex_int,Uub: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_oi(Uu),Uua),Uub) = aa_int_int(if_int(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_int(Uua,Uub)),one_one_int) ).

% ATP.lambda_285
tff(fact_8405_ATP_Olambda__286,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_ok(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_286
tff(fact_8406_ATP_Olambda__287,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_od(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_287
tff(fact_8407_ATP_Olambda__288,axiom,
    ! [Uu: real,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_oh(Uu),Uua),Uub) = aa_int_int(if_int(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_int(Uua,Uub)),one_one_int) ).

% ATP.lambda_288
tff(fact_8408_ATP_Olambda__289,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ol(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_289
tff(fact_8409_ATP_Olambda__290,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_of(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_290
tff(fact_8410_ATP_Olambda__291,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sn(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uu,Uub),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_291
tff(fact_8411_ATP_Olambda__292,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sp(Uu),Uua),Uub) = aa_real_real(if_real(aa_nat_bool(Uu,Uub),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_292
tff(fact_8412_ATP_Olambda__293,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_so(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(Uu,Uub)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_293
tff(fact_8413_ATP_Olambda__294,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_sm(Uu),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(Uu,Uub),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_294
tff(fact_8414_ATP_Olambda__295,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1090353380788292083t_real(aTP_Lamm_jr(Uu),Uua),Uub) = aa_real_real(if_real(aa_Ext2772979874139974381t_bool(Uua,Uub),aa_Ext2541572767299617110t_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_295
tff(fact_8415_ATP_Olambda__296,axiom,
    ! [Uu: fun_Ex7660298398204468809at_nat,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] : aa_Extended_enat_nat(aa_fun943526657271735063at_nat(aTP_Lamm_jn(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Ext2772979874139974381t_bool(Uua,Uub)),aa_Extended_enat_nat(Uu,Uub)),zero_zero_nat) ).

% ATP.lambda_296
tff(fact_8416_ATP_Olambda__297,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_jp(Uu),Uua),Uub) = aa_real_real(if_real(aa_complex_bool(Uua,Uub),aa_complex_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_297
tff(fact_8417_ATP_Olambda__298,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_bool,Uub: complex] : aa_complex_nat(aa_fun9038227978352879617ex_nat(aTP_Lamm_jl(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(Uua,Uub)),aa_complex_nat(Uu,Uub)),zero_zero_nat) ).

% ATP.lambda_298
tff(fact_8418_ATP_Olambda__299,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_jo(Uu),Uua),Uub) = aa_real_real(if_real(aa_real_bool(Uua,Uub),aa_real_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_299
tff(fact_8419_ATP_Olambda__300,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_bool,Uub: real] : aa_real_nat(aa_fun1504556115071737857al_nat(aTP_Lamm_jj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(Uua,Uub)),aa_real_nat(Uu,Uub)),zero_zero_nat) ).

% ATP.lambda_300
tff(fact_8420_ATP_Olambda__301,axiom,
    ! [Uu: fun_real_int,Uua: fun_real_bool,Uub: real] : aa_real_int(aa_fun6550077132417316957al_int(aTP_Lamm_js(Uu),Uua),Uub) = aa_int_int(if_int(aa_real_bool(Uua,Uub),aa_real_int(Uu,Uub)),zero_zero_int) ).

% ATP.lambda_301
tff(fact_8421_ATP_Olambda__302,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_bool,Uub: nat] : aa_nat_int(aa_fun4986539647837908701at_int(aTP_Lamm_jt(Uu),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(Uua,Uub),aa_nat_int(Uu,Uub)),zero_zero_int) ).

% ATP.lambda_302
tff(fact_8422_ATP_Olambda__303,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_jq(Uu),Uua),Uub) = aa_real_real(if_real(aa_int_bool(Uua,Uub),aa_int_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_303
tff(fact_8423_ATP_Olambda__304,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_bool,Uub: int] : aa_int_nat(aa_fun7908831221058585985nt_nat(aTP_Lamm_jm(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(Uua,Uub)),aa_int_nat(Uu,Uub)),zero_zero_nat) ).

% ATP.lambda_304
tff(fact_8424_ATP_Olambda__305,axiom,
    ! [Uu: fun_Ex7660298398204468809at_nat,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] : aa_Extended_enat_nat(aa_fun943526657271735063at_nat(aTP_Lamm_qc(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Ext2772979874139974381t_bool(Uua,Uub)),aa_Extended_enat_nat(Uu,Uub)),one_one_nat) ).

% ATP.lambda_305
tff(fact_8425_ATP_Olambda__306,axiom,
    ! [Uu: fun_Ex3482447378695272101at_int,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] : aa_Extended_enat_int(aa_fun5989047674617314163at_int(aTP_Lamm_qf(Uu),Uua),Uub) = aa_int_int(if_int(aa_Ext2772979874139974381t_bool(Uua,Uub),aa_Extended_enat_int(Uu,Uub)),one_one_int) ).

% ATP.lambda_306
tff(fact_8426_ATP_Olambda__307,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_bool,Uub: complex] : aa_complex_complex(aa_fun1034237174562290143omplex(aTP_Lamm_qi(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(Uua,Uub),aa_complex_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_307
tff(fact_8427_ATP_Olambda__308,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_bool,Uub: complex] : aa_complex_nat(aa_fun9038227978352879617ex_nat(aTP_Lamm_qa(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(Uua,Uub)),aa_complex_nat(Uu,Uub)),one_one_nat) ).

% ATP.lambda_308
tff(fact_8428_ATP_Olambda__309,axiom,
    ! [Uu: fun_complex_int,Uua: fun_complex_bool,Uub: complex] : aa_complex_int(aa_fun4860376958843682909ex_int(aTP_Lamm_qe(Uu),Uua),Uub) = aa_int_int(if_int(aa_complex_bool(Uua,Uub),aa_complex_int(Uu,Uub)),one_one_int) ).

% ATP.lambda_309
tff(fact_8429_ATP_Olambda__310,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_qg(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(Uua,Uub),aa_real_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_310
tff(fact_8430_ATP_Olambda__311,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_bool,Uub: real] : aa_real_nat(aa_fun1504556115071737857al_nat(aTP_Lamm_pz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(Uua,Uub)),aa_real_nat(Uu,Uub)),one_one_nat) ).

% ATP.lambda_311
tff(fact_8431_ATP_Olambda__312,axiom,
    ! [Uu: fun_real_int,Uua: fun_real_bool,Uub: real] : aa_real_int(aa_fun6550077132417316957al_int(aTP_Lamm_qd(Uu),Uua),Uub) = aa_int_int(if_int(aa_real_bool(Uua,Uub),aa_real_int(Uu,Uub)),one_one_int) ).

% ATP.lambda_312
tff(fact_8432_ATP_Olambda__313,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_bool,Uub: nat] : aa_nat_complex(aa_fun8588439219086927967omplex(aTP_Lamm_qh(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uua,Uub),aa_nat_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_313
tff(fact_8433_ATP_Olambda__314,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_bool,Uub: int] : aa_int_nat(aa_fun7908831221058585985nt_nat(aTP_Lamm_qb(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(Uua,Uub)),aa_int_nat(Uu,Uub)),one_one_nat) ).

% ATP.lambda_314
tff(fact_8434_ATP_Olambda__315,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ug(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_315
tff(fact_8435_ATP_Olambda__316,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ue(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_316
tff(fact_8436_ATP_Olambda__317,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ui(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_317
tff(fact_8437_ATP_Olambda__318,axiom,
    ! [Uu: fun_Ex7123704652682714774at_nat,Uua: nat,Uub: extended_enat] : aa_Extended_enat_nat(aa_nat887872048497836987at_nat(aTP_Lamm_iy(Uu),Uua),Uub) = aa_nat_nat(aa_Ext158989295719858759at_nat(Uu,Uub),Uua) ).

% ATP.lambda_318
tff(fact_8438_ATP_Olambda__319,axiom,
    ! [Uu: fun_Ex4633621507750988018at_int,Uua: nat,Uub: extended_enat] : aa_Extended_enat_int(aa_nat5933393065843416087at_int(aTP_Lamm_pe(Uu),Uua),Uub) = aa_nat_int(aa_Ext5204510313065437859at_int(Uu,Uub),Uua) ).

% ATP.lambda_319
tff(fact_8439_ATP_Olambda__320,axiom,
    ! [Uu: fun_Ex6820543082444009806nt_int,Uua: int,Uub: extended_enat] : aa_Extended_enat_int(aa_int4341511938324142267at_int(aTP_Lamm_ig(Uu),Uua),Uub) = aa_int_int(aa_Ext6203864024966492415nt_int(Uu,Uub),Uua) ).

% ATP.lambda_320
tff(fact_8440_ATP_Olambda__321,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_ip(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_321
tff(fact_8441_ATP_Olambda__322,axiom,
    ! [Uu: fun_co7370231399432144932at_int,Uua: nat,Uub: complex] : aa_complex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_pb(Uu),Uua),Uub) = aa_nat_int(aa_com2205869648058525157at_int(Uu,Uub),Uua) ).

% ATP.lambda_322
tff(fact_8442_ATP_Olambda__323,axiom,
    ! [Uu: fun_co333780937270390912nt_int,Uua: int,Uub: complex] : aa_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_ib(Uu),Uua),Uub) = aa_int_int(aa_com3205223359959579713nt_int(Uu,Uub),Uua) ).

% ATP.lambda_323
tff(fact_8443_ATP_Olambda__324,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_jd(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_324
tff(fact_8444_ATP_Olambda__325,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_il(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_325
tff(fact_8445_ATP_Olambda__326,axiom,
    ! [Uu: fun_real_fun_nat_int,Uua: nat,Uub: real] : aa_real_int(aa_nat_fun_real_int(aTP_Lamm_oy(Uu),Uua),Uub) = aa_nat_int(aa_real_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_326
tff(fact_8446_ATP_Olambda__327,axiom,
    ! [Uu: fun_real_fun_int_int,Uua: int,Uub: real] : aa_real_int(aa_int_fun_real_int(aTP_Lamm_ht(Uu),Uua),Uub) = aa_int_int(aa_real_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_327
tff(fact_8447_ATP_Olambda__328,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jh(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_328
tff(fact_8448_ATP_Olambda__329,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ts(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_329
tff(fact_8449_ATP_Olambda__330,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tp(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_330
tff(fact_8450_ATP_Olambda__331,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_tv(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_331
tff(fact_8451_ATP_Olambda__332,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_hx(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_332
tff(fact_8452_ATP_Olambda__333,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_it(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_333
tff(fact_8453_ATP_Olambda__334,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yi(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_yh(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_334
tff(fact_8454_ATP_Olambda__335,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_yg(Uu),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_yf(Uu),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_335
tff(fact_8455_ATP_Olambda__336,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ye(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_yd(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_336
tff(fact_8456_ATP_Olambda__337,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_yc(Uu),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_yb(Uu),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_337
tff(fact_8457_ATP_Olambda__338,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ya(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_xz(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_338
tff(fact_8458_ATP_Olambda__339,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_xy(Uu),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_xx(Uu),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_339
tff(fact_8459_ATP_Olambda__340,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_sw(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_complex),zero_zero_complex)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_340
tff(fact_8460_ATP_Olambda__341,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sv(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_real),zero_zero_real)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_341
tff(fact_8461_ATP_Olambda__342,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_su(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_int),zero_zero_int)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_342
tff(fact_8462_ATP_Olambda__343,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_tw(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_tv(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_343
tff(fact_8463_ATP_Olambda__344,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ty(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tp(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_344
tff(fact_8464_ATP_Olambda__345,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_tt(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ts(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_345
tff(fact_8465_ATP_Olambda__346,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tq(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tp(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_346
tff(fact_8466_ATP_Olambda__347,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aaf(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_347
tff(fact_8467_ATP_Olambda__348,axiom,
    ! [Uu: real,Uua: fun_na2978037997607195054t_real,Uub: nat] : aa_nat_real(aa_fun2748096013305118955t_real(aTP_Lamm_xr(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_Ext2541572767299617110t_real(aa_nat6736956760766371991t_real(Uua,Uub),zero_z5237406670263579293d_enat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_348
tff(fact_8468_ATP_Olambda__349,axiom,
    ! [Uu: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_xq(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_complex_real(aa_nat480766339200631653x_real(Uua,Uub),zero_zero_complex)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_349
tff(fact_8469_ATP_Olambda__350,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_xo(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_350
tff(fact_8470_ATP_Olambda__351,axiom,
    ! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_xn(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_351
tff(fact_8471_ATP_Olambda__352,axiom,
    ! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_xp(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_352
tff(fact_8472_ATP_Olambda__353,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_uo(Uu),Uua),Uub))
    <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tn(Uu),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_real ) ) ).

% ATP.lambda_353
tff(fact_8473_ATP_Olambda__354,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_un(Uu),Uua),Uub))
    <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tm(Uu),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_354
tff(fact_8474_ATP_Olambda__355,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_kr(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_355
tff(fact_8475_ATP_Olambda__356,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ks(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_356
tff(fact_8476_ATP_Olambda__357,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aac(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_357
tff(fact_8477_ATP_Olambda__358,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_abo(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_358
tff(fact_8478_ATP_Olambda__359,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_yl(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu,Uub))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_359
tff(fact_8479_ATP_Olambda__360,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yn(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu,Uub))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_360
tff(fact_8480_ATP_Olambda__361,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_abp(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_361
tff(fact_8481_ATP_Olambda__362,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_cr(Uu),Uua),Uub))
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Uub)),Uua)) ) ) ).

% ATP.lambda_362
tff(fact_8482_ATP_Olambda__363,axiom,
    ! [Uu: set_Extended_enat,Uua: nat,Uub: list_Extended_enat] :
      ( pp(aa_lis1364384421681148007t_bool(aa_nat4017767907309459068t_bool(aTP_Lamm_cq(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(set_Extended_enat2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3941691890525107288d_enat(Uub)),Uua)) ) ) ).

% ATP.lambda_363
tff(fact_8483_ATP_Olambda__364,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_cp(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua)) ) ) ).

% ATP.lambda_364
tff(fact_8484_ATP_Olambda__365,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_cs(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Uub)),Uua)) ) ) ).

% ATP.lambda_365
tff(fact_8485_ATP_Olambda__366,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_ct(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Uub)),Uua)) ) ) ).

% ATP.lambda_366
tff(fact_8486_ATP_Olambda__367,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_cm(Uu),Uua),Uub))
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Uub) = Uua ) ) ) ).

% ATP.lambda_367
tff(fact_8487_ATP_Olambda__368,axiom,
    ! [Uu: set_Extended_enat,Uua: nat,Uub: list_Extended_enat] :
      ( pp(aa_lis1364384421681148007t_bool(aa_nat4017767907309459068t_bool(aTP_Lamm_cl(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(set_Extended_enat2(Uub)),Uu))
        & ( size_s3941691890525107288d_enat(Uub) = Uua ) ) ) ).

% ATP.lambda_368
tff(fact_8488_ATP_Olambda__369,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_ck(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_369
tff(fact_8489_ATP_Olambda__370,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_cn(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Uub)),Uu))
        & ( aa_list_nat_nat(size_size_list_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_370
tff(fact_8490_ATP_Olambda__371,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_co(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu))
        & ( aa_list_int_nat(size_size_list_int,Uub) = Uua ) ) ) ).

% ATP.lambda_371
tff(fact_8491_ATP_Olambda__372,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_abn(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_372
tff(fact_8492_ATP_Olambda__373,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zg(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_373
tff(fact_8493_ATP_Olambda__374,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_nat8857836526276733660t_bool(aTP_Lamm_zl(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),pow_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uu))))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_374
tff(fact_8494_ATP_Olambda__375,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_abs(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uub)) ) ) ).

% ATP.lambda_375
tff(fact_8495_ATP_Olambda__376,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zh(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),aa_nat_nat(suc,Uua))) ) ) ).

% ATP.lambda_376
tff(fact_8496_ATP_Olambda__377,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_yk(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uu),Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_377
tff(fact_8497_ATP_Olambda__378,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ym(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(diffs_real(Uu),Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_378
tff(fact_8498_ATP_Olambda__379,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ul(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_379
tff(fact_8499_ATP_Olambda__380,axiom,
    ! [Uu: extended_enat,Uua: set_Extended_enat,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_set3329637325543298558t_bool(aTP_Lamm_fq(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uua)) ) ) ).

% ATP.lambda_380
tff(fact_8500_ATP_Olambda__381,axiom,
    ! [Uu: real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_fr(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_381
tff(fact_8501_ATP_Olambda__382,axiom,
    ! [Uu: list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_fs(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_382
tff(fact_8502_ATP_Olambda__383,axiom,
    ! [Uu: set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_ft(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua)) ) ) ).

% ATP.lambda_383
tff(fact_8503_ATP_Olambda__384,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_fu(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_384
tff(fact_8504_ATP_Olambda__385,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_fv(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_385
tff(fact_8505_ATP_Olambda__386,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_zc(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uua))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_386
tff(fact_8506_ATP_Olambda__387,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ew(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_387
tff(fact_8507_ATP_Olambda__388,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_yo(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_388
tff(fact_8508_ATP_Olambda__389,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_el(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_389
tff(fact_8509_ATP_Olambda__390,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_em(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_390
tff(fact_8510_ATP_Olambda__391,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_en(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_391
tff(fact_8511_ATP_Olambda__392,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun4759809662886658982t_bool(aTP_Lamm_bs(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & pp(aa_Ext2772979874139974381t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_392
tff(fact_8512_ATP_Olambda__393,axiom,
    ! [Uu: set_complex,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_jk(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_393
tff(fact_8513_ATP_Olambda__394,axiom,
    ! [Uu: set_real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_bt(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_394
tff(fact_8514_ATP_Olambda__395,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_bu(Uu),Uua),Uub))
    <=> ( member_list_nat(Uub,Uu)
        & pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_395
tff(fact_8515_ATP_Olambda__396,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_bv(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uu))
        & pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_396
tff(fact_8516_ATP_Olambda__397,axiom,
    ! [Uu: set_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_bw(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_397
tff(fact_8517_ATP_Olambda__398,axiom,
    ! [Uu: set_int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_bx(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_398
tff(fact_8518_ATP_Olambda__399,axiom,
    ! [Uu: fun_Ex8563670519792538868t_bool,Uua: extended_enat,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(aTP_Lamm_gj(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_Ext2772979874139974381t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_399
tff(fact_8519_ATP_Olambda__400,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_gk(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_real_bool(Uu,Uub)) ) ) ).

% ATP.lambda_400
tff(fact_8520_ATP_Olambda__401,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_gh(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_list_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_401
tff(fact_8521_ATP_Olambda__402,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_gi(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_set_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_402
tff(fact_8522_ATP_Olambda__403,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gl(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_403
tff(fact_8523_ATP_Olambda__404,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_gm(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_404
tff(fact_8524_ATP_Olambda__405,axiom,
    ! [Uu: fun_Ex8563670519792538868t_bool,Uua: extended_enat,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(aTP_Lamm_gd(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_Ext2772979874139974381t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_405
tff(fact_8525_ATP_Olambda__406,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ge(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_real_bool(Uu,Uub)) ) ) ).

% ATP.lambda_406
tff(fact_8526_ATP_Olambda__407,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_gb(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_list_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_407
tff(fact_8527_ATP_Olambda__408,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_gc(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_set_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_408
tff(fact_8528_ATP_Olambda__409,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gf(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_409
tff(fact_8529_ATP_Olambda__410,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_gg(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_410
tff(fact_8530_ATP_Olambda__411,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun338017137003533149t_bool(aTP_Lamm_ej(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & ( aa_Ext2541572767299617110t_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_411
tff(fact_8531_ATP_Olambda__412,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun1105177683714449665t_bool(aTP_Lamm_dz(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & ( aa_Extended_enat_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_412
tff(fact_8532_ATP_Olambda__413,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_ef(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_413
tff(fact_8533_ATP_Olambda__414,axiom,
    ! [Uu: set_complex,Uua: fun_complex_nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun4400319914713427657x_bool(aTP_Lamm_dv(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_414
tff(fact_8534_ATP_Olambda__415,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_eb(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_415
tff(fact_8535_ATP_Olambda__416,axiom,
    ! [Uu: set_real,Uua: fun_real_nat,Uub: real] :
      ( pp(aa_real_bool(aa_fun8134817056309787853l_bool(aTP_Lamm_dr(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_416
tff(fact_8536_ATP_Olambda__417,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_ed(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_417
tff(fact_8537_ATP_Olambda__418,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1783405840085286277t_bool(aTP_Lamm_dt(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_418
tff(fact_8538_ATP_Olambda__419,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_eh(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_419
tff(fact_8539_ATP_Olambda__420,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: int] :
      ( pp(aa_int_bool(aa_fun5627573752281863885t_bool(aTP_Lamm_dx(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_420
tff(fact_8540_ATP_Olambda__421,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun1105177683714449665t_bool(aTP_Lamm_df(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & ( aa_Extended_enat_nat(Uua,Uub) != one_one_nat ) ) ) ).

% ATP.lambda_421
tff(fact_8541_ATP_Olambda__422,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex3482447378695272101at_int,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun821126609344339805t_bool(aTP_Lamm_dp(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & ( aa_Extended_enat_int(Uua,Uub) != one_one_int ) ) ) ).

% ATP.lambda_422
tff(fact_8542_ATP_Olambda__423,axiom,
    ! [Uu: set_complex,Uua: fun_complex_nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun4400319914713427657x_bool(aTP_Lamm_db(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_nat(Uua,Uub) != one_one_nat ) ) ) ).

% ATP.lambda_423
tff(fact_8543_ATP_Olambda__424,axiom,
    ! [Uu: set_complex,Uua: fun_complex_int,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun4637897056364002669x_bool(aTP_Lamm_dl(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_int(Uua,Uub) != one_one_int ) ) ) ).

% ATP.lambda_424
tff(fact_8544_ATP_Olambda__425,axiom,
    ! [Uu: set_real,Uua: fun_real_nat,Uub: real] :
      ( pp(aa_real_bool(aa_fun8134817056309787853l_bool(aTP_Lamm_cx(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_nat(Uua,Uub) != one_one_nat ) ) ) ).

% ATP.lambda_425
tff(fact_8545_ATP_Olambda__426,axiom,
    ! [Uu: set_real,Uua: fun_real_int,Uub: real] :
      ( pp(aa_real_bool(aa_fun2745684120339038577l_bool(aTP_Lamm_dh(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_int(Uua,Uub) != one_one_int ) ) ) ).

% ATP.lambda_426
tff(fact_8546_ATP_Olambda__427,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1783405840085286277t_bool(aTP_Lamm_cz(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_nat(Uua,Uub) != one_one_nat ) ) ) ).

% ATP.lambda_427
tff(fact_8547_ATP_Olambda__428,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8175233552433550377t_bool(aTP_Lamm_dj(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_int(Uua,Uub) != one_one_int ) ) ) ).

% ATP.lambda_428
tff(fact_8548_ATP_Olambda__429,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: int] :
      ( pp(aa_int_bool(aa_fun5627573752281863885t_bool(aTP_Lamm_dd(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_nat(Uua,Uub) != one_one_nat ) ) ) ).

% ATP.lambda_429
tff(fact_8549_ATP_Olambda__430,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: int] :
      ( pp(aa_int_bool(aa_fun2796029427775352177t_bool(aTP_Lamm_dn(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_int(Uua,Uub) != one_one_int ) ) ) ).

% ATP.lambda_430
tff(fact_8550_ATP_Olambda__431,axiom,
    ! [Uu: set_Extended_enat,Uua: set_Extended_enat,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_set3329637325543298558t_bool(aTP_Lamm_ez(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uua)) ) ) ).

% ATP.lambda_431
tff(fact_8551_ATP_Olambda__432,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_fa(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ~ pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_432
tff(fact_8552_ATP_Olambda__433,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_fb(Uu),Uua),Uub))
    <=> ( member_list_nat(Uub,Uu)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_433
tff(fact_8553_ATP_Olambda__434,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_fc(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uu))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua)) ) ) ).

% ATP.lambda_434
tff(fact_8554_ATP_Olambda__435,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_fe(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ~ pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_435
tff(fact_8555_ATP_Olambda__436,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_fd(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ~ pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_436
tff(fact_8556_ATP_Olambda__437,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aap(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uu)),Uua)) ) ).

% ATP.lambda_437
tff(fact_8557_ATP_Olambda__438,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ud(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu)) ) ).

% ATP.lambda_438
tff(fact_8558_ATP_Olambda__439,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_uq(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu)) ) ).

% ATP.lambda_439
tff(fact_8559_ATP_Olambda__440,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_cu(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_440
tff(fact_8560_ATP_Olambda__441,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_zk(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_441
tff(fact_8561_ATP_Olambda__442,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_xh(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_442
tff(fact_8562_ATP_Olambda__443,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xg(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_443
tff(fact_8563_ATP_Olambda__444,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aad(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_444
tff(fact_8564_ATP_Olambda__445,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_tm(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_445
tff(fact_8565_ATP_Olambda__446,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tn(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_446
tff(fact_8566_ATP_Olambda__447,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vf(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_447
tff(fact_8567_ATP_Olambda__448,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_up(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_448
tff(fact_8568_ATP_Olambda__449,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ci(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_449
tff(fact_8569_ATP_Olambda__450,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ze(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_450
tff(fact_8570_ATP_Olambda__451,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aTP_Lamm_yr(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_Ext2541572767299617110t_real(Uua,Uub)),aa_Ext2541572767299617110t_real(Uu,Uub)) ).

% ATP.lambda_451
tff(fact_8571_ATP_Olambda__452,axiom,
    ! [Uu: fun_Ex3482447378695272101at_int,Uua: fun_Ex3482447378695272101at_int,Uub: extended_enat] : aa_Extended_enat_int(aa_fun1695976788104708604at_int(aTP_Lamm_yv(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Extended_enat_int(Uua,Uub)),aa_Extended_enat_int(Uu,Uub)) ).

% ATP.lambda_452
tff(fact_8572_ATP_Olambda__453,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ys(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_453
tff(fact_8573_ATP_Olambda__454,axiom,
    ! [Uu: fun_real_int,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_yw(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(Uua,Uub)),aa_real_int(Uu,Uub)) ).

% ATP.lambda_454
tff(fact_8574_ATP_Olambda__455,axiom,
    ! [Uu: fun_set_nat_real,Uua: fun_set_nat_real,Uub: set_nat] : aa_set_nat_real(aa_fun3569254276315353500t_real(aTP_Lamm_yt(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_set_nat_real(Uua,Uub)),aa_set_nat_real(Uu,Uub)) ).

% ATP.lambda_455
tff(fact_8575_ATP_Olambda__456,axiom,
    ! [Uu: fun_set_nat_int,Uua: fun_set_nat_int,Uub: set_nat] : aa_set_nat_int(aa_fun3600621996379952540at_int(aTP_Lamm_yx(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(Uua,Uub)),aa_set_nat_int(Uu,Uub)) ).

% ATP.lambda_456
tff(fact_8576_ATP_Olambda__457,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_za(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_457
tff(fact_8577_ATP_Olambda__458,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_yy(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uua,Uub)),aa_nat_int(Uu,Uub)) ).

% ATP.lambda_458
tff(fact_8578_ATP_Olambda__459,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_yu(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_459
tff(fact_8579_ATP_Olambda__460,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_yz(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uua,Uub)),aa_int_int(Uu,Uub)) ).

% ATP.lambda_460
tff(fact_8580_ATP_Olambda__461,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_aat(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_461
tff(fact_8581_ATP_Olambda__462,axiom,
    ! [Uu: fun_Ex7660298398204468809at_nat,Uua: fun_Ex7660298398204468809at_nat,Uub: extended_enat] : aa_Extended_enat_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_kt(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Extended_enat_nat(Uua,Uub)),aa_Extended_enat_nat(Uu,Uub)) ).

% ATP.lambda_462
tff(fact_8582_ATP_Olambda__463,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ku(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_463
tff(fact_8583_ATP_Olambda__464,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_kv(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_464
tff(fact_8584_ATP_Olambda__465,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ru(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_465
tff(fact_8585_ATP_Olambda__466,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kx(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu,Uub)) ).

% ATP.lambda_466
tff(fact_8586_ATP_Olambda__467,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_kw(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_467
tff(fact_8587_ATP_Olambda__468,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_hp(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_468
tff(fact_8588_ATP_Olambda__469,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hq(Uu),Uua),Uub) = aa_real_real(plus_plus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_469
tff(fact_8589_ATP_Olambda__470,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ho(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_470
tff(fact_8590_ATP_Olambda__471,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_sf(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_471
tff(fact_8591_ATP_Olambda__472,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_hn(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_472
tff(fact_8592_ATP_Olambda__473,axiom,
    ! [Uu: fun_Ex8563670519792538868t_bool,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun4759809662886658982t_bool(aTP_Lamm_al(Uu),Uua),Uub))
    <=> ( pp(aa_Ext2772979874139974381t_bool(Uu,Uub))
        | pp(aa_Ext2772979874139974381t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_473
tff(fact_8593_ATP_Olambda__474,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_aj(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        | pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_474
tff(fact_8594_ATP_Olambda__475,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_af(Uu),Uua),Uub))
    <=> ( pp(aa_real_bool(Uu,Uub))
        | pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_475
tff(fact_8595_ATP_Olambda__476,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_ag(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
        | pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_476
tff(fact_8596_ATP_Olambda__477,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_ah(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
        | pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_477
tff(fact_8597_ATP_Olambda__478,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ai(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        | pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_478
tff(fact_8598_ATP_Olambda__479,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_ak(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        | pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_479
tff(fact_8599_ATP_Olambda__480,axiom,
    ! [Uu: fun_Ex8563670519792538868t_bool,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun4759809662886658982t_bool(aTP_Lamm_as(Uu),Uua),Uub))
    <=> ( pp(aa_Ext2772979874139974381t_bool(Uu,Uub))
        & pp(aa_Ext2772979874139974381t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_480
tff(fact_8600_ATP_Olambda__481,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_aq(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_481
tff(fact_8601_ATP_Olambda__482,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_am(Uu),Uua),Uub))
    <=> ( pp(aa_real_bool(Uu,Uub))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_482
tff(fact_8602_ATP_Olambda__483,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_an(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
        & pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_483
tff(fact_8603_ATP_Olambda__484,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_ao(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
        & pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_484
tff(fact_8604_ATP_Olambda__485,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ap(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_485
tff(fact_8605_ATP_Olambda__486,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_ar(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_486
tff(fact_8606_ATP_Olambda__487,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cj(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua)) ) ).

% ATP.lambda_487
tff(fact_8607_ATP_Olambda__488,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_xb(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_488
tff(fact_8608_ATP_Olambda__489,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xc(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_489
tff(fact_8609_ATP_Olambda__490,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sj(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_490
tff(fact_8610_ATP_Olambda__491,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_si(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_491
tff(fact_8611_ATP_Olambda__492,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abc(Uu),Uua),Uub) = aa_nat_real(power_power_real(aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_492
tff(fact_8612_ATP_Olambda__493,axiom,
    ! [Uu: real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_fw(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_493
tff(fact_8613_ATP_Olambda__494,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_fx(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_494
tff(fact_8614_ATP_Olambda__495,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_fy(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_495
tff(fact_8615_ATP_Olambda__496,axiom,
    ! [Uu: nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_fz(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_496
tff(fact_8616_ATP_Olambda__497,axiom,
    ! [Uu: int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_ga(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_497
tff(fact_8617_ATP_Olambda__498,axiom,
    ! [Uu: extended_enat,Uua: extended_enat,Uub: nat] : aa_nat_Extended_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_mc(Uu),Uua),Uub) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Uu),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uub)),Uua)) ).

% ATP.lambda_498
tff(fact_8618_ATP_Olambda__499,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_mb(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_499
tff(fact_8619_ATP_Olambda__500,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_md(Uu),Uua),Uub) = aa_real_real(plus_plus_real(Uu),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_500
tff(fact_8620_ATP_Olambda__501,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lv(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).

% ATP.lambda_501
tff(fact_8621_ATP_Olambda__502,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_lu(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_502
tff(fact_8622_ATP_Olambda__503,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_nn(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_503
tff(fact_8623_ATP_Olambda__504,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_no(Uu),Uua),Uub) = aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_504
tff(fact_8624_ATP_Olambda__505,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_nm(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_505
tff(fact_8625_ATP_Olambda__506,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ld(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_506
tff(fact_8626_ATP_Olambda__507,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_le(Uu),Uua),Uub) = aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_507
tff(fact_8627_ATP_Olambda__508,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_lc(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_508
tff(fact_8628_ATP_Olambda__509,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ko(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_509
tff(fact_8629_ATP_Olambda__510,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sh(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_510
tff(fact_8630_ATP_Olambda__511,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sg(Uu),Uua),Uub) = aa_real_real(times_times_real(Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_511
tff(fact_8631_ATP_Olambda__512,axiom,
    ! [Uu: extended_enat,Uua: nat,Uub: nat] : aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_rd(Uu),Uua),Uub) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Uu),aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_512
tff(fact_8632_ATP_Olambda__513,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_re(Uu),Uua),Uub) = aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_513
tff(fact_8633_ATP_Olambda__514,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rg(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_514
tff(fact_8634_ATP_Olambda__515,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_rf(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_515
tff(fact_8635_ATP_Olambda__516,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mx(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_516
tff(fact_8636_ATP_Olambda__517,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mw(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_517
tff(fact_8637_ATP_Olambda__518,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qu(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_518
tff(fact_8638_ATP_Olambda__519,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_xl(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_519
tff(fact_8639_ATP_Olambda__520,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_lb(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_520
tff(fact_8640_ATP_Olambda__521,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_abb(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_521
tff(fact_8641_ATP_Olambda__522,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_la(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_522
tff(fact_8642_ATP_Olambda__523,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qj(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_523
tff(fact_8643_ATP_Olambda__524,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sl(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_524
tff(fact_8644_ATP_Olambda__525,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sk(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_525
tff(fact_8645_ATP_Olambda__526,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_yf(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(fconj(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub),aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uuc)),aa_complex_complex(times_times_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_complex) ).

% ATP.lambda_526
tff(fact_8646_ATP_Olambda__527,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_yh(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(fconj(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub),aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uuc)),aa_real_real(times_times_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_real) ).

% ATP.lambda_527
tff(fact_8647_ATP_Olambda__528,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_xx(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(fconj(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uuc))),aa_complex_complex(times_times_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_complex) ).

% ATP.lambda_528
tff(fact_8648_ATP_Olambda__529,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_xz(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(fconj(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uuc))),aa_real_real(times_times_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_real) ).

% ATP.lambda_529
tff(fact_8649_ATP_Olambda__530,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_yb(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub),aa_complex_complex(times_times_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_complex) ).

% ATP.lambda_530
tff(fact_8650_ATP_Olambda__531,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_yd(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub),aa_real_real(times_times_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_real) ).

% ATP.lambda_531
tff(fact_8651_ATP_Olambda__532,axiom,
    ! [Uu: nat,Uua: fun_na8797796813480665199d_enat,Uub: fun_na8797796813480665199d_enat,Uuc: nat] : aa_nat_Extended_enat(aa_fun2512298231148344956d_enat(aa_fun1703792696854984886d_enat(aTP_Lamm_vm(Uu),Uua),Uub),Uuc) = aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_Extended_enat(Uua,Uuc)),aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),zero_z5237406670263579293d_enat),aa_nat_Extended_enat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_532
tff(fact_8652_ATP_Olambda__533,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_vk(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc)),aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),zero_zero_complex),aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_533
tff(fact_8653_ATP_Olambda__534,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_vq(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_real(Uua,Uuc)),aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),zero_zero_real),aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_534
tff(fact_8654_ATP_Olambda__535,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_vo(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),zero_zero_nat),aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_535
tff(fact_8655_ATP_Olambda__536,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_vi(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc)),aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),zero_zero_int),aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_536
tff(fact_8656_ATP_Olambda__537,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_vs(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc)),aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_complex),aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_537
tff(fact_8657_ATP_Olambda__538,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_vt(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_real(Uua,Uuc)),aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_real),aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_538
tff(fact_8658_ATP_Olambda__539,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_vv(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),one_one_nat),aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_539
tff(fact_8659_ATP_Olambda__540,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_vu(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc)),aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_int),aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_540
tff(fact_8660_ATP_Olambda__541,axiom,
    ! [Uu: nat,Uua: fun_na8797796813480665199d_enat,Uub: fun_na8797796813480665199d_enat,Uuc: nat] : aa_nat_Extended_enat(aa_fun2512298231148344956d_enat(aa_fun1703792696854984886d_enat(aTP_Lamm_vn(Uu),Uua),Uub),Uuc) = aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_Extended_enat(Uua,Uuc)),aa_nat_Extended_enat(Uub,Uuc)) ).

% ATP.lambda_541
tff(fact_8661_ATP_Olambda__542,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_vl(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_542
tff(fact_8662_ATP_Olambda__543,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_vr(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_543
tff(fact_8663_ATP_Olambda__544,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_vp(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_544
tff(fact_8664_ATP_Olambda__545,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_vj(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc)),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_545
tff(fact_8665_ATP_Olambda__546,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_rv(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_set_nat_bool(member_nat(Uuc),Uua),aa_nat_real(Uub,Uuc)),aa_nat_real(Uu,Uuc)) ).

% ATP.lambda_546
tff(fact_8666_ATP_Olambda__547,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex5852502975177633731d_enat,Uub: fun_Ex5852502975177633731d_enat,Uuc: extended_enat] : aa_Ext7969882492699152700d_enat(aa_fun7865826685190571204d_enat(aa_fun6128326792922203780d_enat(aTP_Lamm_kl(Uu),Uua),Uub),Uuc) = aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uuc),Uu),aa_Ext7969882492699152700d_enat(Uua,Uuc)),aa_Ext7969882492699152700d_enat(Uub,Uuc)) ).

% ATP.lambda_547
tff(fact_8667_ATP_Olambda__548,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: fun_Ex1451620480498224165t_real,Uuc: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aa_fun967516747568145260t_real(aTP_Lamm_kk(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uuc),Uu),aa_Ext2541572767299617110t_real(Uua,Uuc)),aa_Ext2541572767299617110t_real(Uub,Uuc)) ).

% ATP.lambda_548
tff(fact_8668_ATP_Olambda__549,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: fun_Ex7660298398204468809at_nat,Uuc: extended_enat] : aa_Extended_enat_nat(aa_fun1810565495048290044at_nat(aa_fun5812849712608808208at_nat(aTP_Lamm_ki(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uuc),Uu)),aa_Extended_enat_nat(Uua,Uuc)),aa_Extended_enat_nat(Uub,Uuc)) ).

% ATP.lambda_549
tff(fact_8669_ATP_Olambda__550,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex3482447378695272101at_int,Uub: fun_Ex3482447378695272101at_int,Uuc: extended_enat] : aa_Extended_enat_int(aa_fun1695976788104708604at_int(aa_fun8556764835550362732at_int(aTP_Lamm_kj(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uuc),Uu),aa_Extended_enat_int(Uua,Uuc)),aa_Extended_enat_int(Uub,Uuc)) ).

% ATP.lambda_550
tff(fact_8670_ATP_Olambda__551,axiom,
    ! [Uu: complex,Uua: fun_co6496365337113880337d_enat,Uub: fun_co6496365337113880337d_enat,Uuc: complex] : aa_com3683005661447327866d_enat(aa_fun4954905782667315452d_enat(aa_fun2568869125039355352d_enat(aTP_Lamm_kh(Uu),Uua),Uub),Uuc) = aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu),aa_com3683005661447327866d_enat(Uua,Uuc)),aa_com3683005661447327866d_enat(Uub,Uuc)) ).

% ATP.lambda_551
tff(fact_8671_ATP_Olambda__552,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_kg(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu),aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_552
tff(fact_8672_ATP_Olambda__553,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_ke(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_nat(Uua,Uuc)),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_553
tff(fact_8673_ATP_Olambda__554,axiom,
    ! [Uu: complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_kf(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu),aa_complex_int(Uua,Uuc)),aa_complex_int(Uub,Uuc)) ).

% ATP.lambda_554
tff(fact_8674_ATP_Olambda__555,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] : aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_qy(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu),aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) ).

% ATP.lambda_555
tff(fact_8675_ATP_Olambda__556,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] : aa_real_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_km(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu)),aa_real_nat(Uua,Uuc)),aa_real_nat(Uub,Uuc)) ).

% ATP.lambda_556
tff(fact_8676_ATP_Olambda__557,axiom,
    ! [Uu: real,Uua: fun_real_int,Uub: fun_real_int,Uuc: real] : aa_real_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_kn(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu),aa_real_int(Uua,Uuc)),aa_real_int(Uub,Uuc)) ).

% ATP.lambda_557
tff(fact_8677_ATP_Olambda__558,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_qx(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu)),aa_int_nat(Uua,Uuc)),aa_int_nat(Uub,Uuc)) ).

% ATP.lambda_558
tff(fact_8678_ATP_Olambda__559,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mm(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ml(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_559
tff(fact_8679_ATP_Olambda__560,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mk(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mj(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_560
tff(fact_8680_ATP_Olambda__561,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_wa(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_complex_complex(uminus1482373934393186551omplex,Uub)),aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_complex),zero_zero_complex))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_561
tff(fact_8681_ATP_Olambda__562,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_wc(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_real_real(uminus_uminus_real,Uub)),aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_real),zero_zero_real))),aa_nat_real(power_power_real(Uua),Uuc)) ).

% ATP.lambda_562
tff(fact_8682_ATP_Olambda__563,axiom,
    ! [Uu: nat,Uua: int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_wb(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_int_int(uminus_uminus_int,Uub)),aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_int),zero_zero_int))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_563
tff(fact_8683_ATP_Olambda__564,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex4633621507750988018at_int,Uub: fun_Ex7682446853002821117t_bool,Uuc: nat] : aa_nat_int(aa_fun6943296219937830324at_int(aa_fun3819077064105665321at_int(aTP_Lamm_pf(Uu),Uua),Uub),Uuc) = groups2878480467620962989at_int(aa_nat5933393065843416087at_int(aTP_Lamm_pe(Uua),Uuc),collec4429806609662206161d_enat(aa_nat6328455298437991426t_bool(aa_fun955218191390214396t_bool(aTP_Lamm_iz(Uu),Uub),Uuc))) ).

% ATP.lambda_564
tff(fact_8684_ATP_Olambda__565,axiom,
    ! [Uu: set_complex,Uua: fun_co7370231399432144932at_int,Uub: fun_co2841405903858895691t_bool,Uuc: nat] : aa_nat_int(aa_fun6041977265744554632at_int(aa_fun3475035881565545205at_int(aTP_Lamm_pc(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_pb(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_bq(Uu),Uub),Uuc))) ).

% ATP.lambda_565
tff(fact_8685_ATP_Olambda__566,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_int,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_int(aa_fun1494899438858969354at_int(aa_fun6942549246588167161at_int(aTP_Lamm_oz(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_nat_fun_real_int(aTP_Lamm_oy(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bi(Uu),Uub),Uuc))) ).

% ATP.lambda_566
tff(fact_8686_ATP_Olambda__567,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex7123704652682714774at_nat,Uub: fun_Ex7682446853002821117t_bool,Uuc: nat] : aa_nat_nat(aa_fun1897775202592251224at_nat(aa_fun594951741515554929at_nat(aTP_Lamm_pt(Uu),Uua),Uub),Uuc) = groups2880970938130013265at_nat(aa_nat887872048497836987at_nat(aTP_Lamm_iy(Uua),Uuc),collec4429806609662206161d_enat(aa_nat6328455298437991426t_bool(aa_fun955218191390214396t_bool(aTP_Lamm_iz(Uu),Uub),Uuc))) ).

% ATP.lambda_567
tff(fact_8687_ATP_Olambda__568,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: nat] : aa_nat_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_pr(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aTP_Lamm_it(Uua),Uuc),collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_iu(Uu),Uub),Uuc))) ).

% ATP.lambda_568
tff(fact_8688_ATP_Olambda__569,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: nat] : aa_nat_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_pp(Uu),Uua),Uub),Uuc) = groups861055069439313189ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_ip(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_bq(Uu),Uub),Uuc))) ).

% ATP.lambda_569
tff(fact_8689_ATP_Olambda__570,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_pn(Uu),Uua),Uub),Uuc) = groups4696554848551431203al_nat(aa_nat_fun_real_nat(aTP_Lamm_il(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bi(Uu),Uub),Uuc))) ).

% ATP.lambda_570
tff(fact_8690_ATP_Olambda__571,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex6820543082444009806nt_int,Uub: fun_Ex7222163901356174497t_bool,Uuc: int] : aa_int_int(aa_fun1757119979448242868nt_int(aa_fun547143500490336389nt_int(aTP_Lamm_pl(Uu),Uua),Uub),Uuc) = groups2878480467620962989at_int(aa_int4341511938324142267at_int(aTP_Lamm_ig(Uua),Uuc),collec4429806609662206161d_enat(aa_int7536572431366371422t_bool(aa_fun8461784944404243964t_bool(aTP_Lamm_ih(Uu),Uub),Uuc))) ).

% ATP.lambda_571
tff(fact_8691_ATP_Olambda__572,axiom,
    ! [Uu: set_complex,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: int] : aa_int_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_pj(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_ib(Uua),Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_ic(Uu),Uub),Uuc))) ).

% ATP.lambda_572
tff(fact_8692_ATP_Olambda__573,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: int] : aa_int_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_ph(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_int_fun_real_int(aTP_Lamm_ht(Uua),Uuc),collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_bk(Uu),Uub),Uuc))) ).

% ATP.lambda_573
tff(fact_8693_ATP_Olambda__574,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: complex] : aa_complex_complex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_ji(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_jh(Uua),Uuc),collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_bn(Uu),Uub),Uuc))) ).

% ATP.lambda_574
tff(fact_8694_ATP_Olambda__575,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: complex] : aa_complex_complex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_je(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_jd(Uua),Uuc),collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_bj(Uu),Uub),Uuc))) ).

% ATP.lambda_575
tff(fact_8695_ATP_Olambda__576,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex7123704652682714774at_nat,Uub: fun_Ex7682446853002821117t_bool,Uuc: nat] : aa_nat_nat(aa_fun1897775202592251224at_nat(aa_fun594951741515554929at_nat(aTP_Lamm_ja(Uu),Uua),Uub),Uuc) = groups2027974829824023292at_nat(aa_nat887872048497836987at_nat(aTP_Lamm_iy(Uua),Uuc),collec4429806609662206161d_enat(aa_nat6328455298437991426t_bool(aa_fun955218191390214396t_bool(aTP_Lamm_iz(Uu),Uub),Uuc))) ).

% ATP.lambda_576
tff(fact_8696_ATP_Olambda__577,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: nat] : aa_nat_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_iv(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_it(Uua),Uuc),collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_iu(Uu),Uub),Uuc))) ).

% ATP.lambda_577
tff(fact_8697_ATP_Olambda__578,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: nat] : aa_nat_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_iq(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_ip(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_bq(Uu),Uub),Uuc))) ).

% ATP.lambda_578
tff(fact_8698_ATP_Olambda__579,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_im(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_il(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bi(Uu),Uub),Uuc))) ).

% ATP.lambda_579
tff(fact_8699_ATP_Olambda__580,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex6820543082444009806nt_int,Uub: fun_Ex7222163901356174497t_bool,Uuc: int] : aa_int_int(aa_fun1757119979448242868nt_int(aa_fun547143500490336389nt_int(aTP_Lamm_ii(Uu),Uua),Uub),Uuc) = groups2025484359314973016at_int(aa_int4341511938324142267at_int(aTP_Lamm_ig(Uua),Uuc),collec4429806609662206161d_enat(aa_int7536572431366371422t_bool(aa_fun8461784944404243964t_bool(aTP_Lamm_ih(Uu),Uub),Uuc))) ).

% ATP.lambda_580
tff(fact_8700_ATP_Olambda__581,axiom,
    ! [Uu: set_complex,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: int] : aa_int_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_id(Uu),Uua),Uub),Uuc) = groups5690904116761175830ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_ib(Uua),Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_ic(Uu),Uub),Uuc))) ).

% ATP.lambda_581
tff(fact_8701_ATP_Olambda__582,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_bool,Uuc: int] : aa_int_int(aa_fun4504283173926863078nt_int(aa_fun5512849559334105941nt_int(aTP_Lamm_hy(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_hx(Uua),Uuc)),collect_nat(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_bo(Uu),Uub),Uuc))) ).

% ATP.lambda_582
tff(fact_8702_ATP_Olambda__583,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: int] : aa_int_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_hu(Uu),Uua),Uub),Uuc) = groups1932886352136224148al_int(aa_int_fun_real_int(aTP_Lamm_ht(Uua),Uuc),collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_bk(Uu),Uub),Uuc))) ).

% ATP.lambda_583
tff(fact_8703_ATP_Olambda__584,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_vh(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_vg(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_584
tff(fact_8704_ATP_Olambda__585,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_ux(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_uw(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_585
tff(fact_8705_ATP_Olambda__586,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_uv(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_uu(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_586
tff(fact_8706_ATP_Olambda__587,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_ut(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_us(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_587
tff(fact_8707_ATP_Olambda__588,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_aak(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_588
tff(fact_8708_ATP_Olambda__589,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_aai(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_589
tff(fact_8709_ATP_Olambda__590,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aag(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(Uub),Uua)),Uuc)) ).

% ATP.lambda_590
tff(fact_8710_ATP_Olambda__591,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aah(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(Uua),Uub)),Uuc)) ).

% ATP.lambda_591
tff(fact_8711_ATP_Olambda__592,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nc(Uu),Uua),Uub),Uuc) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uu),Uub)) ).

% ATP.lambda_592
tff(fact_8712_ATP_Olambda__593,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ne(Uu),Uua),Uub),Uuc) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_593
tff(fact_8713_ATP_Olambda__594,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_na(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),Uuc))),aa_nat_int(power_power_int(Uu),Uub)) ).

% ATP.lambda_594
tff(fact_8714_ATP_Olambda__595,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_zm(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_595
tff(fact_8715_ATP_Olambda__596,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mj(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uua),Uu)),Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one)))),Uuc))) ).

% ATP.lambda_596
tff(fact_8716_ATP_Olambda__597,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ml(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uua),Uu)),Uuc)),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one)))),Uuc))) ).

% ATP.lambda_597
tff(fact_8717_ATP_Olambda__598,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_vc(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(comm_s2602460028002588243omplex(Uu),Uuc))),aa_nat_complex(comm_s2602460028002588243omplex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_598
tff(fact_8718_ATP_Olambda__599,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ve(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(comm_s7457072308508201937r_real(Uu),Uuc))),aa_nat_real(comm_s7457072308508201937r_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_599
tff(fact_8719_ATP_Olambda__600,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_vd(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uu),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_600
tff(fact_8720_ATP_Olambda__601,axiom,
    ! [Uu: extended_enat,Uua: extended_enat,Uub: nat,Uuc: nat] : aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aa_Ext8984936454148280526d_enat(aTP_Lamm_uz(Uu),Uua),Uub),Uuc) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_Extended_enat(power_8040749407984259932d_enat(Uu),Uuc))),aa_nat_Extended_enat(power_8040749407984259932d_enat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_601
tff(fact_8721_ATP_Olambda__602,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_uy(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_602
tff(fact_8722_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_vb(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_603
tff(fact_8723_ATP_Olambda__604,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ur(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_604
tff(fact_8724_ATP_Olambda__605,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_va(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(power_power_int(Uu),Uuc))),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_605
tff(fact_8725_ATP_Olambda__606,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nk(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_complex(power_power_complex(Uu),Uuc)) ).

% ATP.lambda_606
tff(fact_8726_ATP_Olambda__607,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_nl(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(power_power_real(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(power_power_real(Uu),Uuc)) ).

% ATP.lambda_607
tff(fact_8727_ATP_Olambda__608,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_nj(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_int(power_power_int(Uu),Uuc)) ).

% ATP.lambda_608
tff(fact_8728_ATP_Olambda__609,axiom,
    ! [Uu: set_complex,Uua: fun_re3010880487650781423x_bool,Uub: real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_jb(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_609
tff(fact_8729_ATP_Olambda__610,axiom,
    ! [Uu: set_complex,Uua: fun_na5941084329111710283x_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_jf(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_610
tff(fact_8730_ATP_Olambda__611,axiom,
    ! [Uu: set_nat,Uua: fun_Ex7682446853002821117t_bool,Uub: extended_enat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_Ext6309394871002001526t_bool(aa_fun7698690945434549304t_bool(aTP_Lamm_iw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_Ext6309394871002001526t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_611
tff(fact_8731_ATP_Olambda__612,axiom,
    ! [Uu: set_nat,Uua: fun_co2841405903858895691t_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_in(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_612
tff(fact_8732_ATP_Olambda__613,axiom,
    ! [Uu: set_nat,Uua: fun_re9121486324101951821t_bool,Uub: real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_ij(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_613
tff(fact_8733_ATP_Olambda__614,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_bool,Uub: int,Uuc: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_ir(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_614
tff(fact_8734_ATP_Olambda__615,axiom,
    ! [Uu: set_int,Uua: fun_Ex7222163901356174497t_bool,Uub: extended_enat,Uuc: int] :
      ( pp(aa_int_bool(aa_Ext3185103904533377306t_bool(aa_fun4313485577018776448t_bool(aTP_Lamm_ie(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_Ext3185103904533377306t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_615
tff(fact_8735_ATP_Olambda__616,axiom,
    ! [Uu: set_int,Uua: fun_co2381122952212249071t_bool,Uub: complex,Uuc: int] :
      ( pp(aa_int_bool(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_hz(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_com2211550117870696024t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_616
tff(fact_8736_ATP_Olambda__617,axiom,
    ! [Uu: set_int,Uua: fun_re8661203372455305201t_bool,Uub: real,Uuc: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_hr(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_617
tff(fact_8737_ATP_Olambda__618,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_bool,Uub: nat,Uuc: int] :
      ( pp(aa_int_bool(aa_nat_fun_int_bool(aa_fun7617171277304647676t_bool(aTP_Lamm_hv(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_nat_fun_int_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_618
tff(fact_8738_ATP_Olambda__619,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex7682446853002821117t_bool,Uub: nat,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_nat6328455298437991426t_bool(aa_fun955218191390214396t_bool(aTP_Lamm_iz(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & pp(aa_nat_bool(aa_Ext6309394871002001526t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_619
tff(fact_8739_ATP_Olambda__620,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex7222163901356174497t_bool,Uub: int,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_int7536572431366371422t_bool(aa_fun8461784944404243964t_bool(aTP_Lamm_ih(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & pp(aa_int_bool(aa_Ext3185103904533377306t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_620
tff(fact_8740_ATP_Olambda__621,axiom,
    ! [Uu: set_complex,Uua: fun_co1378794544173401837x_bool,Uub: complex,Uuc: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aa_fun2898175553265993724x_bool(aTP_Lamm_br(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_com1553036987509291606x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_621
tff(fact_8741_ATP_Olambda__622,axiom,
    ! [Uu: set_complex,Uua: fun_co2841405903858895691t_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_bq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_622
tff(fact_8742_ATP_Olambda__623,axiom,
    ! [Uu: set_complex,Uua: fun_co2381122952212249071t_bool,Uub: int,Uuc: complex] :
      ( pp(aa_complex_bool(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_ic(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_int_bool(aa_com2211550117870696024t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_623
tff(fact_8743_ATP_Olambda__624,axiom,
    ! [Uu: set_real,Uua: fun_re7357811994743134493t_bool,Uub: extended_enat,Uuc: real] :
      ( pp(aa_real_bool(aa_Ext4143593917056713754l_bool(aa_fun3666023268991729404l_bool(aTP_Lamm_bl(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_Ext2772979874139974381t_bool(aa_rea3571252896447469150t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_624
tff(fact_8744_ATP_Olambda__625,axiom,
    ! [Uu: set_real,Uua: fun_re3010880487650781423x_bool,Uub: complex,Uuc: real] :
      ( pp(aa_real_bool(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_bj(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_625
tff(fact_8745_ATP_Olambda__626,axiom,
    ! [Uu: set_real,Uua: fun_re9121486324101951821t_bool,Uub: nat,Uuc: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bi(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_626
tff(fact_8746_ATP_Olambda__627,axiom,
    ! [Uu: set_real,Uua: fun_re8661203372455305201t_bool,Uub: int,Uuc: real] :
      ( pp(aa_real_bool(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_bk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_627
tff(fact_8747_ATP_Olambda__628,axiom,
    ! [Uu: set_nat,Uua: fun_na938974098958486209t_bool,Uub: extended_enat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_Ext6309394871002001526t_bool(aa_fun9145731736113502972t_bool(aTP_Lamm_bp(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_Ext2772979874139974381t_bool(aa_nat6328455298437991426t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_628
tff(fact_8748_ATP_Olambda__629,axiom,
    ! [Uu: set_nat,Uua: fun_na5941084329111710283x_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_bn(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_629
tff(fact_8749_ATP_Olambda__630,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_bool,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_bm(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_630
tff(fact_8750_ATP_Olambda__631,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_bool,Uub: int,Uuc: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_bo(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_int_bool(aa_nat_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_631
tff(fact_8751_ATP_Olambda__632,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_bool,Uub: nat,Uuc: int] :
      ( pp(aa_int_bool(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_iu(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_632
tff(fact_8752_ATP_Olambda__633,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nd(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_633
tff(fact_8753_ATP_Olambda__634,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nf(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(power_power_real(Uu),Uuc)),aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_634
tff(fact_8754_ATP_Olambda__635,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nb(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu),Uuc)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_635
tff(fact_8755_ATP_Olambda__636,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nh(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_636
tff(fact_8756_ATP_Olambda__637,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ni(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(power_power_real(Uu),Uuc)),aa_nat_real(power_power_real(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_637
tff(fact_8757_ATP_Olambda__638,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ng(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu),Uuc)),aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_638
tff(fact_8758_ATP_Olambda__639,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_um(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_639
tff(fact_8759_ATP_Olambda__640,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_zr(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_640
tff(fact_8760_ATP_Olambda__641,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_zp(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_641
tff(fact_8761_ATP_Olambda__642,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_zt(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_642
tff(fact_8762_ATP_Olambda__643,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_zv(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_643
tff(fact_8763_ATP_Olambda__644,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_abz(Uu),Uua),Uub),Uuc))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua) ) ) ).

% ATP.lambda_644
tff(fact_8764_ATP_Olambda__645,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: fun_Ex7660298398204468809at_nat,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun1105177683714449665t_bool(aa_fun6406732541880856067t_bool(aTP_Lamm_dg(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Extended_enat_nat(Uua,Uuc)),aa_Extended_enat_nat(Uub,Uuc)) != one_one_nat ) ) ) ).

% ATP.lambda_645
tff(fact_8765_ATP_Olambda__646,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex3482447378695272101at_int,Uub: fun_Ex3482447378695272101at_int,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun821126609344339805t_bool(aa_fun2211662805375314179t_bool(aTP_Lamm_dq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_Extended_enat_int(Uua,Uuc)),aa_Extended_enat_int(Uub,Uuc)) != one_one_int ) ) ) ).

% ATP.lambda_646
tff(fact_8766_ATP_Olambda__647,axiom,
    ! [Uu: set_complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun4400319914713427657x_bool(aa_fun3291692084277223911x_bool(aTP_Lamm_dc(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(Uua,Uuc)),aa_complex_nat(Uub,Uuc)) != one_one_nat ) ) ) ).

% ATP.lambda_647
tff(fact_8767_ATP_Olambda__648,axiom,
    ! [Uu: set_complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun4637897056364002669x_bool(aa_fun3778671508893124399x_bool(aTP_Lamm_dm(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_complex_int(Uua,Uuc)),aa_complex_int(Uub,Uuc)) != one_one_int ) ) ) ).

% ATP.lambda_648
tff(fact_8768_ATP_Olambda__649,axiom,
    ! [Uu: set_real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] :
      ( pp(aa_real_bool(aa_fun8134817056309787853l_bool(aa_fun1579482225532230253l_bool(aTP_Lamm_cy(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(Uua,Uuc)),aa_real_nat(Uub,Uuc)) != one_one_nat ) ) ) ).

% ATP.lambda_649
tff(fact_8769_ATP_Olambda__650,axiom,
    ! [Uu: set_real,Uua: fun_real_int,Uub: fun_real_int,Uuc: real] :
      ( pp(aa_real_bool(aa_fun2745684120339038577l_bool(aa_fun2217247558787703221l_bool(aTP_Lamm_di(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(Uua,Uuc)),aa_real_int(Uub,Uuc)) != one_one_int ) ) ) ).

% ATP.lambda_650
tff(fact_8770_ATP_Olambda__651,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1783405840085286277t_bool(aa_fun5267048652646918913t_bool(aTP_Lamm_da(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uua,Uuc)),aa_nat_nat(Uub,Uuc)) != one_one_nat ) ) ) ).

% ATP.lambda_651
tff(fact_8771_ATP_Olambda__652,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun8175233552433550377t_bool(aa_fun3357394936337705033t_bool(aTP_Lamm_dk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uua,Uuc)),aa_nat_int(Uub,Uuc)) != one_one_int ) ) ) ).

% ATP.lambda_652
tff(fact_8772_ATP_Olambda__653,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      ( pp(aa_int_bool(aa_fun5627573752281863885t_bool(aa_fun3624795673971170029t_bool(aTP_Lamm_de(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(Uua,Uuc)),aa_int_nat(Uub,Uuc)) != one_one_nat ) ) ) ).

% ATP.lambda_653
tff(fact_8773_ATP_Olambda__654,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: fun_int_int,Uuc: int] :
      ( pp(aa_int_bool(aa_fun2796029427775352177t_bool(aa_fun1715141957661956149t_bool(aTP_Lamm_do(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uua,Uuc)),aa_int_int(Uub,Uuc)) != one_one_int ) ) ) ).

% ATP.lambda_654
tff(fact_8774_ATP_Olambda__655,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: fun_Ex1451620480498224165t_real,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun338017137003533149t_bool(aa_fun6293361260138588547t_bool(aTP_Lamm_ek(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & ( aa_real_real(plus_plus_real(aa_Ext2541572767299617110t_real(Uua,Uuc)),aa_Ext2541572767299617110t_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_655
tff(fact_8775_ATP_Olambda__656,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: fun_Ex7660298398204468809at_nat,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun1105177683714449665t_bool(aa_fun6406732541880856067t_bool(aTP_Lamm_ea(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(Uua,Uuc)),aa_Extended_enat_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_656
tff(fact_8776_ATP_Olambda__657,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_eg(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_real_real(plus_plus_real(aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_657
tff(fact_8777_ATP_Olambda__658,axiom,
    ! [Uu: set_complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun4400319914713427657x_bool(aa_fun3291692084277223911x_bool(aTP_Lamm_dw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(Uua,Uuc)),aa_complex_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_658
tff(fact_8778_ATP_Olambda__659,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_ec(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_real_real(plus_plus_real(aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_659
tff(fact_8779_ATP_Olambda__660,axiom,
    ! [Uu: set_real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] :
      ( pp(aa_real_bool(aa_fun8134817056309787853l_bool(aa_fun1579482225532230253l_bool(aTP_Lamm_ds(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(Uua,Uuc)),aa_real_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_660
tff(fact_8780_ATP_Olambda__661,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_ee(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_real_real(plus_plus_real(aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_661
tff(fact_8781_ATP_Olambda__662,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1783405840085286277t_bool(aa_fun5267048652646918913t_bool(aTP_Lamm_du(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uua,Uuc)),aa_nat_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_662
tff(fact_8782_ATP_Olambda__663,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_ei(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_real_real(plus_plus_real(aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_663
tff(fact_8783_ATP_Olambda__664,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      ( pp(aa_int_bool(aa_fun5627573752281863885t_bool(aa_fun3624795673971170029t_bool(aTP_Lamm_dy(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Uua,Uuc)),aa_int_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_664
tff(fact_8784_ATP_Olambda__665,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_xj(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_665
tff(fact_8785_ATP_Olambda__666,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xi(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_666
tff(fact_8786_ATP_Olambda__667,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_wv(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_667
tff(fact_8787_ATP_Olambda__668,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_wx(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,Uuc)),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_668
tff(fact_8788_ATP_Olambda__669,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_wt(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_669
tff(fact_8789_ATP_Olambda__670,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_uu(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uuc)),aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_670
tff(fact_8790_ATP_Olambda__671,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_uw(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,Uuc)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_671
tff(fact_8791_ATP_Olambda__672,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_vg(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_672
tff(fact_8792_ATP_Olambda__673,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_us(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_673
tff(fact_8793_ATP_Olambda__674,axiom,
    ! [Uu: set_nat,Uua: fun_Ex7123704652682714774at_nat,Uub: fun_Ex7682446853002821117t_bool,Uuc: extended_enat] : aa_Extended_enat_nat(aa_fun6947057244751970126at_nat(aa_fun635179117725900299at_nat(aTP_Lamm_ps(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_Ext158989295719858759at_nat(Uua,Uuc)),collect_nat(aa_Ext6309394871002001526t_bool(aa_fun7698690945434549304t_bool(aTP_Lamm_iw(Uu),Uub),Uuc))) ).

% ATP.lambda_674
tff(fact_8794_ATP_Olambda__675,axiom,
    ! [Uu: set_nat,Uua: fun_Ex4633621507750988018at_int,Uub: fun_Ex7682446853002821117t_bool,Uuc: extended_enat] : aa_Extended_enat_int(aa_fun2769206225242773418at_int(aa_fun1492286042523573003at_int(aTP_Lamm_pd(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_Ext5204510313065437859at_int(Uua,Uuc)),collect_nat(aa_Ext6309394871002001526t_bool(aa_fun7698690945434549304t_bool(aTP_Lamm_iw(Uu),Uub),Uuc))) ).

% ATP.lambda_675
tff(fact_8795_ATP_Olambda__676,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: int] : aa_int_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_pq(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_ir(Uu),Uub),Uuc))) ).

% ATP.lambda_676
tff(fact_8796_ATP_Olambda__677,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: complex] : aa_complex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_po(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_in(Uu),Uub),Uuc))) ).

% ATP.lambda_677
tff(fact_8797_ATP_Olambda__678,axiom,
    ! [Uu: set_nat,Uua: fun_co7370231399432144932at_int,Uub: fun_co2841405903858895691t_bool,Uuc: complex] : aa_complex_int(aa_fun4872694516007147366ex_int(aa_fun2110580046701903059ex_int(aTP_Lamm_pa(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_com2205869648058525157at_int(Uua,Uuc)),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_in(Uu),Uub),Uuc))) ).

% ATP.lambda_678
tff(fact_8798_ATP_Olambda__679,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_pm(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_ij(Uu),Uub),Uuc))) ).

% ATP.lambda_679
tff(fact_8799_ATP_Olambda__680,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_int,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_int(aa_fun1626300625876492518al_int(aa_fun1560246819506852821al_int(aTP_Lamm_ox(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_real_fun_nat_int(Uua,Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_ij(Uu),Uub),Uuc))) ).

% ATP.lambda_680
tff(fact_8800_ATP_Olambda__681,axiom,
    ! [Uu: set_int,Uua: fun_Ex6820543082444009806nt_int,Uub: fun_Ex7222163901356174497t_bool,Uuc: extended_enat] : aa_Extended_enat_int(aa_fun1926933649907246598at_int(aa_fun1321938580149485067at_int(aTP_Lamm_pk(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_Ext6203864024966492415nt_int(Uua,Uuc),collect_int(aa_Ext3185103904533377306t_bool(aa_fun4313485577018776448t_bool(aTP_Lamm_ie(Uu),Uub),Uuc))) ).

% ATP.lambda_681
tff(fact_8801_ATP_Olambda__682,axiom,
    ! [Uu: set_int,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: complex] : aa_complex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_pi(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_hz(Uu),Uub),Uuc))) ).

% ATP.lambda_682
tff(fact_8802_ATP_Olambda__683,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: real] : aa_real_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_pg(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_hr(Uu),Uub),Uuc))) ).

% ATP.lambda_683
tff(fact_8803_ATP_Olambda__684,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: nat] : aa_nat_complex(aa_fun2700696402887535882omplex(aa_fun8019143150332516725omplex(aTP_Lamm_jg(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_jf(Uu),Uub),Uuc))) ).

% ATP.lambda_684
tff(fact_8804_ATP_Olambda__685,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: real] : aa_real_complex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_jc(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_jb(Uu),Uub),Uuc))) ).

% ATP.lambda_685
tff(fact_8805_ATP_Olambda__686,axiom,
    ! [Uu: set_nat,Uua: fun_Ex7123704652682714774at_nat,Uub: fun_Ex7682446853002821117t_bool,Uuc: extended_enat] : aa_Extended_enat_nat(aa_fun6947057244751970126at_nat(aa_fun635179117725900299at_nat(aTP_Lamm_ix(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_Ext158989295719858759at_nat(Uua,Uuc)),collect_nat(aa_Ext6309394871002001526t_bool(aa_fun7698690945434549304t_bool(aTP_Lamm_iw(Uu),Uub),Uuc))) ).

% ATP.lambda_686
tff(fact_8806_ATP_Olambda__687,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: int] : aa_int_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_is(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_ir(Uu),Uub),Uuc))) ).

% ATP.lambda_687
tff(fact_8807_ATP_Olambda__688,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: complex] : aa_complex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_io(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_in(Uu),Uub),Uuc))) ).

% ATP.lambda_688
tff(fact_8808_ATP_Olambda__689,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_ik(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_ij(Uu),Uub),Uuc))) ).

% ATP.lambda_689
tff(fact_8809_ATP_Olambda__690,axiom,
    ! [Uu: set_int,Uua: fun_Ex6820543082444009806nt_int,Uub: fun_Ex7222163901356174497t_bool,Uuc: extended_enat] : aa_Extended_enat_int(aa_fun1926933649907246598at_int(aa_fun1321938580149485067at_int(aTP_Lamm_if(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_Ext6203864024966492415nt_int(Uua,Uuc),collect_int(aa_Ext3185103904533377306t_bool(aa_fun4313485577018776448t_bool(aTP_Lamm_ie(Uu),Uub),Uuc))) ).

% ATP.lambda_690
tff(fact_8810_ATP_Olambda__691,axiom,
    ! [Uu: set_int,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: complex] : aa_complex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_ia(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_hz(Uu),Uub),Uuc))) ).

% ATP.lambda_691
tff(fact_8811_ATP_Olambda__692,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_bool,Uuc: nat] : aa_nat_int(aa_fun3504929462025808522at_int(aa_fun3325927984641084153at_int(aTP_Lamm_hw(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_nat_fun_int_int(Uua,Uuc),collect_int(aa_nat_fun_int_bool(aa_fun7617171277304647676t_bool(aTP_Lamm_hv(Uu),Uub),Uuc))) ).

% ATP.lambda_692
tff(fact_8812_ATP_Olambda__693,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: real] : aa_real_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_hs(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_hr(Uu),Uub),Uuc))) ).

% ATP.lambda_693
tff(fact_8813_ATP_Olambda__694,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_lg(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_694
tff(fact_8814_ATP_Olambda__695,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lf(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_695
tff(fact_8815_ATP_Olambda__696,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_qv(Uu),Uua),Uub),Uuc) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_696
tff(fact_8816_ATP_Olambda__697,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_wp(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_wo(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_697
tff(fact_8817_ATP_Olambda__698,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_wn(Uu),Uua),Uub),Uuc),Uud) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_wm(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_698
tff(fact_8818_ATP_Olambda__699,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_wl(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_wk(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_699
tff(fact_8819_ATP_Olambda__700,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_aaj(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_aai(Uua),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(times_times_real(Uub),aa_real_real(divide_divide_real(aa_nat_real(power_power_real(Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_700
tff(fact_8820_ATP_Olambda__701,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_wy(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_wx(Uua),Uuc),Uud)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu))),aa_nat_real(power_power_real(Uub),Uud)) ).

% ATP.lambda_701
tff(fact_8821_ATP_Olambda__702,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_ww(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_wv(Uua),Uuc),Uud),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu))),aa_nat_complex(power_power_complex(Uub),Uud)) ).

% ATP.lambda_702
tff(fact_8822_ATP_Olambda__703,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_wu(Uu),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_wt(Uua),Uuc),Uud)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu))),aa_nat_int(power_power_int(Uub),Uud)) ).

% ATP.lambda_703
tff(fact_8823_ATP_Olambda__704,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_wi(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uud)),Uua)),one_one_complex)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_704
tff(fact_8824_ATP_Olambda__705,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_wj(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Uud)),Uua)),one_one_real)),Uud)),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_705
tff(fact_8825_ATP_Olambda__706,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_vw(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uu)),Uua)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_706
tff(fact_8826_ATP_Olambda__707,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_vy(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Uu)),Uua)),Uud)),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_707
tff(fact_8827_ATP_Olambda__708,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_vx(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,Uua)),Uud)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Uub)),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_708
tff(fact_8828_ATP_Olambda__709,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_vz(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(gbinomial_real(aa_real_real(uminus_uminus_real,Uua)),Uud)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,Uub)),Uud))),aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_709
tff(fact_8829_ATP_Olambda__710,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_wm(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_710
tff(fact_8830_ATP_Olambda__711,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_wo(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(Uua),Uuc)) ).

% ATP.lambda_711
tff(fact_8831_ATP_Olambda__712,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_wk(Uu),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_int(power_power_int(Uub),Uud))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_712
tff(fact_8832_ATP_Olambda__713,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abf(Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uuc),Uud))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uud),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uu)))
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,bit0(one)))))))
           => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I3)),X_13))
            <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Uub),I3)) ) )
        & ( ( Uuc = Uud )
         => ! [X5: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(Uua)))
             => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
        & ( ( Uuc != Uud )
         => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),Uud))
            & ! [X5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uu)))
               => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),X5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),X5))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_713
tff(fact_8833_ATP_Olambda__714,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_xe(Uu),Uua) = Uu ).

% ATP.lambda_714
tff(fact_8834_ATP_Olambda__715,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_xd(Uu),Uua) = Uu ).

% ATP.lambda_715
tff(fact_8835_ATP_Olambda__716,axiom,
    ! [Uu: extended_enat] : aa_Ext7969882492699152700d_enat(aTP_Lamm_ch,Uu) = Uu ).

% ATP.lambda_716
tff(fact_8836_ATP_Olambda__717,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_cg,Uu) = Uu ).

% ATP.lambda_717
tff(fact_8837_ATP_Olambda__718,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_cf,Uu) = Uu ).

% ATP.lambda_718
tff(fact_8838_ATP_Olambda__719,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_cd,Uu) = Uu ).

% ATP.lambda_719
tff(fact_8839_ATP_Olambda__720,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ce,Uu) = Uu ).

% ATP.lambda_720
tff(fact_8840_ATP_Olambda__721,axiom,
    ! [Uu: extended_enat] : aa_Ext7969882492699152700d_enat(aTP_Lamm_cc,Uu) = zero_z5237406670263579293d_enat ).

% ATP.lambda_721
tff(fact_8841_ATP_Olambda__722,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_cb,Uu) = zero_zero_complex ).

% ATP.lambda_722
tff(fact_8842_ATP_Olambda__723,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ca,Uu) = zero_zero_real ).

% ATP.lambda_723
tff(fact_8843_ATP_Olambda__724,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_rt,Uu) = zero_zero_complex ).

% ATP.lambda_724
tff(fact_8844_ATP_Olambda__725,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_go,Uu) = zero_zero_real ).

% ATP.lambda_725
tff(fact_8845_ATP_Olambda__726,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_by,Uu) = zero_zero_nat ).

% ATP.lambda_726
tff(fact_8846_ATP_Olambda__727,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_rs,Uu) = zero_zero_int ).

% ATP.lambda_727
tff(fact_8847_ATP_Olambda__728,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_bz,Uu) = zero_zero_int ).

% ATP.lambda_728
tff(fact_8848_ATP_Olambda__729,axiom,
    ! [Uu: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_be,Uu))
    <=> $false ) ).

% ATP.lambda_729
tff(fact_8849_ATP_Olambda__730,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_bf,Uu))
    <=> $false ) ).

% ATP.lambda_730
tff(fact_8850_ATP_Olambda__731,axiom,
    ! [Uu: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_bc,Uu))
    <=> $false ) ).

% ATP.lambda_731
tff(fact_8851_ATP_Olambda__732,axiom,
    ! [Uu: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_bd,Uu))
    <=> $false ) ).

% ATP.lambda_732
tff(fact_8852_ATP_Olambda__733,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_bg,Uu))
    <=> $false ) ).

% ATP.lambda_733
tff(fact_8853_ATP_Olambda__734,axiom,
    ! [Uu: int] :
      ( pp(aa_int_bool(aTP_Lamm_bh,Uu))
    <=> $false ) ).

% ATP.lambda_734
tff(fact_8854_ATP_Olambda__735,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_zf,Uu))
    <=> $true ) ).

% ATP.lambda_735

% Helper facts (55)
tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa_bool_bool(fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa_bool_bool(fNot,P))
      | ~ pp(P) ) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(fconj(P,Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fdisj(P,Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(fdisj(P,Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(fdisj(P,Q)) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(if_int(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(if_int(fTrue,X),Y) = X ).

tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ).

tff(help_fEx_1_1_fEx_001t__Nat__Onat_U,axiom,
    ! [P: fun_nat_bool,X: nat] :
      ( ~ pp(aa_nat_bool(P,X))
      | pp(aa_fun_nat_bool_bool(fEx_nat,P)) ) ).

tff(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(if_real(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(if_real(fTrue,X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ( X != Y )
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fTrue,X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( X != Y )
      | pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(if_Extended_enat(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(if_Extended_enat(fTrue,X),Y) = X ).

tff(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] : if_VEBT_VEBT(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] : if_VEBT_VEBT(fTrue,X,Y) = X ).

tff(help_fequal_2_1_fequal_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != Y )
      | pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,X),Y))
      | ( X = Y ) ) ).

tff(help_fAll_1_1_fAll_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ~ pp(fAll_VEBT_VEBT(P))
      | pp(aa_VEBT_VEBT_bool(P,X)) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X != Y )
      | pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(fequal_set_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(fequal_set_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( X != Y )
      | pp(aa_list_nat_bool(fequal_list_nat(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ~ pp(aa_list_nat_bool(fequal_list_nat(X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__fun_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: fun_int_int,Y: fun_int_int] : if_fun_int_int(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__fun_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: fun_int_int,Y: fun_int_int] : if_fun_int_int(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fTrue,X,Y) = X ).

tff(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fTrue,X,Y) = X ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_bool_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_3745307990312268622T_VEBT(P,Q),R) = aa_bool_bool(P,aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool_001t__HOL__Obool_U,axiom,
    ! [P: fun_VE2066785035971159626l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBS_6222329278419335549l_bool(P,Q),R) = aa_bool_bool(aa_VEB8047906998784800507l_bool(P,R),aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_fu7168253649841883962l_bool,Q: fun_VE3763956218935105199t_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_2885587777813823575T_VEBT(P,Q),R) = aa_fun_nat_bool_bool(P,aa_VEB4832260192069861672t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_bo1841576699660343469l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEB8047906998784800507l_bool(cOMBB_3438363824644082366T_VEBT(P,Q),R) = aa_boo8970175072276269406l_bool(P,aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool_U,axiom,
    ! [P: fun_VE7346385682144900493T_bool,Q: set_VEBT_VEBT,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBC_771164880915498576T_bool(P,Q),R) = aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(P,R),Q) ).

% Conjectures (1)
tff(conj_0,conjecture,
    nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na),vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,mi),na))),i) = vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,mi),na)) ).

%------------------------------------------------------------------------------