%------------------------------------------------------------------------------
% File     : ITP207_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Heap_Time_Monad 00386_012441
% 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    : 0018_Heap_Time_Monad_00386_012441 [Des22]

% Status   : Theorem
% Rating   : 0.00 v8.2.0, 0.33 v8.1.0
% Syntax   : Number of formulae    : 12354 (4050 unt;3399 typ;   0 def)
%            Number of atoms       : 19637 (8324 equ)
%            Maximal formula atoms :   17 (   2 avg)
%            Number of connectives : 11661 (1557   ~; 330   |; 858   &)
%                                         (2075 <=>;6841  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   20 (   4 avg)
%            Maximal term depth    :   12 (   2 avg)
%            Number of FOOLs       :  946 ( 565 fml; 381 var)
%            Number of X terms     :  509 (   0  []; 477 ite;  32 let)
%            Number of types       :  797 ( 796 usr)
%            Number of type conns  : 3319 (2086   >;1233   *;   0   +;   0  <<)
%            Number of predicates  :  197 ( 194 usr;   2 prp; 0-4 aty)
%            Number of functors    : 2417 (2417 usr; 525 con; 0-8 aty)
%            Number of variables   : 20747 (20544   !; 203   ?;20747   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 13:57:18.659
%------------------------------------------------------------------------------
% Could-be-implicit typings (796)
tff(ty_n_t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J,type,
    fun_Pr5219672437278659779atural: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_Pr6966570558394395095nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J_J,type,
    fun_fu2502439290952149605atural: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_J,type,
    fun_fu8946524575271555235nteger: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J_J,type,
    fun_Pr6582898042194347943atural: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_J,type,
    fun_Pr8371818785129130299nteger: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_Pr5010838083610845651nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    fun_fu9160953773549289085nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    fun_Pr8826142978676617911nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J,type,
    fun_fu7873099233167337571atural: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_fu8902681562042749453nteger: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_Pr944821188406467171e_term: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_Pr4557384415951816845e_term: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_fu5813366174458671467nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_Pr5872836680101629291e_term: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_Pr2541390607678853719nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_J,type,
    fun_fu3460006408547937207nteger: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_J,type,
    fun_Pr5283128996861011131nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_fu7248590935147425627e_term: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_fu2505754153367914865e_term: $tType ).

tff(ty_n_t__fun_It__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_Mt__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J,type,
    fun_fu2492676192234506515it_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_M_Eo_J_J,type,
    fun_Pr6341106346053136302_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_fu836511555612062035e_term: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    fun_Pr8015448952463279699atural: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_Pr6160931176198666718nteger: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J,type,
    fun_Pr1972435576755982779atural: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_Pr2985126522952997572nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_Mt__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J,type,
    fun_fu9056938625493858259it_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_Pr3284083897292419283nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_M_Eo_J_J,type,
    fun_Pr782666833218042862_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    fun_fu465145433024125141nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    fun_Pr4353654455642104759nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_fu2822266754163309965nteger: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_Pr4759914407354907917e_term: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    produc1319942482725812455at_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__fun_It__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
    fun_Pr8463018485500633518_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_fu1456393871774969959e_term: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_fu3093988009420988519e_term: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
    fun_se5282722391638144526_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_Eo_J_Mt__fun_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
    fun_fu1322491050725084261_nat_o: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_Mt__fun_It__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
    fun_op1633239139951409690_nat_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_Pr1987159835869338959nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_Pr4781553364444837327nt_int: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J,type,
    set_Pr620054291978045485atural: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    set_Pr3486312109254157132nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    fun_fu2716251949043459987at_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_M_Eo_J,type,
    fun_Pr3733920118687433658ural_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_M_Eo_J,type,
    fun_Pr7015759342434529115eger_o: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J_Mt__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J_M_Eo_J_J,type,
    fun_fu6174467509129201352char_o: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_Mt__fun_It__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
    fun_op636363337204153321_nat_o: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_Mt__fun_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
    fun_op2843162390950497001_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_fu4835546211937200935e_term: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_fu1176768540439120679e_term: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_Pr2054267358141571796nteger: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_Pr8197042229416165454nteger: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    produc9140211490168418807atural: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    produc2453742332425399446nteger: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    produc7223384021016661839e_term: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    produc592931078481472324e_term: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_M_Eo_J_J,type,
    fun_fu3694549533497745878ural_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    fun_fu7695273699059480439eger_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J_J,type,
    fun_Pr8801881646629708820term_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J_J,type,
    fun_Pr4680327687137491167term_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_fu4545303443005895275nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_Pr7653582204007996523e_term: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    fun_Pr7644814098969791837it_nat: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_Mt__fun_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
    fun_op5061235789045561272_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    fun_fu5924095598813543062at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J_J,type,
    fun_fu6439581347498178894nt_int: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_Mt__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_M_Eo_J_J,type,
    fun_fu3498901270904692718char_o: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J_Mt__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J_J_M_Eo_J_J,type,
    fun_fu2997722009910848836eral_o: $tType ).

tff(ty_n_t__fun_It__fun_Itf__a_Mt__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_J_Mt__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J,type,
    fun_fu6488529940522614227it_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    fun_Pr8191437870481986254it_nat: $tType ).

tff(ty_n_t__fun_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J_J,type,
    fun_pr6291093682560481277atural: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    set_Pr6700895561065273913nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
    fun_Pr3540825087398332590_int_o: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J_J_J,type,
    fun_fu4284519319507362051iteral: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    produc3043004577601930755nt_int: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    produc4155239895104388691e_term: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J,type,
    fun_a_4942204877778174258it_nat: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_Mt__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_M_Eo_J_J,type,
    fun_fu3650339520157847284char_o: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_Mt__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J_M_Eo_J_J,type,
    fun_fu1915598979789041386eral_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    fun_Pr2403869735082759103it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    fun_fu6865642474282033866_int_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J_J,type,
    fun_Pr5526332658708858896term_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    fun_fu963894927305104083et_nat: $tType ).

tff(ty_n_t__fun_It__fun_Itf__a_Mt__Heap____Time____Monad__OHeap_Itf__a_J_J_Mt__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J,type,
    fun_fu4855298819597708819it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_fu9036519777977489245e_term: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    fun_fu204026027145793479at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J_J,type,
    fun_fu719511775830429311nt_int: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J_J,type,
    fun_fu8824737095519370695iteral: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    fun_Pr5651933991979716257it_nat: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__fun_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    fun_a_6373226321756256586it_nat: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J,type,
    fun_a_3426355041879316131it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    fun_fu6383124537394581533et_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
    fun_fu1569576118606662777et_int: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Nat__Onat_J,type,
    produc8298518842471466016rm_nat: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Int__Oint_J,type,
    produc4120667822962269308rm_int: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Nat__Onat_J,type,
    produc3529374061956318379rm_nat: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Int__Oint_J,type,
    produc8574895079301897479rm_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_Pr1552678306225191715nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_Pr8696298990344678139nt_int: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_Mt__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_M_Eo_J_J,type,
    fun_fu1143750687482120046char_o: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_Mt__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_M_Eo_J_J,type,
    fun_fu9053574190653347440eral_o: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    set_Pr1200682188532975830nteger: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_M_Eo_J,type,
    fun_Pr1347628893830317649eger_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    fun_fu3373238862511017284_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J_J,type,
    fun_fu5323616038969315652_int_o: $tType ).

tff(ty_n_t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__fun_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J,type,
    fun_he8369595706569872389it_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    fun_Pr7273243183861550738it_nat: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__fun_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    fun_a_3171520138701809467it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_J,type,
    fun_fu2847332191311216467_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_fu4800041728187187127nteger: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J,type,
    fun_Pr160617923510306311atural: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Nat__Onat_J,type,
    produc1648056901703654172rm_nat: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Int__Oint_J,type,
    produc6693577919049233272rm_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_fu4810317505167811835e_term: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J,type,
    fun_fu1976046603244687703iteral: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_na6030372222970483132e_term: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_na2562132801372711111e_term: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_in1403664741593307672e_term: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_in7158797356850311459e_term: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Code____Numeral__Onatural_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    produc7476732711998919211atural: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    produc7509114972854644512nteger: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    produc3611471421753121338e_term: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    fun_fu3422731660242565741eger_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J_J,type,
    fun_Pr8608829099060101225term_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J,type,
    fun_fu7188715730271225026term_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J,type,
    fun_fu4736795817735658359term_o: $tType ).

tff(ty_n_t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__fun_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J,type,
    fun_he8849654208132081782it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    fun_fu5458086908000294626et_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_J_J,type,
    fun_fu6384561554008368101_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_Mt__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_M_Eo_J_J,type,
    fun_fu8851414294642953504char_o: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_Mt__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_M_Eo_J_J,type,
    fun_fu3765068114029489514eral_o: $tType ).

tff(ty_n_t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    fun_he5234484606919320760it_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_na6101041082579861176e_term: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_in237891044757944084e_term: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_Mt__fun_It__Option__Ooption_It__Num__Onum_J_M_Eo_J_J,type,
    fun_op6018817604785332104_num_o: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Num__Onum_J_Mt__fun_It__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
    fun_op298641349230941576_nat_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    fun_Pr285225112257423215e_term: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    fun_Pr5098669352780829690e_term: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J,type,
    fun_fu6378075912484669126term_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_fu6741006549690681129nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_fu5303860655635331075nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    fun_fu9079770784915425324et_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
    fun_fu4266222366127506568et_int: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J,type,
    fun_fu9120652711261886663iteral: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    set_Pr4166680992137330533nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
    fun_Pr1196100989263358594_int_o: $tType ).

tff(ty_n_t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    fun_he1921015881043195817it_nat: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_Co1561183069634336733nteger: $tType ).

tff(ty_n_t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    fun_he4944555133437423336it_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    fun_Pr7763680713840462443e_term: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    produc3911288613690379145it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
    fun_fu6829939267127632275at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J_J,type,
    fun_fu2855959522417824915at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J_J,type,
    fun_fu2189424204742085595at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J_Mt__fun_It__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
    fun_fu8651063156380458899at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J_J,type,
    fun_fu6187811966698790363at_int: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_Mt__fun_It__Option__Ooption_It__Num__Onum_J_M_Eo_J_J,type,
    fun_op8016730479996091735_num_o: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Num__Onum_J_Mt__fun_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
    fun_op7041624247167716439_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    fun_fu3898734532474705027nt_int: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_Mt__fun_It__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_M_Eo_J_J,type,
    fun_fu4098043175606838510char_o: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_Mt__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_M_Eo_J_J,type,
    fun_fu414282169915291036eral_o: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    produc4788131091073775919nt_int: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    produc5464280974217902119e_term: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    fun_fu2978539155105838238_int_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J_J,type,
    fun_Pr7145817153658775868term_o: $tType ).

tff(ty_n_t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    fun_he8889679731534893401it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_fu3378010530705604573at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
    fun_fu7137239131779924701at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_fu3776778612278285441nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J_Mt__fun_It__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_fu9114370601171615197at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_fu1817862469383530113nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
    fun_fu6666991544514956483at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Nat__Onat_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Nat__Onat_J_J,type,
    fun_fu8621421401018972867nt_nat: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J,type,
    fun_fu6635290411792845739iteral: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    fun_na5667031169817407269it_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_M_Eo_J_J,type,
    fun_Pr3592279944642289454ural_o: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    produc4453839368661128058it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_J,type,
    fun_fu3617688454659378439_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_J,type,
    fun_fu2341038929883010567_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_fu7641778686208158983_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu3968995010428139015_int_o: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    fun_op479301348043730329_nat_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_Pr2477239776880934807at_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_Pr1308109350235310159nt_int: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    set_Pr4329608150637261639at_nat: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc9217583074051984497at_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    fun_Pr3059359313727237556_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__fun_It__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_M_Eo_J_J,type,
    fun_fu74290633640755532char_o: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_Mt__fun_It__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_M_Eo_J_J,type,
    fun_fu2809877164214477290eral_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    fun_Pr74676805955934211atural: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    fun_Pr6222699239247526825at_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    fun_Pr4436041713351579233nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_Pr5876161306168176259nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    fun_fu4638078822479986034_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Rat__Orat_Mt__Rat__Orat_J_M_Eo_J_J,type,
    fun_fu4306744791715064858_rat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J_J,type,
    fun_fu6699046866436802030_int_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    fun_se7472145016457562254_nat_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J_J,type,
    fun_se3135557758678015630_int_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J_J,type,
    fun_o_5935606648521830764t_char: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Nat__Onat_J,type,
    produc3811788542593197729rm_nat: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Int__Oint_J,type,
    produc8857309559938776829rm_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    fun_fu2380148858382964977_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J_J,type,
    fun_fu5088349107139067777_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_fu6709995119303243583nteger: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    fun_na5926011011515597590it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_Mt__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_J_J,type,
    fun_fu8575605362271143641atural: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_fu6327154928193479288nteger: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
    fun_Pr367382672041210069at_nat: $tType ).

tff(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    option8956607266484857688it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_fu717549361983437348nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    fun_fu918106826131887214nt_int: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    fun_op9133723461703387112_nat_o: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc3843707927480180839at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_J,type,
    fun_fu27622379790049391unit_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Product____Type__Ounit_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_J,type,
    fun_fu6154516897781741631_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J,type,
    fun_fu1966733495413849543iteral: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__fun_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_fu4073184632390761289at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
    fun_fu8732410533422943817at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J_Mt__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_fu9148247276982919825at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J_Mt__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J_J,type,
    fun_fu3143578834292514029nt_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
    fun_fu4584101141160326545at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_fu7802804735324696557nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_fu3180206027003385489at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J_J,type,
    fun_fu6398909621167755501nt_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
    fun_fu7839431928035568017at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_fu1834763485345162221nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_fu6909826129937320757nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_M_Eo_J,type,
    fun_Pr8032424337255632745_nat_o: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    set_Pr457366540195662369at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_fu3064074204415222357nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
    fun_fu1098312133097393581at_rat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    fun_Pr557098546508347797at_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    fun_na1668100129766719005at_nat: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J,type,
    fun_o_8622092458504374209t_char: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    fun_se5639961202347202350_nat_o: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Nat__Onat_J,type,
    produc5342793037436079856rm_nat: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Int__Oint_J,type,
    produc1164942017926883148rm_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_Mt__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_M_Eo_J_J,type,
    fun_fu9045186299510176150at_o_o: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_Co2470855966376689427nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_Mt__fun_It__Product____Type__Ounit_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_J,type,
    fun_fu7877188687433499331unit_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu4592178885959468770_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    fun_fu8427683764667544036nt_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_na1605103330871484349e_term: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_in4421688504851875865e_term: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    produc4675096598859438275atural: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
    fun_Pr4403862266400233414at_nat: $tType ).

tff(ty_n_t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    option3562590408128118217it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__fun_It__fun_It__String__Oliteral_Mt__String__Oliteral_J_M_Eo_J_J,type,
    fun_fu6323475733073529544eral_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J,type,
    fun_fu7717421471775461377term_o: $tType ).

tff(ty_n_t__Set__Oset_It__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_J,type,
    set_Pr8693737435421807431at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_fu7333075202678876331nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_fu6811445973280110947nteger: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    produc8664842809031399944it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_Mt__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_fu9087321537282282473_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_fu8532066432991188095_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J_Mt__fun_It__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu619080566302588373_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu63825462011493995_int_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J,type,
    fun_o_8831979434090536342t_char: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J_J_J,type,
    fun_o_8412685193225684464iteral: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc732395585841259969at_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    fun_Pr4252341946692247051at_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    fun_Pr2465684420796299459nt_int: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    fun_se374060482380105870_nat_o: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J_J,type,
    set_Pr1932065953672099015od_o_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    fun_fu3275390434952296369_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Int__Oint_J_J,type,
    fun_fu5207212825597307859nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_M_Eo_J,type,
    fun_Pr5746006151868627512_nat_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_na3603975746771071116e_term: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    fun_in94812177438132456e_term: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_fu5510336631606277565nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_fu1763652423599918571nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_fu6212260319156258755_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_J,type,
    fun_fu1657029809911482051_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu7692897720060892355_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_J,type,
    fun_fu2875010803360890307_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_fu8123456035635431619_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_J,type,
    fun_fu3568225526390654915_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu6598766258096663491_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_J,type,
    fun_fu1780879341396661443_nat_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_na8713714802964096592at_nat: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    produc3346328345660746739nteger: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    fun_Pr1826760651301808490et_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J,type,
    fun_fu2195365108911689727iteral: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__fun_It__Int__Oint_Mt__Rat__Orat_J_M_Eo_J_J,type,
    fun_fu7677565902479959693_rat_o: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Onatural_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    fun_Co6386927884524458043e_term: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J_J,type,
    fun_Co3544761417380347684eger_o: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    fun_Co3460408610320954864e_term: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
    fun_fu3293275151183473192it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J,type,
    fun_fu3265955482920998386term_o: $tType ).

tff(ty_n_t__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,
    produc859450856879609959at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Product____Type__Ounit_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_fu5406395769199844267_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_J,type,
    fun_fu6254727761009756655unit_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Product____Type__Ounit_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu8485898235817490163_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_J,type,
    fun_fu934241560069853935unit_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_Mt__fun_It__Product____Type__Ounit_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_fu1214092143650121711_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_J,type,
    fun_fu2062424135460034099unit_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_Mt__fun_It__Product____Type__Ounit_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu4586251570762536175_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_J,type,
    fun_fu6257966931869675755unit_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J,type,
    fun_o_4966981456184207595t_char: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J_J,type,
    fun_o_564418488415493701iteral: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__String__Oliteral_J_J_J_J_J_J_J_J,type,
    fun_o_4077924883752292140iteral: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_Mt__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
    fun_fu6371730626855558661it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    fun_fu8391177546919232046_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
    fun_fu7503566767892366382_int_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    fun_Pr2028404973800837934_nat_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    fun_Pr9018117161326028078_int_o: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    produc2934264451710624999od_o_o: $tType ).

tff(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    produc3260487557148687353it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__Rat__Orat_J_Mt__fun_It__Rat__Orat_Mt__fun_It__Product____Type__Ounit_Mt__Rat__Orat_J_J_J,type,
    fun_fu1171974497074579908it_rat: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J_J_J,type,
    fun_fu511840035978624972it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__Int__Oint_J_Mt__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Int__Oint_J_J_J,type,
    fun_fu8548799023038560624it_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    fun_fu1345883967705819239_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_J,type,
    fun_fu6579618355643354601at_rat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_fu5935951186182374353at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Int__Oint_J_Mt__fun_It__fun_It__Nat__Onat_Mt__Int__Oint_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
    fun_fu960360787657196261at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_fu6092417073193843705nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    fun_fu2849355231831568730at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J,type,
    fun_fu359272086899841974at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J,type,
    fun_fu3625494130629553242at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    fun_fu6089003341714010202at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J,type,
    fun_fu4103912687242499418at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    fun_fu726873822431485118eger_o: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    produc7993323856043421654atural: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    produc5240993858131070144nteger: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    produc8915214464240719482atural: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    produc6162884466328367972nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J_Mt__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
    fun_fu2987389051837124479on_num: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Option__Ooption_It__Num__Onum_J_J_Mt__fun_It__Nat__Onat_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
    fun_fu1574286537503582443on_num: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__Int__Oint_Mt__Rat__Orat_J_J,type,
    fun_fu7460908759652350970nt_rat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    fun_Pr6405077315951356852et_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    fun_Pr3914994171019630096et_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Code____Numeral__Ointeger_J,type,
    fun_Pr6596356882939945227nteger: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    fun_na4143055713577223196at_nat: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    fun_in2204242111582549184nt_int: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_Co6597186368970283863nteger: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__String__Ochar_J_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    fun_li7251496822767205277t_char: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    fun_in5730379987311621695e_term: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
    fun_fu6410142044385539954it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
    fun_fu8835698356539894734it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_Mt__fun_It__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_M_Eo_J_J,type,
    fun_fu5789129120561279854ar_o_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J,type,
    fun_o_4601425329145410496t_char: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J,type,
    fun_o_2475340183403050522iteral: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_Mt__fun_It__fun_It__Int__Oint_M_Eo_J_M_Eo_J_J,type,
    fun_fu9090758901314268781nt_o_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_fu2454420574033104442at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Set__Oset_It__Int__Oint_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_fu3739269683040691718nt_int: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__String__Oliteral_J_J_J_J_J_J_J,type,
    fun_o_7929707447824500097iteral: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_fu1806770557662746761_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_fu6075382464776002713_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Int__Oint_J_Mt__fun_It__fun_It__Nat__Onat_Mt__Int__Oint_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_fu7440396458814700513_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu1481857817119020493_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_fu6453395667394816515nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Rat__Orat_J_Mt__fun_It__fun_It__Int__Oint_Mt__Rat__Orat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu7714913403879666293_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Nat__Onat_J_Mt__fun_It__fun_It__Int__Oint_Mt__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu2760153274138146437_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu4125167268176844237_int_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J_J_J_J_J_J_J,type,
    fun_o_754185029415132912un_o_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_Pr2118491707810114563at_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_Pr2478283820680522243nt_int: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_Mt__fun_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_M_Eo_J_J,type,
    fun_se6885878543466214468_nat_o: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Code____Numeral__Ointeger_J_Mt__fun_It__List__Olist_It__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    fun_li279074556804398960eger_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_Pr3455331679158536364_nat_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_J,type,
    fun_na5914473681949881652_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    fun_fu6229700951322188702unit_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__fun_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    fun_se6503204438271284984_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Rat__Orat_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Rat__Orat_J_J,type,
    fun_fu3811435807551234611at_rat: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    fun_fu1590586450949517859at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J,type,
    fun_fu3492966305048685787at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    fun_fu6924846375782198887atural: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_J,type,
    fun_fu1961541339144114791atural: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_fu2499394402704830801nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_fu6921880696561781257nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J_J,type,
    fun_fu3090746827843086623atural: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_fu3628599891403802633nteger: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_na8856381646659805649nteger: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J_J,type,
    fun_Co5080488584548667985nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_fu171175841697012900at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    fun_fu6904464733620061952at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_fu8900680393080819712nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Num__Onum_Mt__Int__Oint_J_Mt__fun_It__fun_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    fun_fu6196024946523381971eger_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_fu6204623786860714916at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_fu3354657885645732096nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_fu6436183194761443139_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu5646853623633916995_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_fu8304507701009772099_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu7493154764834273859_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_fu927436805355353155_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_fu138107234227827011_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_fu5053878551662330229at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    fun_fu3043648374899647533nt_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    fun_na5060536216179792737_nat_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Ounit_J_Mt__fun_It__List__Olist_It__Product____Type__Ounit_J_M_Eo_J_J,type,
    fun_se4193343538020515080unit_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Code____Numeral__Ointeger_J,type,
    fun_se2775971902884351953nteger: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    fun_Pr27309308411849096et_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    fun_na6574806417004769082atural: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_na3822476419092417572nteger: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    fun_in7496697025202066910atural: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_in4744367027289715400nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_Mt__fun_It__fun_It__String__Oliteral_M_Eo_J_M_Eo_J_J,type,
    fun_fu9204177235980457629al_o_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    fun_na4518284863240432007_nat_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J_J,type,
    fun_na6120508140265747663_int_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
    fun_o_2320692194582262805t_char: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J,type,
    fun_o_2790778467162637167iteral: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_fu7075445697382584196_int_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__String__Oliteral_J_J_J_J_J_J,type,
    fun_o_3780185944647023446iteral: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Rat__Orat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_J,type,
    fun_fu4710970720072501148at_rat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_fu4413992040155564420at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Rat__Orat_J_Mt__fun_It__Rat__Orat_Mt__fun_It__Int__Oint_Mt__Rat__Orat_J_J_J,type,
    fun_fu114989186063767636nt_rat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J_J,type,
    fun_fu9041382543001606716nt_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_fu923337204662071120nt_int: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__fun_It__Rat__Orat_Mt__fun_It__Rat__Orat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_J_J,type,
    fun_ra8505079968215247908at_rat: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Onatural_Mt__fun_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    fun_Co6813818151677833714atural: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_Co1981900195452921617nteger: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J_J_J_J_J_J,type,
    fun_o_2727025911028299589un_o_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    fun_fu1047064357265694805_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J,type,
    fun_fu152868422178311599um_num: $tType ).

tff(ty_n_t__fun_It__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J_Mt__fun_It__Num__Onum_Mt__Int__Oint_J_J,type,
    fun_fu1947565687807238081um_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Num__Onum_Mt__Num__Onum_J_Mt__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
    fun_fu4719417816141531311on_num: $tType ).

tff(ty_n_t__fun_It__fun_It__Num__Onum_Mt__Int__Oint_J_Mt__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
    fun_fu7964671872712412353on_num: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Option__Ooption_It__Num__Onum_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_fu1502755112258691281at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
    fun_fu5130958774822480081on_num: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_Co6473412210723208761nteger: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Code____Numeral__Ointeger_J_Mt__List__Olist_It__Code____Numeral__Ointeger_J_J,type,
    fun_li4850912443864144983nteger: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_na4138050610385705875at_nat: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_in1696471567231710611nt_int: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_Pr5488025237498180813et_nat: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
    set_Pr2522554150109002629et_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    fun_Pr3947133745332865710_nat_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J_M_Eo_J,type,
    fun_Pr4126283446944868854_int_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_M_Eo_J,type,
    fun_Pr8359263388657976682ural_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J,type,
    fun_Pr7211993375558241536eger_o: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    fun_Co1766037875893669450eger_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    fun_fu6871373343706178705unit_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_Mt__Int__Oint_J_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    fun_fu7766707205226367213unit_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Num__Onum_Mt__Int__Oint_J_Mt__fun_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_fu3380082687118895412nteger: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    produc6653097349344004940it_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Num__Onum_Mt__Int__Oint_J_Mt__fun_It__fun_It__Num__Onum_Mt__Int__Oint_J_M_Eo_J_J,type,
    fun_fu5908277796454533188_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_fu5776950848406856154at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    fun_fu3286867703475129398at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_fu5586244933174592126at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    fun_fu3096161788242865370at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    fun_fu7773166507867613914nt_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_fu5283083362935887158nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    fun_fu9201817095025044880_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_fu570715051317967998at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    fun_fu7304003943241017050at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    fun_fu2757636626010989786nt_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_fu267553481079263030nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    fun_fu2566930710778725758nt_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_fu76847565846999002nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    fun_fu4208408993564413984_int_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    fun_se2389560805064864136_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    fun_fu2408719596384163971_nat_o: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc7248412053542808358at_nat: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    produc6945250483304103390nt_int: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    produc759720530913461378nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Option__Ooption_It__Num__Onum_J_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_fu2897328509465618819_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_M_Eo_J_Mt__fun_It__Nat__Onat_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
    fun_fu2797749360310127211on_num: $tType ).

tff(ty_n_t__fun_It__Product____Type__Ounit_Mt__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_J,type,
    fun_Pr292408699144966967t_unit: $tType ).

tff(ty_n_t__fun_It__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_M_Eo_J,type,
    fun_fu7823346190879031666char_o: $tType ).

tff(ty_n_t__fun_It__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_Mt__fun_It__String__Oliteral_M_Eo_J_J,type,
    fun_fu6209626299996703082eral_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Rat__Orat_J,type,
    fun_se588918216276836300at_rat: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
    fun_se4097146774670233028at_nat: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Int__Oint_J,type,
    fun_se9142667792015812128at_int: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Rat__Orat_J,type,
    fun_se593059424582005380nt_rat: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Nat__Onat_J,type,
    fun_se4101287982975402108nt_nat: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Int__Oint_J,type,
    fun_se9146809000320981208nt_int: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    fun_ra7314870381713115802nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Code____Numeral__Ointeger_J,type,
    fun_Pr6026251430685297nteger: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    fun_Pr3513738403344720850et_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    fun_Pr1023655258412994094et_int: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Num__Onum_J_Mt__fun_It__Option__Ooption_It__Num__Onum_J_M_Eo_J_J,type,
    fun_op493924057549976310_num_o: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__String__Ochar_J_Mt__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_J,type,
    fun_li5260340055336554888char_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_in8804710076054006414at_nat: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    fun_o_6199005644470072298t_char: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J,type,
    fun_o_3955463192764026948iteral: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_Pr6984107991879650211_nat_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_Pr4738963024209204295_int_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Rat__Orat_M_Eo_J_J,type,
    fun_Pr7537129237977221859_rat_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    fun_na3970208099681795815_nat_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    fun_in2375804543815556931_nat_o: $tType ).

tff(ty_n_t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Product____Type__Ounit_J,type,
    fun_he5123621388056232515t_unit: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__String__Oliteral_J_J_J_J_J,type,
    fun_o_2312947153046382251iteral: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__String__Ochar_J_Mt__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_J,type,
    fun_se8116641925208253410char_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Ounit_J_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    fun_se8207921111072574094unit_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J_J_J_J_J,type,
    fun_o_9201541227772858138un_o_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Ounit_M_Eo_J_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    fun_fu182299653934541849unit_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    fun_na607345901702507418atural: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_na7078387940644931716nteger: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__Code____Numeral__Ointeger_J_J,type,
    fun_li1966387620041874891nteger: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Code____Numeral__Ointeger_J_Mt__List__Olist_It__String__Ochar_J_J,type,
    fun_li4211551766102051787t_char: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Onatural_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_J,type,
    fun_Co740390889742897050atural: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_Co6226381647958292740nteger: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_Co3011819262099091808nteger: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Code____Numeral__Ointeger_Mt__Num__Onum_J_J,type,
    fun_Co8521658056209057742er_num: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_J,type,
    fun_Co3583066429914935172er_nat: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Code____Numeral__Ointeger_Mt__Int__Oint_J_J,type,
    fun_Co1092983284983208416er_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_fu2798942646354680517_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_fu5838931486547983824_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_M_Eo_J_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_J,type,
    fun_fu3129017019860109481atural: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_M_Eo_J_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_fu7898940055406642654nteger: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_fu2231310910465984013_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_fu5271299750659287320_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_M_Eo_J_Mt__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J_J,type,
    fun_fu1591967865782657961atural: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_M_Eo_J_Mt__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_fu6361890901329191134nteger: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_li5450038453877631591at_nat: $tType ).

tff(ty_n_t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    list_s1210847774152347623at_nat: $tType ).

tff(ty_n_t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    list_s8839374986383574687nt_int: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    fun_li7366011709773453460_nat_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_M_Eo_J_J,type,
    fun_na4138138649119996554_nat_o: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Nat__Onat_J_Mt__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
    fun_li1025265955262268170_nat_o: $tType ).

tff(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_se7855581050983116737at_nat: $tType ).

tff(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    set_se6260736226359567993nt_int: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc7819656566062154093et_nat: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    produc2115011035271226405et_int: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc7822875418678951345atural: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc8923325533196201883nteger: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    fun_se4782206882984618170_nat_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
    fun_se7249947736802449410_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_M_Eo_J,type,
    fun_fu6891447142836604759at_o_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
    fun_se4323335347534645604_nat_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_Mt__fun_It__List__Olist_It__Int__Oint_J_M_Eo_J_J,type,
    fun_se4632862981408181932_int_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_M_Eo_J_J,type,
    fun_na1108963418173300452_nat_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    fun_na6948449159361463396_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Num__Onum_Mt__Num__Onum_J_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J,type,
    fun_fu8155524643419131615um_num: $tType ).

tff(ty_n_t__fun_It__fun_It__Num__Onum_Mt__Num__Onum_J_Mt__fun_It__Num__Onum_Mt__Int__Oint_J_J,type,
    fun_fu726849872193282289um_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Num__Onum_Mt__Int__Oint_J_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J,type,
    fun_fu5322398707847370225um_num: $tType ).

tff(ty_n_t__fun_It__fun_It__Num__Onum_Mt__Int__Oint_J_Mt__fun_It__Num__Onum_Mt__Int__Oint_J_J,type,
    fun_fu7117095973476296707um_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    fun_fu8328862064278946247at_rat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_fu7002938529518989751at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    fun_fu4512855384587262995at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Int__Oint_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    fun_fu8464300013002972271at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Rat__Orat_J_Mt__fun_It__Int__Oint_Mt__Rat__Orat_J_J,type,
    fun_fu1008360527747082367nt_rat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Nat__Onat_J_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    fun_fu8905809029841901679nt_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Nat__Onat_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_fu6415725884910174923nt_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_fu1143798476471108391nt_int: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__fun_It__Rat__Orat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_J,type,
    fun_ra3995598502166916815at_rat: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_J,type,
    fun_ra5513060666784058567at_rat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Rat__Orat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_J,type,
    fun_na1148497095348402375at_rat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_na3244735409550470327at_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
    fun_na7654559027617327379at_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
    fun_na2382631619178260847at_int: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
    fun_in6793278964266839919at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J,type,
    fun_fu1769128043447206010eger_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    fun_se2170385732985038782_nat_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_Mt__fun_It__Set__Oset_It__Int__Oint_J_M_Eo_J_J,type,
    fun_se2051630309975736070_int_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    fun_na3711450577076729278_nat_o: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Onatural_Mt__fun_It__Code____Numeral__Onatural_M_Eo_J_J,type,
    fun_Co7302518231560758906ural_o: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    fun_Co2081214437782444048eger_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
    fun_na7595719129879163854on_num: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__Product____Type__Ounit_J_M_Eo_J_J,type,
    fun_na2792118505502381397unit_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    fun_se4446312468244607761unit_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Ounit_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    fun_Pr1646868699942875873_nat_o: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    fun_op8989173174992673105on_num: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Product____Type__Ounit_J_M_Eo_J_J,type,
    fun_na7876804994713067643unit_o: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__String__Ochar_J_Mt__fun_It__String__Oliteral_M_Eo_J_J,type,
    fun_li9104157641623487077eral_o: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
    fun_li3177621261580724543t_char: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J,type,
    fun_o_507274782360720025iteral: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_ra4503298312962071866nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Rat__Orat_J,type,
    fun_Pr8317260874932935276at_rat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    fun_Pr2602117396471556196at_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Int__Oint_J,type,
    fun_Pr7647638413817135296at_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Rat__Orat_J,type,
    fun_Pr1652463973008025892nt_rat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
    fun_Pr982841511892225912nt_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_na1476630412504255114at_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_na1173468842265550146nt_int: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_in4514472496968388910at_nat: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_in4211310926729683942nt_int: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__String__Oliteral_J_J_J_J,type,
    fun_o_7984170265123672448iteral: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    fun_na819858167508000762st_nat: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J_J_J_J,type,
    fun_o_4499574304626203759un_o_o: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_J,type,
    fun_a_2990225006182978344t_unit: $tType ).

tff(ty_n_t__fun_It__Product____Type__Ounit_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    fun_Pr1520749766685617582unit_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Ounit_Mt__Heap____Time____Monad__OHeap_Itf__a_J_J,type,
    fun_Pr5215307506330550056Heap_a: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Code____Numeral__Ointeger_J_Mt__String__Oliteral_J,type,
    fun_li9041563744671036928iteral: $tType ).

tff(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    filter1242075044329608583at_nat: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_se1179353343636554400at_nat: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    fun_se7912642235559603452at_int: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_se876191773397849432nt_int: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    fun_se2640714827120536920at_int: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_se4827636401813558708nt_int: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_nu2537414170016844299nteger: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    fun_na106483476154090400at_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J,type,
    fun_na6839772368077139452at_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J,type,
    fun_na8135612474337863287nteger: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J,type,
    fun_in6555721293707029592at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_fu7145897819039579925_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_fu5477517762487726877_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_Mt__Int__Oint_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_fu8207122332086543297_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_M_Eo_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    fun_fu3162257917499486297at_rat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_M_Eo_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_fu318872218122388049at_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_M_Eo_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    fun_fu7052161110045437101at_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Rat__Orat_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_fu3473604023002724957_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_fu1805223966450871909_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_fu4534828536049688329_int_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_M_Eo_J_Mt__fun_It__Int__Oint_Mt__Rat__Orat_J_J,type,
    fun_fu3922030663826099033nt_rat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_M_Eo_J_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    fun_fu1078644964449000785nt_nat: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_M_Eo_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_fu7811933856372049837nt_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_J,type,
    fun_na7928631014926962227_nat_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_J,type,
    fun_in5570076352459579423_int_o: $tType ).

tff(ty_n_t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mtf__a_J,type,
    fun_he6342020308413958580unit_a: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__fun_It__Product____Type__Ounit_Mt__Rat__Orat_J_J,type,
    fun_ra5613769993463817465it_rat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
    fun_na381414567679918841it_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_J,type,
    fun_na6185503771353564761char_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Int__Oint_J_J,type,
    fun_in5522822332083694073it_int: $tType ).

tff(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    list_P6011104703257516679at_nat: $tType ).

tff(ty_n_t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_Eo_J,type,
    fun_he8413527710729349838unit_o: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_Pr1261947904930325089at_nat: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    set_Pr958786334691620121nt_int: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    fun_Pr4796943543331071514_nat_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    fun_Pr3238374608701176162_int_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_o_8988054265482332260nt_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
    fun_na7234455742697417626_nat_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__Int__Oint_J_M_Eo_J_J,type,
    fun_na7306406234920378942_int_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Code____Numeral__Ointeger_J,type,
    fun_se7586849273513198688nteger: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    fun_Co7398788559141407565atural: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    fun_Co8499238673658658103nteger: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_se382663106265794036_nat_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_se7360890175450123928_int_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_se867122708194164540_int_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    fun_na8563630686737515764_nat_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Int__Oint_J_M_Eo_J_J,type,
    fun_na8635581178960477080_int_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    fun_in7136481858371106712_nat_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    fun_in4891666191172872833eger_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_M_Eo_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_fu4001317426178549763_nat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_M_Eo_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_fu3883539188510166275_int_o: $tType ).

tff(ty_n_t__fun_It__String__Oliteral_Mt__fun_It__String__Oliteral_M_Eo_J_J,type,
    fun_li4177005666941401710eral_o: $tType ).

tff(ty_n_t__fun_It__String__Oliteral_Mt__List__Olist_It__String__Ochar_J_J,type,
    fun_li7473841323753414984t_char: $tType ).

tff(ty_n_t__fun_It__Product____Type__Ounit_Mt__Code____Numeral__Onatural_J,type,
    fun_Pr6278675377255837256atural: $tType ).

tff(ty_n_t__fun_It__Product____Type__Ounit_Mt__Code____Numeral__Ointeger_J,type,
    fun_Pr1820225556744648189nteger: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__String__Ochar_J_Mt__String__Oliteral_J,type,
    fun_li5857714827575272844iteral: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J,type,
    fun_o_5833298648030073966iteral: $tType ).

tff(ty_n_t__fun_It__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_M_Eo_J,type,
    fun_fu2157805571543427467ar_o_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Ounit_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_Pr4402817529748954391_nat_o: $tType ).

tff(ty_n_t__fun_It__Product____Type__Ounit_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_Pr2157672562078508475_int_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    fun_na5971062356810558811unit_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    fun_in8984369272479023031unit_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__String__Oliteral_J_J_J,type,
    fun_o_4805661730443325909iteral: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    set_list_list_nat: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_M_Eo_J,type,
    fun_list_list_nat_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J_J_J,type,
    fun_o_8391042909487769924un_o_o: $tType ).

tff(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
    set_set_list_nat: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_list_set_nat: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    produc6271795597528267376eger_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_M_Eo_J,type,
    fun_set_list_nat_o: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    fun_list_set_nat_o: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Code____Numeral__Ointeger_J_M_Eo_J,type,
    fun_li1325434182413301700eger_o: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__fun_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
    fun_rat_fun_rat_rat: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    fun_rat_fun_nat_rat: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__fun_It__Int__Oint_Mt__Rat__Orat_J_J,type,
    fun_rat_fun_int_rat: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_Mt__Int__Oint_J_J,type,
    fun_num_fun_num_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
    fun_nat_fun_rat_rat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    fun_nat_fun_nat_rat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_nat_fun_nat_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    fun_nat_fun_nat_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    fun_nat_fun_int_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_nat_fun_int_int: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_int_fun_nat_nat: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    fun_int_fun_nat_int: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Rat__Orat_J_J,type,
    fun_int_fun_int_rat: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    fun_int_fun_int_int: $tType ).

tff(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_set_set_nat: $tType ).

tff(ty_n_t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J,type,
    heap_T5738788834812785303t_unit: $tType ).

tff(ty_n_t__fun_It__String__Ochar_Mt__fun_It__String__Ochar_M_Eo_J_J,type,
    fun_char_fun_char_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    fun_set_set_nat_o: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Ounit_J_J,type,
    set_li873732205721079667t_unit: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Num__Onum_J_Mt__Num__Onum_J,type,
    fun_option_num_num: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Num__Onum_J_Mt__Nat__Onat_J,type,
    fun_option_num_nat: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Num__Onum_J_Mt__Int__Oint_J,type,
    fun_option_num_int: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    fun_num_option_num: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    fun_nat_option_num: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J,type,
    fun_list_char_nat: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Product____Type__Ounit_J_M_Eo_J,type,
    fun_li6269451478509088136unit_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    fun_int_option_num: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__Heap____Time____Monad__OHeap_Itf__a_J_J,type,
    fun_a_6093291656547882009Heap_a: $tType ).

tff(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
    set_set_Product_unit: $tType ).

tff(ty_n_t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    heap_e7401611519738050253t_unit: $tType ).

tff(ty_n_t__fun_It__String__Ochar_Mt__Code____Numeral__Ointeger_J,type,
    fun_ch2863552307440546731nteger: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Product____Type__Ounit_J_M_Eo_J,type,
    fun_se4342725320470837934unit_o: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__String__Ochar_J,type,
    fun_Co8487158583234504619r_char: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    product_prod_num_num: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    product_prod_nat_num: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    product_prod_nat_nat: $tType ).

tff(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    product_prod_int_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    fun_nat_list_nat: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    fun_list_nat_nat: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Nat__Onat_J_Mt__Int__Oint_J,type,
    fun_list_nat_int: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    set_Product_prod_o_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Rat__Orat_J,type,
    fun_set_nat_rat: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    fun_set_nat_nat: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J,type,
    fun_set_nat_int: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Code____Numeral__Onatural_J,type,
    fun_num_Code_natural: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Code____Numeral__Ointeger_J,type,
    fun_num_Code_integer: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    fun_nat_set_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J,type,
    fun_nat_Code_natural: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J,type,
    fun_nat_Code_integer: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J,type,
    fun_int_Code_natural: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J,type,
    fun_int_Code_integer: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Onatural_Mt__Nat__Onat_J,type,
    fun_Code_natural_nat: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__Num__Onum_J,type,
    fun_Code_integer_num: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J,type,
    fun_Code_integer_nat: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__Int__Oint_J,type,
    fun_Code_integer_int: $tType ).

tff(ty_n_t__fun_It__fun_It__Rat__Orat_Mt__Rat__Orat_J_M_Eo_J,type,
    fun_fun_rat_rat_o: $tType ).

tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    fun_fun_int_int_o: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__fun_It__Rat__Orat_M_Eo_J_J,type,
    fun_rat_fun_rat_o: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
    fun_num_fun_num_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_nat_fun_nat_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_nat_fun_int_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_int_fun_nat_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_int_fun_int_o: $tType ).

tff(ty_n_t__fun_It__String__Oliteral_Mt__String__Oliteral_J,type,
    fun_literal_literal: $tType ).

tff(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
    sum_sum_nat_nat: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
    set_list_char: $tType ).

tff(ty_n_t__Filter__Ofilter_It__Set__Oset_It__Nat__Onat_J_J,type,
    filter_set_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Ounit_Mt__Rat__Orat_J,type,
    fun_Product_unit_rat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
    fun_Product_unit_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Ounit_Mt__Int__Oint_J,type,
    fun_Product_unit_int: $tType ).

tff(ty_n_t__fun_It__Option__Ooption_It__Num__Onum_J_M_Eo_J,type,
    fun_option_num_o: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
    fun_list_char_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__String__Oliteral_J_J,type,
    fun_o_fun_o_literal: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    fun_o_option_num: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J_J,type,
    fun_o_fun_o_fun_o_o: $tType ).

tff(ty_n_t__Predicate__Opred_It__Product____Type__Ounit_J,type,
    pred_Product_unit: $tType ).

tff(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    list_list_nat: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    set_list_nat: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
    set_list_int: $tType ).

tff(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    list_set_nat: $tType ).

tff(ty_n_t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    list_Code_integer: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    fun_list_nat_o: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Int__Oint_J_M_Eo_J,type,
    fun_list_int_o: $tType ).

tff(ty_n_t__fun_It__Typerep__Otyperep_Mt__Nat__Onat_J,type,
    fun_typerep_nat: $tType ).

tff(ty_n_t__fun_It__String__Ochar_Mt__String__Ochar_J,type,
    fun_char_char: $tType ).

tff(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

tff(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    set_set_int: $tType ).

tff(ty_n_t__Set__Oset_It__Code____Numeral__Onatural_J,type,
    set_Code_natural: $tType ).

tff(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    set_Code_integer: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    fun_set_nat_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
    fun_set_int_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__String__Oliteral_J,type,
    fun_nat_literal: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Onatural_M_Eo_J,type,
    fun_Code_natural_o: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
    fun_Code_integer_o: $tType ).

tff(ty_n_t__List__Olist_It__Product____Type__Ounit_J,type,
    list_Product_unit: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
    set_Product_unit: $tType ).

tff(ty_n_t__fun_It__String__Ochar_Mt__Nat__Onat_J,type,
    fun_char_nat: $tType ).

tff(ty_n_t__fun_It__Product____Type__Ounit_M_Eo_J,type,
    fun_Product_unit_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__String__Ochar_J,type,
    fun_nat_char: $tType ).

tff(ty_n_t__Heap____Time____Monad__OHeap_Itf__a_J,type,
    heap_Time_Heap_a: $tType ).

tff(ty_n_t__List__Olist_It__Typerep__Otyperep_J,type,
    list_typerep: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
    fun_rat_rat: $tType ).

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

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

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

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

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

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

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

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

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

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

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

tff(ty_n_t__Set__Oset_It__String__Oliteral_J,type,
    set_literal: $tType ).

tff(ty_n_t__fun_It__String__Oliteral_M_Eo_J,type,
    fun_literal_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__String__Oliteral_J,type,
    fun_o_literal: $tType ).

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

tff(ty_n_t__List__Olist_It__String__Ochar_J,type,
    list_char: $tType ).

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

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J,type,
    fun_o_fun_o_o: $tType ).

tff(ty_n_t__Set__Oset_It__String__Ochar_J,type,
    set_char: $tType ).

tff(ty_n_t__fun_It__String__Ochar_M_Eo_J,type,
    fun_char_o: $tType ).

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

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

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

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

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

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

tff(ty_n_t__Code____Numeral__Onatural,type,
    code_natural: $tType ).

tff(ty_n_t__Code____Numeral__Ointeger,type,
    code_integer: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_M_Eo_J,type,
    fun_rat_o: $tType ).

tff(ty_n_t__fun_It__Num__Onum_M_Eo_J,type,
    fun_num_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_M_Eo_J,type,
    fun_nat_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_M_Eo_J,type,
    fun_int_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Nat__Onat_J,type,
    fun_o_nat: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Int__Oint_J,type,
    fun_o_int: $tType ).

tff(ty_n_t__Product____Type__Ounit,type,
    product_unit: $tType ).

tff(ty_n_t__fun_Itf__a_M_Eo_J,type,
    fun_a_o: $tType ).

tff(ty_n_t__Typerep__Otyperep,type,
    typerep: $tType ).

tff(ty_n_t__String__Oliteral,type,
    literal: $tType ).

tff(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

tff(ty_n_t__fun_I_Eo_M_Eo_J,type,
    fun_o_o: $tType ).

tff(ty_n_t__String__Ochar,type,
    char: $tType ).

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

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

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

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

tff(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (2603)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_se7855581050983116737at_nat > fun_se4782206882984618170_nat_o ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_se6260736226359567993nt_int > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: product_unit > fun_fu182299653934541849unit_o ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: set_Pr1261947904930325089at_nat > fun_fu2380148858382964977_nat_o ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: set_Pr958786334691620121nt_int > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: set_Pr1261947904930325089at_nat > fun_na5060536216179792737_nat_o ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: set_char > fun_na6185503771353564761char_o ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: set_Product_unit > fun_na2792118505502381397unit_o ).

tff(sy_c_ATP_058Lamm__aaq____,type,
    aTP_Lamm_aaq: set_set_nat > fun_na6948449159361463396_nat_o ).

tff(sy_c_ATP_058Lamm__aar____,type,
    aTP_Lamm_aar: set_list_nat > fun_na4138138649119996554_nat_o ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: set_Product_unit > fun_na2792118505502381397unit_o ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: set_set_nat > fun_na6948449159361463396_nat_o ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: set_list_nat > fun_na4138138649119996554_nat_o ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: set_Pr1261947904930325089at_nat > fun_na5060536216179792737_nat_o ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: set_char > fun_na6185503771353564761char_o ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: nat > fun_se4632862981408181932_int_o ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: nat > fun_se4193343538020515080unit_o ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: nat > fun_se6503204438271284984_nat_o ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: nat > fun_se6885878543466214468_nat_o ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: nat > fun_se8116641925208253410char_o ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: nat > fun_se4323335347534645604_nat_o ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aci____,type,
    aTP_Lamm_aci: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__acj____,type,
    aTP_Lamm_acj: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: fun_int_int ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: fun_num_fun_num_int ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: fun_int_nat ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: fun_nat_set_nat ).

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

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

tff(sy_c_ATP_058Lamm__acy____,type,
    aTP_Lamm_acy: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__acz____,type,
    aTP_Lamm_acz: nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_int > fun_int_o ).

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

tff(sy_c_ATP_058Lamm__adb____,type,
    aTP_Lamm_adb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__adc____,type,
    aTP_Lamm_adc: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__add____,type,
    aTP_Lamm_add: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__ade____,type,
    aTP_Lamm_ade: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__adf____,type,
    aTP_Lamm_adf: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__adg____,type,
    aTP_Lamm_adg: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__adh____,type,
    aTP_Lamm_adh: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__adi____,type,
    aTP_Lamm_adi: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__adj____,type,
    aTP_Lamm_adj: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__adk____,type,
    aTP_Lamm_adk: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__adl____,type,
    aTP_Lamm_adl: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__adm____,type,
    aTP_Lamm_adm: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__adn____,type,
    aTP_Lamm_adn: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__ado____,type,
    aTP_Lamm_ado: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__adp____,type,
    aTP_Lamm_adp: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__adq____,type,
    aTP_Lamm_adq: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__adr____,type,
    aTP_Lamm_adr: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ads____,type,
    aTP_Lamm_ads: fun_na819858167508000762st_nat ).

tff(sy_c_ATP_058Lamm__adt____,type,
    aTP_Lamm_adt: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__adu____,type,
    aTP_Lamm_adu: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__adv____,type,
    aTP_Lamm_adv: fun_Pr2118491707810114563at_nat ).

tff(sy_c_ATP_058Lamm__adw____,type,
    aTP_Lamm_adw: fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__adx____,type,
    aTP_Lamm_adx: rat > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__ady____,type,
    aTP_Lamm_ady: fun_char_fun_char_o ).

tff(sy_c_ATP_058Lamm__adz____,type,
    aTP_Lamm_adz: fun_li3177621261580724543t_char ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: heap_Time_Heap_a > fun_Pr5215307506330550056Heap_a ).

tff(sy_c_ATP_058Lamm__aea____,type,
    aTP_Lamm_aea: fun_o_5935606648521830764t_char ).

tff(sy_c_ATP_058Lamm__aeb____,type,
    aTP_Lamm_aeb: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aec____,type,
    aTP_Lamm_aec: fun_nat_int ).

tff(sy_c_ATP_058Lamm__aed____,type,
    aTP_Lamm_aed: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__aee____,type,
    aTP_Lamm_aee: fun_se4782206882984618170_nat_o ).

tff(sy_c_ATP_058Lamm__aef____,type,
    aTP_Lamm_aef: fun_list_char_o ).

tff(sy_c_ATP_058Lamm__aeg____,type,
    aTP_Lamm_aeg: list_Code_integer > fun_o_4077924883752292140iteral ).

tff(sy_c_ATP_058Lamm__aeh____,type,
    aTP_Lamm_aeh: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__aei____,type,
    aTP_Lamm_aei: fun_o_754185029415132912un_o_o ).

tff(sy_c_ATP_058Lamm__aej____,type,
    aTP_Lamm_aej: fun_o_754185029415132912un_o_o ).

tff(sy_c_ATP_058Lamm__aek____,type,
    aTP_Lamm_aek: fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__ael____,type,
    aTP_Lamm_ael: fun_o_754185029415132912un_o_o ).

tff(sy_c_ATP_058Lamm__aem____,type,
    aTP_Lamm_aem: fun_o_754185029415132912un_o_o ).

tff(sy_c_ATP_058Lamm__aen____,type,
    aTP_Lamm_aen: fun_o_754185029415132912un_o_o ).

tff(sy_c_ATP_058Lamm__aeo____,type,
    aTP_Lamm_aeo: fun_o_754185029415132912un_o_o ).

tff(sy_c_ATP_058Lamm__aep____,type,
    aTP_Lamm_aep: fun_o_754185029415132912un_o_o ).

tff(sy_c_ATP_058Lamm__aeq____,type,
    aTP_Lamm_aeq: fun_o_754185029415132912un_o_o ).

tff(sy_c_ATP_058Lamm__aer____,type,
    aTP_Lamm_aer: fun_o_754185029415132912un_o_o ).

tff(sy_c_ATP_058Lamm__aes____,type,
    aTP_Lamm_aes: fun_pr6291093682560481277atural ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_he8413527710729349838unit_o > fun_fu9056938625493858259it_nat ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_he8413527710729349838unit_o > fun_fu2492676192234506515it_nat ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_he1921015881043195817it_nat ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_he5234484606919320760it_nat ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: product_unit > fun_he4944555133437423336it_nat ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: a > fun_he8889679731534893401it_nat ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: product_unit > fun_he4944555133437423336it_nat ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: a > fun_he8889679731534893401it_nat ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_he6342020308413958580unit_a > fun_he1921015881043195817it_nat ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_he5123621388056232515t_unit > fun_he5234484606919320760it_nat ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: char > fun_char_char ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: num > fun_nat_num ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_Pr5876161306168176259nteger > fun_fu4800041728187187127nteger ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: fun_char_char ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: nat > fun_he5234484606919320760it_nat ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_int_int > fun_fu7117095973476296707um_int ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_int_option_num > fun_fu7964671872712412353on_num ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_int_num > fun_fu5322398707847370225um_num ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_option_num_int > fun_fu1947565687807238081um_int ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_op8989173174992673105on_num > fun_fu2987389051837124479on_num ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_option_num_num > fun_fu152868422178311599um_num ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_num_int > fun_fu726849872193282289um_int ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_num_option_num > fun_fu4719417816141531311on_num ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_num_num > fun_fu8155524643419131615um_num ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_num_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_Pr5746006151868627512_nat_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_num_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_Pr5746006151868627512_nat_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_Co7398788559141407565atural ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_int_int ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: set_Pr1200682188532975830nteger > fun_fu3422731660242565741eger_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_Pr620054291978045485atural > fun_fu3694549533497745878ural_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: set_Pr3486312109254157132nteger > fun_fu7695273699059480439eger_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: set_Pr6700895561065273913nt_int > fun_fu6865642474282033866_int_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: set_Pr4166680992137330533nt_int > fun_fu2978539155105838238_int_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_rat_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_int_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_se4782206882984618170_nat_o > fun_fu2380148858382964977_nat_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_se7249947736802449410_int_o > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: set_se7855581050983116737at_nat > fun_fu2380148858382964977_nat_o ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: set_se6260736226359567993nt_int > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_Co7398788559141407565atural ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_int_int ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_Co7398788559141407565atural ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_int_int ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: code_natural > fun_na607345901702507418atural ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: rat > fun_nat_fun_rat_rat ).

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

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

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: num > fun_na4138050610385705875at_nat ).

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

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_na1668100129766719005at_nat > fun_Pr557098546508347797at_nat ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_na5914473681949881652_nat_o > fun_Pr3455331679158536364_nat_o ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_o_int > fun_fu6453395667394816515nt_int ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_int_o > fun_fu1481857817119020493_int_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_int_int > fun_fu6092417073193843705nt_int ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_Pr982841511892225912nt_int > fun_fu717549361983437348nt_int ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_in4211310926729683942nt_int > fun_fu918106826131887214nt_int ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_Pr2478283820680522243nt_int > fun_fu3898734532474705027nt_int ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_fu3275390434952296369_nat_o > fun_fu6384561554008368101_nat_o ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_Pr2028404973800837934_nat_o > fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_Pr982841511892225912nt_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_a_6093291656547882009Heap_a > fun_a_3171520138701809467it_nat ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: fun_a_6093291656547882009Heap_a > fun_a_3426355041879316131it_nat ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: heap_Time_Heap_a > fun_fu4855298819597708819it_nat ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: fun_a_2990225006182978344t_unit > fun_a_6373226321756256586it_nat ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: fun_a_2990225006182978344t_unit > fun_a_4942204877778174258it_nat ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: heap_Time_Heap_a > fun_fu6488529940522614227it_nat ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: int > fun_int_fun_int_o ).

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

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: nat > fun_num_option_num ).

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

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_op8989173174992673105on_num > fun_fu1574286537503582443on_num ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_option_num_o > fun_fu2897328509465618819_nat_o ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_option_num_nat > fun_fu1502755112258691281at_nat ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_o_option_num > fun_fu2797749360310127211on_num ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_o_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_o_nat > fun_fu318872218122388049at_nat ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_nat_option_num > fun_fu5130958774822480081on_num ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_nat_o > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_nat_o ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_nat_o ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_int_fun_int_int ).

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

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: rat > fun_nat_fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_se7586849273513198688nteger > fun_fu5510336631606277565nteger ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_set_nat_rat > fun_fu3811435807551234611at_rat ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_set_nat_int > fun_fu3492966305048685787at_int ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_se2775971902884351953nteger > fun_fu5303860655635331075nteger ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

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

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

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

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: int > fun_fu1143798476471108391nt_int ).

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

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_se4097146774670233028at_nat > fun_fu6666991544514956483at_nat ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_se4101287982975402108nt_nat > fun_fu8621421401018972867nt_nat ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_in4211310926729683942nt_int ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: nat > fun_nat_Code_integer ).

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

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: set_Pr1261947904930325089at_nat > fun_se4782206882984618170_nat_o ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: set_Pr958786334691620121nt_int > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: num > fun_num_o ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_nat_Code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_nat_Code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_nat_Code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_nat_Code_natural > fun_na607345901702507418atural ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_nat_Code_integer > fun_nat_Code_integer ).

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

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

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_nat_Code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: nat > fun_fu8575605362271143641atural ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: nat > fun_fu8575605362271143641atural ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: code_natural > fun_Co740390889742897050atural ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_nat_int ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_int_int ).

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

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

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_nat_fun_nat_int > fun_se7912642235559603452at_int ).

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

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_nat_fun_nat_int > fun_se7912642235559603452at_int ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_nat_fun_int_int > fun_se2640714827120536920at_int ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_nat_fun_int_int > fun_se876191773397849432nt_int ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_int_fun_nat_int > fun_se876191773397849432nt_int ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_int_fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_int_fun_nat_int > fun_se2640714827120536920at_int ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_int_int > fun_nat_fun_int_int ).

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

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_nat_int > fun_na7654559027617327379at_int ).

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

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_nat_Code_natural > fun_na607345901702507418atural ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_nat_Code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_nat_rat > fun_nat_fun_rat_rat ).

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

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

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_Pr5283128996861011131nteger ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_Pr6582898042194347943atural ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_Pr8371818785129130299nteger ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_Pr8826142978676617911nt_int ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_Pr4353654455642104759nt_int ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: int > fun_nat_fun_nat_int ).

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

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

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

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

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: nat > fun_fu8575605362271143641atural ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: code_natural > fun_nat_Code_natural ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_int_fun_nat_int ).

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

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

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

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: set_Pr1261947904930325089at_nat > fun_se4782206882984618170_nat_o ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: set_Pr958786334691620121nt_int > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: literal > fun_literal_o ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: num > fun_num_o ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_nat_fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_nat_fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: code_integer > fun_na8856381646659805649nteger ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: code_integer > fun_na8856381646659805649nteger ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: rat > fun_nat_fun_nat_rat ).

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

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

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: nat > fun_fu1961541339144114791atural ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: int > fun_fu3090746827843086623atural ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: int > fun_fu8905809029841901679nt_nat ).

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

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: nat > fun_fu1961541339144114791atural ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: int > fun_fu3090746827843086623atural ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: nat > fun_fu1961541339144114791atural ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: int > fun_fu3090746827843086623atural ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: nat > fun_fu1961541339144114791atural ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: int > fun_fu3090746827843086623atural ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: set_nat > fun_fu927436805355353155_nat_o ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: set_int > fun_fu7493154764834273859_int_o ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: set_int > fun_fu5646853623633916995_int_o ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: set_int > fun_fu4584101141160326545at_int ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: set_nat > fun_fu8304507701009772099_nat_o ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: set_nat > fun_fu7802804735324696557nt_int ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: set_int > fun_fu9148247276982919825at_nat ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: fun_nat_fun_int_nat > fun_int_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: set_nat > fun_fu3143578834292514029nt_nat ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: set_nat > fun_fu4073184632390761289at_nat ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: set_nat > fun_fu4073184632390761289at_nat ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: set_nat > fun_fu1834763485345162221nt_int ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: set_int > fun_fu7839431928035568017at_int ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: set_int > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: set_int > fun_fu6909826129937320757nt_int ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: set_int > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: set_int > fun_fu6909826129937320757nt_int ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: set_nat > fun_fu6212260319156258755_nat_o ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: set_nat > fun_fu6829939267127632275at_nat ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_se1179353343636554400at_nat > fun_na106483476154090400at_nat ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: set_set_nat > fun_fu1657029809911482051_nat_o ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: set_set_nat > fun_fu3378010530705604573at_nat ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: set_int > fun_fu7692897720060892355_int_o ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: set_int > fun_fu2189424204742085595at_int ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_se876191773397849432nt_int > fun_in6555721293707029592at_int ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: set_set_nat > fun_fu2875010803360890307_nat_o ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: set_set_nat > fun_fu3776778612278285441nt_int ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: set_set_nat > fun_fu3568225526390654915_nat_o ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: set_set_nat > fun_fu9114370601171615197at_nat ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_na106483476154090400at_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: set_nat > fun_fu8123456035635431619_nat_o ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: set_nat > fun_fu8651063156380458899at_nat ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: set_set_nat > fun_fu1780879341396661443_nat_o ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: set_set_nat > fun_fu1817862469383530113nt_int ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_in6555721293707029592at_int > fun_se876191773397849432nt_int ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: set_int > fun_fu6598766258096663491_int_o ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: set_int > fun_fu6187811966698790363at_int ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: set_int > fun_fu9148247276982919825at_nat ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: set_nat > fun_fu3143578834292514029nt_nat ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: set_nat > fun_fu4073184632390761289at_nat ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: set_nat > fun_fu4073184632390761289at_nat ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: set_nat > fun_fu8732410533422943817at_int ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: set_nat > fun_fu8732410533422943817at_int ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: set_int > fun_fu4584101141160326545at_int ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: set_nat > fun_fu7802804735324696557nt_int ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: set_nat > fun_fu1834763485345162221nt_int ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: set_int > fun_fu7839431928035568017at_int ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: set_int > fun_fu6909826129937320757nt_int ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: set_int > fun_fu6909826129937320757nt_int ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: set_nat > fun_fu6829939267127632275at_nat ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: set_set_nat > fun_fu3378010530705604573at_nat ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: set_nat > fun_fu2855959522417824915at_int ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_se7912642235559603452at_int > fun_na6839772368077139452at_int ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: set_set_nat > fun_fu7137239131779924701at_int ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: set_int > fun_fu2189424204742085595at_int ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: set_set_nat > fun_fu3776778612278285441nt_int ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: set_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: set_nat > fun_fu6075382464776002713_nat_o ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: set_int > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: set_int > fun_fu2760153274138146437_int_o ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: set_nat > fun_fu8207122332086543297_nat_o ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: set_nat > fun_fu7440396458814700513_nat_o ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: set_int > fun_fu4534828536049688329_int_o ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: set_int > fun_fu4125167268176844237_int_o ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: set_nat > fun_fu5838931486547983824_nat_o ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: set_nat > fun_fu8532066432991188095_nat_o ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: set_int > fun_fu5271299750659287320_int_o ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: set_int > fun_fu63825462011493995_int_o ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: set_int > fun_fu3473604023002724957_int_o ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: set_int > fun_fu7714913403879666293_int_o ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: set_nat > fun_fu2798942646354680517_nat_o ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: set_nat > fun_fu9087321537282282473_nat_o ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: set_int > fun_fu2231310910465984013_int_o ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: set_int > fun_fu619080566302588373_int_o ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: set_nat > fun_fu2798942646354680517_nat_o ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: set_nat > fun_fu9087321537282282473_nat_o ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: set_int > fun_fu2231310910465984013_int_o ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: set_int > fun_fu619080566302588373_int_o ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: set_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: set_nat > fun_fu6075382464776002713_nat_o ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: set_int > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: set_int > fun_fu2760153274138146437_int_o ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: set_nat > fun_fu8207122332086543297_nat_o ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: set_nat > fun_fu7440396458814700513_nat_o ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: set_int > fun_fu4534828536049688329_int_o ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: set_int > fun_fu4125167268176844237_int_o ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: set_nat > fun_fu5838931486547983824_nat_o ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: set_nat > fun_fu8532066432991188095_nat_o ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: set_int > fun_fu5271299750659287320_int_o ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: set_int > fun_fu63825462011493995_int_o ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: set_int > fun_fu3473604023002724957_int_o ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: set_int > fun_fu7714913403879666293_int_o ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_int_nat > fun_fu1078644964449000785nt_nat ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_nat_int > fun_fu7052161110045437101at_int ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_nat_Code_integer > fun_fu7898940055406642654nteger ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_int_Code_integer > fun_fu6361890901329191134nteger ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_nat_Code_natural > fun_fu3129017019860109481atural ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_int_Code_natural > fun_fu1591967865782657961atural ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_nat_nat > fun_fu318872218122388049at_nat ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_int_int > fun_fu7811933856372049837nt_int ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_int_Code_integer > fun_int_o ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: fun_int_Code_natural > fun_int_o ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_nat_int > fun_nat_o ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_nat_Code_integer > fun_nat_o ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_nat_rat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_nat_Code_natural > fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: fun_int_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: fun_int_Code_integer > fun_int_o ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_int_Code_natural > fun_int_o ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_nat_Code_integer > fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_nat_Code_natural > fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_nat_int > fun_nat_o ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_int_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_int_nat > fun_fu1078644964449000785nt_nat ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_nat_Code_integer > fun_fu7898940055406642654nteger ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: fun_int_Code_integer > fun_fu6361890901329191134nteger ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_nat_Code_natural > fun_fu3129017019860109481atural ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_int_Code_natural > fun_fu1591967865782657961atural ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_nat_nat > fun_fu318872218122388049at_nat ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_nat_int > fun_fu7052161110045437101at_int ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_int_int > fun_fu7811933856372049837nt_int ).

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

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_Pr4796943543331071514_nat_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_se4782206882984618170_nat_o > fun_fu2380148858382964977_nat_o ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_se7249947736802449410_int_o > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: fun_Pr4796943543331071514_nat_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: set_se7855581050983116737at_nat > fun_se7472145016457562254_nat_o ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: set_se6260736226359567993nt_int > fun_se3135557758678015630_int_o ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: set_nat > fun_fu7641778686208158983_nat_o ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: set_int > fun_fu3968995010428139015_int_o ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: set_Pr1261947904930325089at_nat > fun_fu3617688454659378439_nat_o ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: set_Pr1261947904930325089at_nat > fun_fu2341038929883010567_nat_o ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: set_Product_unit > fun_fu7766707205226367213unit_o ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: set_nat > fun_fu8207122332086543297_nat_o ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: set_Product_unit > fun_fu6871373343706178705unit_o ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: set_int > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: set_Product_unit > fun_fu6229700951322188702unit_o ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: set_nat > fun_fu5838931486547983824_nat_o ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: set_int > fun_fu5271299750659287320_int_o ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: set_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: set_int > fun_fu4534828536049688329_int_o ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: set_set_nat > fun_fu1047064357265694805_nat_o ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: fun_se5639961202347202350_nat_o ).

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

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_na106483476154090400at_nat ).

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

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: int > fun_Product_unit_int ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_in6555721293707029592at_int ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: int > fun_list_nat_int ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: code_integer > fun_Pr1820225556744648189nteger ).

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

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: rat > fun_Product_unit_rat ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: code_integer > fun_se7586849273513198688nteger ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: set_Pr958786334691620121nt_int > fun_na6120508140265747663_int_o ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: set_Product_unit > fun_na7876804994713067643unit_o ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: set_set_nat > fun_na3711450577076729278_nat_o ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: set_list_nat > fun_na1108963418173300452_nat_o ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: set_int > fun_na8635581178960477080_int_o ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: set_Pr1261947904930325089at_nat > fun_na4518284863240432007_nat_o ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: set_nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: set_Product_unit > fun_fu7877188687433499331unit_o ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: set_Product_unit > fun_fu7877188687433499331unit_o ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: set_Product_unit > fun_fu6371730626855558661it_nat ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: set_Product_unit > fun_fu934241560069853935unit_o ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: set_int > fun_fu8485898235817490163_int_o ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: set_int > fun_fu8835698356539894734it_nat ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: set_int > fun_fu4586251570762536175_int_o ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: set_Product_unit > fun_fu6257966931869675755unit_o ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: set_Product_unit > fun_fu3043648374899647533nt_nat ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: set_int > fun_fu2566930710778725758nt_nat ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: set_Product_unit > fun_fu6254727761009756655unit_o ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: set_nat > fun_fu5406395769199844267_nat_o ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: set_nat > fun_fu6410142044385539954it_nat ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: set_nat > fun_fu2757636626010989786nt_nat ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: set_nat > fun_fu1214092143650121711_nat_o ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: set_Product_unit > fun_fu2062424135460034099unit_o ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: set_Product_unit > fun_fu5053878551662330229at_nat ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: set_int > fun_fu5586244933174592126at_nat ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: set_nat > fun_fu5776950848406856154at_nat ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: set_Product_unit > fun_fu27622379790049391unit_o ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: set_set_nat > fun_fu6154516897781741631_nat_o ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: set_set_nat > fun_fu3293275151183473192it_nat ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_Product_unit_nat ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: fun_set_nat_nat ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: fun_list_nat_nat ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: fun_Product_unit_nat > fun_Product_unit_nat ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: fun_set_nat_nat > fun_set_nat_nat ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: fun_list_nat_nat > fun_list_nat_nat ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: product_unit > fun_fu511840035978624972it_nat ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: int > fun_fu9041382543001606716nt_nat ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: product_unit > fun_fu8548799023038560624it_int ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: product_unit > fun_fu6709995119303243583nteger ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: nat > fun_fu7333075202678876331nteger ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: int > fun_fu6811445973280110947nteger ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: product_unit > fun_fu1171974497074579908it_rat ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: nat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: int > fun_fu114989186063767636nt_rat ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: set_Product_unit > fun_na2792118505502381397unit_o ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: set_set_nat > fun_na6948449159361463396_nat_o ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: set_list_nat > fun_na4138138649119996554_nat_o ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: set_Pr1261947904930325089at_nat > fun_na5060536216179792737_nat_o ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: set_char > fun_na6185503771353564761char_o ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: product_unit > fun_se8207921111072574094unit_o ).

tff(sy_c_ATP_058Lamm__zw____,type,
    aTP_Lamm_zw: product_prod_nat_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: produc3843707927480180839at_nat > fun_se5282722391638144526_nat_o ).

tff(sy_c_ATP_058Lamm__zy____,type,
    aTP_Lamm_zy: set_Pr1261947904930325089at_nat > fun_se7472145016457562254_nat_o ).

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: set_Pr958786334691620121nt_int > fun_se3135557758678015630_int_o ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: rat > int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: rat > int ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
    archimedean_frac_rat: rat > rat ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: rat > int ).

tff(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
    bNF_Cardinal_ctwo: set_Product_prod_o_o ).

tff(sy_c_BNF__Cardinal__Arithmetic_Oczero_001_Eo,type,
    bNF_Cardinal_czero_o: set_Product_prod_o_o ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Nat__Onat,type,
    bNF_Ca3793111618940312692of_nat: set_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001_Eo,type,
    bNF_Ca8331644756375544342r_on_o: ( set_o * set_Product_prod_o_o ) > $o ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Def_Oeq__onp_001t__List__Olist_It__String__Ochar_J,type,
    bNF_eq_onp_list_char: fun_list_char_o > fun_li5260340055336554888char_o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J,type,
    bNF_re5337651037176608865t_char: ( fun_o_fun_o_o * fun_fu6174467509129201352char_o * fun_o_5935606648521830764t_char * fun_o_5935606648521830764t_char ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J_J,type,
    bNF_re7219000028618014437iteral: ( fun_o_fun_o_o * fun_fu2997722009910848836eral_o * fun_o_5935606648521830764t_char * fun_o_8412685193225684464iteral ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J,type,
    bNF_re8108790427976964573t_char: ( fun_o_fun_o_o * fun_fu3498901270904692718char_o ) > fun_fu6174467509129201352char_o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J,type,
    bNF_re2080241665920251617iteral: ( fun_o_fun_o_o * fun_fu1915598979789041386eral_o ) > fun_fu2997722009910848836eral_o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J,type,
    bNF_re2075097589415080885t_char: ( fun_o_fun_o_o * fun_fu3650339520157847284char_o ) > fun_fu3498901270904692718char_o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J,type,
    bNF_re2270469785393322297iteral: ( fun_o_fun_o_o * fun_fu9053574190653347440eral_o ) > fun_fu1915598979789041386eral_o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J,type,
    bNF_re8572817166782503389t_char: ( fun_o_fun_o_o * fun_fu1143750687482120046char_o ) > fun_fu3650339520157847284char_o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J,type,
    bNF_re7488407634572369889iteral: ( fun_o_fun_o_o * fun_fu3765068114029489514eral_o ) > fun_fu9053574190653347440eral_o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
    bNF_re7288172596490189065t_char: ( fun_o_fun_o_o * fun_fu8851414294642953504char_o ) > fun_fu1143750687482120046char_o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J,type,
    bNF_re6877862493807443341iteral: ( fun_o_fun_o_o * fun_fu414282169915291036eral_o ) > fun_fu3765068114029489514eral_o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_001t__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    bNF_re8717008942347199453t_char: ( fun_o_fun_o_o * fun_fu4098043175606838510char_o ) > fun_fu8851414294642953504char_o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_001t__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J,type,
    bNF_re7699176496855487201iteral: ( fun_o_fun_o_o * fun_fu2809877164214477290eral_o ) > fun_fu414282169915291036eral_o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_001t__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
    bNF_re7974392315488110685t_char: ( fun_o_fun_o_o * fun_fu74290633640755532char_o ) > fun_fu4098043175606838510char_o ).

tff(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_001t__fun_It__String__Oliteral_Mt__String__Oliteral_J,type,
    bNF_re4041988539332897761iteral: ( fun_o_fun_o_o * fun_fu6323475733073529544eral_o ) > fun_fu2809877164214477290eral_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    bNF_re3379532845092657523nteger: ( fun_in4891666191172872833eger_o * fun_in4891666191172872833eger_o ) > fun_fu726873822431485118eger_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re3804157879324367682nt_int: ( fun_in4891666191172872833eger_o * fun_int_fun_int_o * fun_int_int * fun_Code_integer_int ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Num__Onum_001t__Num__Onum,type,
    bNF_re6718328864250387230um_num: ( fun_in4891666191172872833eger_o * fun_num_fun_num_o * fun_int_num * fun_Code_integer_num ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re9130850084932126070nteger: ( fun_in4891666191172872833eger_o * fun_fu726873822431485118eger_o * fun_int_fun_int_int * fun_Co1981900195452921617nteger ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    bNF_re982302072995117890nteger: ( fun_int_fun_int_o * fun_in4891666191172872833eger_o * fun_int_int * fun_int_Code_integer ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re4712519889275205905nt_int: ( fun_int_fun_int_o * fun_int_fun_int_o ) > fun_fu4208408993564413984_int_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum_001t__Num__Onum,type,
    bNF_re7626690874201225453um_num: ( fun_int_fun_int_o * fun_num_fun_num_o * fun_int_num * fun_int_num ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re6250860962936578807nt_int: ( fun_int_fun_int_o * fun_Pr9018117161326028078_int_o ) > fun_fu6699046866436802030_int_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re2214769303045360666nt_rat: ( fun_int_fun_int_o * fun_Pr7537129237977221859_rat_o ) > fun_fu7677565902479959693_rat_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re5717476210804479843nt_int: ( fun_int_fun_int_o * fun_fu4208408993564413984_int_o * fun_int_fun_int_int * fun_int_fun_int_int ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re4808202430149235895nt_int: ( fun_int_fun_int_o * fun_fu6699046866436802030_int_o * fun_in1696471567231710611nt_int * fun_in1696471567231710611nt_int ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
    bNF_re2486238095283335024nt_rat: ( fun_int_fun_int_o * fun_fu7677565902479959693_rat_o * fun_in1696471567231710611nt_int * fun_int_fun_int_rat ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
    bNF_re5828396963709814525t_char: ( fun_li279074556804398960eger_o * fun_li5260340055336554888char_o * fun_li4211551766102051787t_char * fun_li4211551766102051787t_char ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral,type,
    bNF_re3035051291559180600iteral: ( fun_li279074556804398960eger_o * fun_li9104157641623487077eral_o * fun_li4211551766102051787t_char * fun_li9041563744671036928iteral ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001_Eo_001_Eo,type,
    bNF_re6668474094832959905ar_o_o: ( fun_li5260340055336554888char_o * fun_o_fun_o_o ) > fun_fu5789129120561279854ar_o_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    bNF_re901197953848981245nteger: ( fun_li5260340055336554888char_o * fun_li279074556804398960eger_o * fun_li1966387620041874891nteger * fun_li1966387620041874891nteger ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
    bNF_re1407854171455731685t_char: ( fun_li5260340055336554888char_o * fun_li5260340055336554888char_o ) > fun_fu74290633640755532char_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re76688982720252355at_nat: ( fun_li5260340055336554888char_o * fun_nat_fun_nat_o * fun_list_char_nat * fun_list_char_nat ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001t__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_001t__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
    bNF_re6109064546783844641char_o: ( fun_li5260340055336554888char_o * fun_fu5789129120561279854ar_o_o * fun_li5260340055336554888char_o * fun_li5260340055336554888char_o ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001t__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_001t__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
    bNF_re1514760611062797473t_char: ( fun_li5260340055336554888char_o * fun_fu74290633640755532char_o * fun_li7251496822767205277t_char * fun_li7251496822767205277t_char ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral_001_Eo_001_Eo,type,
    bNF_re8730477052009823892al_o_o: ( fun_li9104157641623487077eral_o * fun_o_fun_o_o ) > fun_fu9204177235980457629al_o_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
    bNF_re4991257583289732696t_char: ( fun_li9104157641623487077eral_o * fun_li5260340055336554888char_o * fun_li3177621261580724543t_char * fun_li7473841323753414984t_char ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral,type,
    bNF_re5453467372598631581iteral: ( fun_li9104157641623487077eral_o * fun_li9104157641623487077eral_o ) > fun_fu6323475733073529544eral_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral_001t__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_001t__fun_It__String__Oliteral_M_Eo_J,type,
    bNF_re9191414151699148995eral_o: ( fun_li9104157641623487077eral_o * fun_fu9204177235980457629al_o_o * fun_li5260340055336554888char_o * fun_li4177005666941401710eral_o ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re5653821019739307937at_nat: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > fun_fu9201817095025044880_nat_o ).

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

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re5162689864697376059at_nat: ( fun_nat_fun_nat_o * fun_fu9201817095025044880_nat_o * fun_nat_fun_nat_nat * fun_nat_fun_nat_nat ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    bNF_re6501075790457514782nteger: ( fun_num_fun_num_o * fun_in4891666191172872833eger_o ) > fun_fu6196024946523381971eger_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re1822329894187522285nt_int: ( fun_num_fun_num_o * fun_int_fun_int_o ) > fun_fu5908277796454533188_int_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__fun_It__Num__Onum_Mt__Int__Oint_J_001t__fun_It__Num__Onum_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re4520166952630509772nteger: ( fun_num_fun_num_o * fun_fu6196024946523381971eger_o * fun_num_fun_num_int * fun_nu2537414170016844299nteger ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__fun_It__Num__Onum_Mt__Int__Oint_J_001t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
    bNF_re6378542964492772123um_int: ( fun_num_fun_num_o * fun_fu5908277796454533188_int_o * fun_num_fun_num_int * fun_num_fun_num_int ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
    bNF_re8699439704749558557nt_o_o: ( fun_Pr9018117161326028078_int_o * fun_o_fun_o_o * fun_Pr3238374608701176162_int_o * fun_Pr3238374608701176162_int_o ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7145576690424134365nt_int: ( fun_Pr9018117161326028078_int_o * fun_Pr9018117161326028078_int_o ) > fun_fu5323616038969315652_int_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re4734134812813710757nt_int: ( fun_Pr9018117161326028078_int_o * fun_fu5323616038969315652_int_o * fun_Pr1308109350235310159nt_int * fun_Pr1308109350235310159nt_int ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
    bNF_re1494630372529172596at_o_o: ( fun_Pr7537129237977221859_rat_o * fun_o_fun_o_o * fun_Pr3238374608701176162_int_o * fun_rat_o ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re8279943556446156061nt_rat: ( fun_Pr7537129237977221859_rat_o * fun_Pr7537129237977221859_rat_o ) > fun_fu4306744791715064858_rat_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
    bNF_re6323452265856231590at_rat: ( fun_Pr7537129237977221859_rat_o * fun_fu4306744791715064858_rat_o * fun_Pr1308109350235310159nt_int * fun_rat_fun_rat_rat ) > $o ).

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

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

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

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

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

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

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

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re7934895593101944656at_int: ( fun_Pr2028404973800837934_nat_o * fun_Pr4738963024209204295_int_o * fun_Pr2118491707810114563at_nat * fun_Pr7647638413817135296at_int ) > $o ).

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

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

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

tff(sy_c_BNF__Wellorder__Constructions_OordIso_001_Eo_001_Eo,type,
    bNF_We2654380646378065620so_o_o: set_Pr1932065953672099015od_o_o ).

tff(sy_c_BNF__Wellorder__Constructions_OordIso_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_We5258908940166488438at_nat: set_Pr4329608150637261639at_nat ).

tff(sy_c_BNF__Wellorder__Constructions_OordLess_001_Eo_001t__Nat__Onat,type,
    bNF_We8182288985678559134_o_nat: set_Pr457366540195662369at_nat ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
    bNF_We3818239936649020644el_nat: fun_se4782206882984618170_nat_o ).

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

tff(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001t__Nat__Onat_001t__Nat__Onat,type,
    basic_876126793109182934at_nat: ( fun_nat_nat * fun_nat_nat ) > fun_Pr2602117396471556196at_nat ).

tff(sy_c_Basic__BNFs_Opred__prod_001t__Int__Oint_001t__Int__Oint,type,
    basic_1567116559311922317nt_int: ( fun_int_o * fun_int_o ) > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Basic__BNFs_Opred__prod_001t__Nat__Onat_001t__Nat__Onat,type,
    basic_570253317919918037at_nat: ( fun_nat_o * fun_nat_o ) > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_Basic__BNFs_Orel__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    basic_4387203522000727145nt_int: ( fun_int_fun_int_o * fun_int_fun_int_o ) > fun_Pr9018117161326028078_int_o ).

tff(sy_c_Basic__BNFs_Orel__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    basic_5328504652464829177at_nat: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

tff(sy_c_Binomial_Ogbinomial_001t__Code____Numeral__Ointeger,type,
    gbinom8545251970709558553nteger: code_integer > fun_nat_Code_integer ).

tff(sy_c_Binomial_Ogbinomial_001t__Code____Numeral__Onatural,type,
    gbinom7368847122466276068atural: code_natural > fun_nat_Code_natural ).

tff(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
    gbinomial_int: fun_int_fun_nat_int ).

tff(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: num > fun_num_option_num ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: fun_int_int ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
    bit_ri6519982836138164636nteger: nat > fun_Co8499238673658658103nteger ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
    bit_se3949692690581998587nteger: fun_Co1981900195452921617nteger ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Onatural,type,
    bit_se2773287842338716102atural: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
    bit_se725231765392027082nd_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Ointeger,type,
    bit_se3928097537394005634nteger: nat > fun_Co8499238673658658103nteger ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Onatural,type,
    bit_se2751692689150723149atural: ( nat * code_natural ) > code_natural ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
    bit_se8568078237143864401it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
    bit_se1345352211410354436nteger: nat > fun_Co8499238673658658103nteger ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Onatural,type,
    bit_se168947363167071951atural: ( nat * code_natural ) > code_natural ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
    bit_se2159334234014336723it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Code____Numeral__Ointeger,type,
    bit_se2119862282449309892nteger: fun_nat_Code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Code____Numeral__Onatural,type,
    bit_se943457434206027407atural: fun_nat_Code_natural ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: fun_nat_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Ointeger,type,
    bit_se1080825931792720795nteger: fun_Co1981900195452921617nteger ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Onatural,type,
    bit_se9127793120404214118atural: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
    bit_se1409905431419307370or_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Ointeger,type,
    bit_se7788150548672797655nteger: nat > fun_Co8499238673658658103nteger ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Onatural,type,
    bit_se6611745700429515170atural: ( nat * code_natural ) > code_natural ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
    bit_se545348938243370406it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se2793503036327961859nteger: nat > fun_Co8499238673658658103nteger ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Onatural,type,
    bit_se1617098188084679374atural: ( nat * code_natural ) > code_natural ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
    bit_se7879613467334960850it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
    bit_se7882103937844011126it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Code____Numeral__Ointeger,type,
    bit_se1745604003318907178nteger: nat > fun_Co8499238673658658103nteger ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Code____Numeral__Onatural,type,
    bit_se569199155075624693atural: ( nat * code_natural ) > code_natural ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
    bit_se2923211474154528505it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se8260200283734997820nteger: nat > fun_Co8499238673658658103nteger ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Onatural,type,
    bit_se7083795435491715335atural: ( nat * code_natural ) > code_natural ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
    bit_se4203085406695923979it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Code____Numeral__Ointeger,type,
    bit_se3222712562003087583nteger: fun_Co1981900195452921617nteger ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Code____Numeral__Onatural,type,
    bit_se2046307713759805098atural: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Ointeger,type,
    bit_se9216721137139052372nteger: code_integer > fun_nat_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Onatural,type,
    bit_se8040316288895769887atural: code_natural > fun_nat_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: fun_int_fun_nat_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: fun_nat_fun_nat_o ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: nat > fun_num_option_num ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
    bit_un7362597486090784418nd_num: num > fun_num_option_num ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: num > fun_num_option_num ).

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

tff(sy_c_Code__Numeral_Ocr__integer,type,
    code_cr_integer: int > fun_Code_integer_o ).

tff(sy_c_Code__Numeral_Ocr__natural,type,
    code_cr_natural: nat > fun_Code_natural_o ).

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

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

tff(sy_c_Code__Numeral_Odup,type,
    code_dup: fun_Co8499238673658658103nteger ).

tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: fun_Code_integer_int ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun_int_Code_integer ).

tff(sy_c_Code__Numeral_Ointeger__of__nat,type,
    code_integer_of_nat: fun_nat_Code_integer ).

tff(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: num > code_integer ).

tff(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: fun_Code_integer_nat ).

tff(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
    code_nat_of_natural: fun_Code_natural_nat ).

tff(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
    code_natural_of_nat: fun_nat_Code_natural ).

tff(sy_c_Code__Numeral_Onegative,type,
    code_negative: fun_num_Code_integer ).

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: fun_Code_integer_num ).

tff(sy_c_Code__Numeral_Opcr__integer,type,
    code_pcr_integer: fun_in4891666191172872833eger_o ).

tff(sy_c_Code__Numeral_Osub,type,
    code_sub: fun_nu2537414170016844299nteger ).

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
    complete_Inf_Inf_nat: fun_set_nat_nat ).

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Product____Type__Ounit,type,
    comple2584293577114468500t_unit: set_Product_unit > product_unit ).

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

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

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
    complete_Sup_Sup_nat: fun_set_nat_nat ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Product____Type__Ounit,type,
    comple4687483117567863418t_unit: set_Product_unit > product_unit ).

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

tff(sy_c_Countable_Onat__to__rat__surj,type,
    nat_to_rat_surj: fun_nat_rat ).

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: fun_Pr982841511892225912nt_int ).

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

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

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
    unique5706413561485394159nteger: fun_Pr7211993375558241536eger_o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
    unique6319869463603278526ux_int: fun_Pr3238374608701176162_int_o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
    unique6322359934112328802ux_nat: fun_Pr4796943543331071514_nat_o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
    unique3479559517661332726nteger: ( num * num ) > produc8923325533196201883nteger ).

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

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

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
    unique4921790084139445826nteger: num > fun_Pr5876161306168176259nteger ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
    unique5024387138958732305ep_int: num > fun_Pr2478283820680522243nt_int ).

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

tff(sy_c_Equiv__Relations_Oequivp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    equiv_749116872481872345at_nat: fun_Pr2028404973800837934_nat_o > $o ).

tff(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Int__Oint,type,
    euclid4774559944035922753ze_int: fun_int_nat ).

tff(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Nat__Onat,type,
    euclid3398187327856392827nt_nat: fun_nat_nat ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
    comm_s8582702949713902594nteger: code_integer > fun_nat_Code_integer ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Onatural,type,
    comm_s7406298101470620109atural: code_natural > fun_nat_Code_natural ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
    comm_s4660882817536571857er_int: fun_int_fun_nat_int ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
    comm_s4663373288045622133er_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
    semiri3624122377584611663nteger: fun_nat_Code_integer ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Onatural,type,
    semiri2447717529341329178atural: fun_nat_Code_natural ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
    semiri1406184849735516958ct_int: fun_nat_int ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
    semiri1408675320244567234ct_nat: fun_nat_nat ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
    semiri773545260158071498ct_rat: fun_nat_rat ).

tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
    inverse_inverse_rat: fun_rat_rat ).

tff(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
    at_top_nat: filter_nat ).

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

tff(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    eventu1038000079068216329at_nat: ( fun_Pr4796943543331071514_nat_o * filter1242075044329608583at_nat ) > $o ).

tff(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    filter3212408913953519116et_nat: ( fun_nat_set_nat * filter_set_nat * filter_nat ) > $o ).

tff(sy_c_Filter_Ofinite__subsets__at__top_001t__Nat__Onat,type,
    finite3254316476582989077op_nat: set_nat > filter_set_nat ).

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

tff(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
    finite_card_int: set_int > nat ).

tff(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Int__Oint_J,type,
    finite_card_list_int: set_list_int > nat ).

tff(sy_c_Finite__Set_Ocard_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    finite7325466520557071688st_nat: set_list_list_nat > nat ).

tff(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
    finite_card_list_nat: set_list_nat > nat ).

tff(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    finite249151656366948015at_nat: set_li5450038453877631591at_nat > nat ).

tff(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Product____Type__Ounit_J,type,
    finite3744894192721311803t_unit: set_li873732205721079667t_unit > nat ).

tff(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    finite5631907774883551598et_nat: set_list_set_nat > nat ).

tff(sy_c_Finite__Set_Ocard_001t__List__Olist_It__String__Ochar_J,type,
    finite1726956870759998265t_char: set_list_char > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
    finite_card_nat: fun_set_nat_nat ).

tff(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    finite6756421564338198497nt_int: fun_se4101287982975402108nt_nat ).

tff(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite711546835091564841at_nat: fun_se4097146774670233028at_nat ).

tff(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    finite3771342082235030671at_nat: set_Pr4329608150637261639at_nat > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
    finite410649719033368117t_unit: set_Product_unit > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Int__Oint_J,type,
    finite_card_set_int: set_set_int > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    finite2364142230527598318st_nat: set_set_list_nat > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
    finite_card_set_nat: set_set_nat > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    finite4053189226111446337nt_int: set_se6260736226359567993nt_int > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    finite4356350796350151305at_nat: set_se7855581050983116737at_nat > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    finite22775419188893461t_unit: set_set_Product_unit > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    finite1149291290879098388et_nat: set_set_set_nat > nat ).

tff(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
    finite_card_char: set_char > nat ).

tff(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
    finite_finite_int: set_int > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
    finite3922522038869484883st_int: set_list_int > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
    finite8100373058378681591st_nat: set_list_nat > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    finite500796754983035824at_nat: set_li5450038453877631591at_nat > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__String__Ochar_J,type,
    finite3619747586194699384t_char: set_list_char > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: fun_set_nat_o ).

tff(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    finite2998713641127702882nt_int: fun_se7249947736802449410_int_o ).

tff(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite6177210948735845034at_nat: fun_se4782206882984618170_nat_o ).

tff(sy_c_Finite__Set_Ofinite_001t__Product____Type__Ounit,type,
    finite4290736615968046902t_unit: set_Product_unit > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
    finite_finite_rat: set_rat > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
    finite6197958912794628473et_int: set_set_int > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
    finite1152437895449049373et_nat: set_set_nat > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    finite8744585540193469122nt_int: set_se6260736226359567993nt_int > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    finite9047747110432174090at_nat: set_se7855581050983116737at_nat > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__String__Ochar,type,
    finite_finite_char: set_char > $o ).

tff(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
    bij_betw_nat_nat: ( fun_nat_nat * set_nat ) > fun_set_nat_o ).

tff(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    comp_C7449957260575251196nteger: fun_Co8499238673658658103nteger > fun_fu1763652423599918571nteger ).

tff(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    comp_C3531382070062128313er_num: ( fun_Co8499238673658658103nteger * fun_num_Code_integer ) > fun_num_Code_integer ).

tff(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    comp_C1043215000170864188nteger: ( fun_Co8499238673658658103nteger * fun_Pr6596356882939945227nteger ) > fun_Pr6596356882939945227nteger ).

tff(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_C4049556595663050210at_nat: ( fun_Code_integer_nat * fun_nat_Code_integer ) > fun_nat_nat ).

tff(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__String__Ochar_001t__Code____Numeral__Ointeger,type,
    comp_C3856975138083468656nteger: ( fun_Co8487158583234504619r_char * fun_Co8499238673658658103nteger ) > fun_Co8487158583234504619r_char ).

tff(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
    comp_C7233611567195229654nteger: ( fun_Co1981900195452921617nteger * fun_Co8499238673658658103nteger ) > fun_Co1981900195452921617nteger ).

tff(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    comp_C7848664941760151970nteger: ( fun_Co1561183069634336733nteger * fun_Co8499238673658658103nteger ) > fun_Co1561183069634336733nteger ).

tff(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    comp_i1585864551200866970er_int: ( fun_int_Code_integer * fun_int_int ) > fun_int_Code_integer ).

tff(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Code____Numeral__Onatural_001t__Int__Oint,type,
    comp_i6850693561888474213al_int: ( fun_int_Code_natural * fun_int_int ) > fun_int_Code_natural ).

tff(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    comp_int_int_int: ( fun_int_int * fun_int_int ) > fun_int_int ).

tff(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat,type,
    comp_int_int_nat: ( fun_int_int * fun_nat_int ) > fun_nat_int ).

tff(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    comp_i4330622015127193916nt_int: ( fun_int_int * fun_Pr982841511892225912nt_int ) > fun_Pr982841511892225912nt_int ).

tff(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
    comp_int_nat_int: ( fun_int_nat * fun_int_int ) > fun_int_nat ).

tff(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_int_nat_nat: ( fun_int_nat * fun_nat_int ) > fun_nat_nat ).

tff(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Num__Onum_001t__Int__Oint,type,
    comp_int_num_int: ( fun_int_num * fun_int_int ) > fun_int_num ).

tff(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Num__Onum_001t__Nat__Onat,type,
    comp_int_num_nat: ( fun_int_num * fun_nat_int ) > fun_nat_num ).

tff(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    comp_i5941103828406351439nt_int: ( fun_in4211310926729683942nt_int * fun_Pr982841511892225912nt_int ) > fun_Pr2478283820680522243nt_int ).

tff(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Rat__Orat_001t__Int__Oint,type,
    comp_int_rat_int: ( fun_int_rat * fun_int_int ) > fun_int_rat ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001_Eo_001t__Nat__Onat,type,
    comp_nat_o_nat: ( fun_nat_o * fun_nat_nat ) > fun_nat_o ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    comp_n3898172953802868194er_nat: ( fun_nat_Code_integer * fun_nat_nat ) > fun_nat_Code_integer ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Nat__Onat,type,
    comp_n9163001964490475437al_nat: ( fun_nat_Code_natural * fun_nat_nat ) > fun_nat_Code_natural ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__Int__Oint,type,
    comp_nat_int_int: ( fun_nat_int * fun_int_nat ) > fun_int_int ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
    comp_nat_int_nat: ( fun_nat_int * fun_nat_nat ) > fun_nat_int ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
    comp_nat_nat_int: ( fun_nat_nat * fun_int_nat ) > fun_int_nat ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_nat_nat_nat: ( fun_nat_nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_n2711240574518091068at_nat: ( fun_nat_nat * fun_Pr2602117396471556196at_nat ) > fun_Pr2602117396471556196at_nat ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Num__Onum_001t__Int__Oint,type,
    comp_nat_num_int: ( fun_nat_num * fun_int_nat ) > fun_int_num ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Num__Onum_001t__Nat__Onat,type,
    comp_nat_num_nat: ( fun_nat_num * fun_nat_nat ) > fun_nat_num ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_n829915685812334579at_nat: ( fun_na1476630412504255114at_nat * fun_Pr2602117396471556196at_nat ) > fun_Pr2118491707810114563at_nat ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Rat__Orat_001t__Nat__Onat,type,
    comp_nat_rat_nat: ( fun_nat_rat * fun_nat_nat ) > fun_nat_rat ).

tff(sy_c_Fun_Ocomp_001t__Num__Onum_001t__Nat__Onat_001t__Int__Oint,type,
    comp_num_nat_int: ( fun_num_nat * fun_int_num ) > fun_int_nat ).

tff(sy_c_Fun_Ocomp_001t__Num__Onum_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_num_nat_nat: ( fun_num_nat * fun_nat_num ) > fun_nat_nat ).

tff(sy_c_Fun_Ocomp_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint,type,
    comp_num_num_int: ( fun_num_num * fun_int_num ) > fun_int_num ).

tff(sy_c_Fun_Ocomp_001t__Num__Onum_001t__Num__Onum_001t__Nat__Onat,type,
    comp_num_num_nat: ( fun_num_num * fun_nat_num ) > fun_nat_num ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    comp_P3428181540658402820nteger: ( fun_Pr6596356882939945227nteger * fun_Pr5876161306168176259nteger ) > fun_Pr6596356882939945227nteger ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    comp_P5356232565778667509nt_int: ( fun_Pr982841511892225912nt_int * fun_Pr2478283820680522243nt_int ) > fun_Pr982841511892225912nt_int ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_P1567445206330693457at_nat: ( fun_Pr2602117396471556196at_nat * fun_Pr2118491707810114563at_nat ) > fun_Pr2602117396471556196at_nat ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    comp_P3118722334806803912t_unit: ( fun_Pr7644814098969791837it_nat * fun_he4944555133437423336it_nat ) > fun_he5234484606919320760it_nat ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P4516649579262022418e_term: ( fun_Pr2054267358141571796nteger * fun_Pr2541390607678853719nteger * produc3611471421753121338e_term ) > produc8923325533196201883nteger ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P6784595103646691119e_term: ( fun_Pr4759914407354907917e_term * fun_Pr2541390607678853719nteger * produc3611471421753121338e_term ) > fun_Co3460408610320954864e_term ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P3263061180182656590e_term: ( fun_Pr1552678306225191715nt_int * fun_Pr3284083897292419283nt_int * produc5464280974217902119e_term ) > product_prod_int_int ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P7738237167614049274e_term: ( fun_Pr7653582204007996523e_term * fun_Pr3284083897292419283nt_int * produc5464280974217902119e_term ) > fun_in5730379987311621695e_term ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P2694320290640119590e_term: ( fun_Pr6160931176198666718nteger * fun_Pr6966570558394395095nteger * produc592931078481472324e_term ) > produc8923325533196201883nteger ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P9209726524401416377e_term: ( fun_Pr4557384415951816845e_term * fun_Pr6966570558394395095nteger * produc592931078481472324e_term ) > fun_Pr5098669352780829690e_term ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P812235655064240294e_term: ( fun_Pr8015448952463279699atural * fun_Pr5219672437278659779atural * produc7223384021016661839e_term ) > produc7822875418678951345atural ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P823182482625092890e_term: ( fun_Pr944821188406467171e_term * fun_Pr5219672437278659779atural * produc7223384021016661839e_term ) > fun_Pr285225112257423215e_term ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P8873622951661117094e_term: ( fun_Pr1987159835869338959nt_int * fun_Pr5010838083610845651nt_int * produc4155239895104388691e_term ) > product_prod_int_int ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P4791264009094562086e_term: ( fun_Pr5872836680101629291e_term * fun_Pr5010838083610845651nt_int * produc4155239895104388691e_term ) > fun_Pr7763680713840462443e_term ).

tff(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    comp_P8552900652980440422t_unit: ( fun_Pr2403869735082759103it_nat * fun_he8889679731534893401it_nat ) > fun_he1921015881043195817it_nat ).

tff(sy_c_Fun_Ocomp_001t__String__Ochar_001t__String__Ochar_001t__Code____Numeral__Ointeger,type,
    comp_c4562247934908617188nteger: ( fun_char_char * fun_Co8487158583234504619r_char ) > fun_Co8487158583234504619r_char ).

tff(sy_c_Fun_Ocomp_001t__String__Ochar_001t__String__Ochar_001t__String__Ochar,type,
    comp_char_char_char: ( fun_char_char * fun_char_char ) > fun_char_char ).

tff(sy_c_Fun_Ocomp_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
    comp_f228016600672082096nteger: ( fun_fu1763652423599918571nteger * fun_Co1981900195452921617nteger ) > fun_Co1981900195452921617nteger ).

tff(sy_c_Fun_Ocomp_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    comp_f6739083541385432444nteger: ( fun_fu4800041728187187127nteger * fun_Co1981900195452921617nteger ) > fun_Co1561183069634336733nteger ).

tff(sy_c_Fun_Ocomp_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    comp_f3848972703483044950nteger: ( fun_fu4800041728187187127nteger * fun_fu1763652423599918571nteger ) > fun_fu4800041728187187127nteger ).

tff(sy_c_Fun_Oid_001_Eo,type,
    id_o: fun_o_o ).

tff(sy_c_Fun_Oid_001t__Int__Oint,type,
    id_int: fun_int_int ).

tff(sy_c_Fun_Oid_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    id_list_Code_integer: fun_li4850912443864144983nteger ).

tff(sy_c_Fun_Oid_001t__Nat__Onat,type,
    id_nat: fun_nat_nat ).

tff(sy_c_Fun_Oid_001t__Num__Onum,type,
    id_num: fun_num_num ).

tff(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: fun_nat_nat > fun_set_nat_o ).

tff(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
    inj_on_nat_char: fun_nat_char > fun_set_nat_o ).

tff(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J_J,type,
    map_fu5639709166104062110iteral: ( fun_o_o * fun_fu4284519319507362051iteral * fun_o_5935606648521830764t_char ) > fun_o_8412685193225684464iteral ).

tff(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J,type,
    map_fu3469583328629365786iteral: ( fun_o_o * fun_fu8824737095519370695iteral ) > fun_fu4284519319507362051iteral ).

tff(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J,type,
    map_fu39305780192311794iteral: ( fun_o_o * fun_fu1976046603244687703iteral ) > fun_fu8824737095519370695iteral ).

tff(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J,type,
    map_fu4012007721304947738iteral: ( fun_o_o * fun_fu9120652711261886663iteral ) > fun_fu1976046603244687703iteral ).

tff(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J,type,
    map_fu7405565234982245702iteral: ( fun_o_o * fun_fu6635290411792845739iteral ) > fun_fu9120652711261886663iteral ).

tff(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001t__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_001t__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J,type,
    map_fu4664413887100446234iteral: ( fun_o_o * fun_fu1966733495413849543iteral ) > fun_fu6635290411792845739iteral ).

tff(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001t__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_001t__fun_It__String__Oliteral_Mt__String__Oliteral_J,type,
    map_fu8523206220227787418iteral: ( fun_o_o * fun_fu2195365108911689727iteral ) > fun_fu1966733495413849543iteral ).

tff(sy_c_Fun_Omap__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    map_fu2599414010547811884nteger: ( fun_Code_integer_int * fun_int_Code_integer ) > fun_fu6921880696561781257nteger ).

tff(sy_c_Fun_Omap__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Num__Onum_001t__Num__Onum,type,
    map_fu1227494855608507351um_num: ( fun_Code_integer_int * fun_num_num * fun_int_num ) > fun_Code_integer_num ).

tff(sy_c_Fun_Omap__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    map_fu1034890151506530863nteger: ( fun_Code_integer_int * fun_fu6921880696561781257nteger * fun_int_fun_int_int ) > fun_Co1981900195452921617nteger ).

tff(sy_c_Fun_Omap__fun_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
    map_fu1239815594074539274atural: ( fun_Code_natural_nat * fun_nat_Code_natural ) > fun_fu6924846375782198887atural ).

tff(sy_c_Fun_Omap__fun_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    map_fu7613509917203888380atural: ( fun_Code_natural_nat * fun_fu6924846375782198887atural * fun_nat_fun_nat_nat ) > fun_Co6813818151677833714atural ).

tff(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    map_fu3424225382358772769nt_rat: ( fun_int_int * fun_Pr1652463973008025892nt_rat ) > fun_fu7460908759652350970nt_rat ).

tff(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
    map_fu7945985449245211511nt_rat: ( fun_int_int * fun_fu7460908759652350970nt_rat * fun_in1696471567231710611nt_int ) > fun_int_fun_int_rat ).

tff(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
    map_fu4826362097070443709at_o_o: ( fun_in4514472496968388910at_nat * fun_o_o ) > fun_fu7075445697382584196_int_o ).

tff(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    map_fu2345160673673942751at_nat: ( fun_in4514472496968388910at_nat * fun_nat_nat * fun_Pr2602117396471556196at_nat ) > fun_int_nat ).

tff(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    map_fu3667384564859982768at_int: ( fun_in4514472496968388910at_nat * fun_Pr7647638413817135296at_int ) > fun_fu3064074204415222357nt_int ).

tff(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001t__fun_It__Int__Oint_M_Eo_J,type,
    map_fu3191548434056052140_int_o: ( fun_in4514472496968388910at_nat * fun_fu7075445697382584196_int_o * fun_Pr2028404973800837934_nat_o ) > fun_int_fun_int_o ).

tff(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    map_fu8294830438409148311nt_int: ( fun_in4514472496968388910at_nat * fun_fu3064074204415222357nt_int * fun_Pr2477239776880934807at_nat ) > fun_int_fun_int_int ).

tff(sy_c_Fun_Omap__fun_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral,type,
    map_fu9222038735678016191iteral: ( fun_li4850912443864144983nteger * fun_li5857714827575272844iteral * fun_li4211551766102051787t_char ) > fun_li9041563744671036928iteral ).

tff(sy_c_Fun_Omap__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    map_fu8638147718074629079nteger: ( fun_num_num * fun_int_Code_integer ) > fun_fu3380082687118895412nteger ).

tff(sy_c_Fun_Omap__fun_001t__Num__Onum_001t__Num__Onum_001t__fun_It__Num__Onum_Mt__Int__Oint_J_001t__fun_It__Num__Onum_Mt__Code____Numeral__Ointeger_J,type,
    map_fu8540202848259802501nteger: ( fun_num_num * fun_fu3380082687118895412nteger * fun_num_fun_num_int ) > fun_nu2537414170016844299nteger ).

tff(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
    map_fu898904425404107465nt_o_o: ( fun_ra4503298312962071866nt_int * fun_o_o * fun_Pr3238374608701176162_int_o ) > fun_rat_o ).

tff(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    map_fu5673905371560938248nt_rat: ( fun_ra4503298312962071866nt_int * fun_Pr1652463973008025892nt_rat ) > fun_fu1098312133097393581at_rat ).

tff(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
    map_fu5968964422797239547at_rat: ( fun_ra4503298312962071866nt_int * fun_fu1098312133097393581at_rat * fun_Pr1308109350235310159nt_int ) > fun_rat_fun_rat_rat ).

tff(sy_c_Fun_Omap__fun_001t__String__Oliteral_001t__List__Olist_It__String__Ochar_J_001_Eo_001_Eo,type,
    map_fu6698795530259886495ar_o_o: ( fun_li7473841323753414984t_char * fun_o_o ) > fun_fu6209626299996703082eral_o ).

tff(sy_c_Fun_Omap__fun_001t__String__Oliteral_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral,type,
    map_fu7398138089906338514iteral: ( fun_li7473841323753414984t_char * fun_li5857714827575272844iteral ) > fun_fu2195365108911689727iteral ).

tff(sy_c_Fun_Omap__fun_001t__String__Oliteral_001t__List__Olist_It__String__Ochar_J_001t__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_001t__fun_It__String__Oliteral_M_Eo_J,type,
    map_fu6041574301308154232eral_o: ( fun_li7473841323753414984t_char * fun_fu6209626299996703082eral_o * fun_li5260340055336554888char_o ) > fun_li4177005666941401710eral_o ).

tff(sy_c_Fun__Def_Ois__measure_001t__Int__Oint,type,
    fun_is_measure_int: fun_int_nat > $o ).

tff(sy_c_Fun__Def_Omax__strict,type,
    fun_max_strict: set_Pr4329608150637261639at_nat ).

tff(sy_c_Fun__Def_Omax__weak,type,
    fun_max_weak: set_Pr4329608150637261639at_nat ).

tff(sy_c_Fun__Def_Omin__strict,type,
    fun_min_strict: set_Pr4329608150637261639at_nat ).

tff(sy_c_Fun__Def_Omin__weak,type,
    fun_min_weak: set_Pr4329608150637261639at_nat ).

tff(sy_c_Fun__Def_Opair__leq,type,
    fun_pair_leq: set_Pr8693737435421807431at_nat ).

tff(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set_Pr8693737435421807431at_nat ).

tff(sy_c_Fun__Def_Oreduction__pair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_re2478310338295953701at_nat: produc1319942482725812455at_nat > $o ).

tff(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
    gcd_Gcd_int: set_int > int ).

tff(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
    gcd_Gcd_nat: fun_set_nat_nat ).

tff(sy_c_GCD_OGcd__class_OLcm_001t__Int__Oint,type,
    gcd_Lcm_int: set_int > int ).

tff(sy_c_GCD_OGcd__class_OLcm_001t__Nat__Onat,type,
    gcd_Lcm_nat: fun_set_nat_nat ).

tff(sy_c_GCD_Obezw,type,
    bezw: nat > fun_na1173468842265550146nt_int ).

tff(sy_c_GCD_Obezw__rel,type,
    bezw_rel: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_GCD_Ogcd__class_Ogcd_001t__Code____Numeral__Ointeger,type,
    gcd_gcd_Code_integer: fun_Co1981900195452921617nteger ).

tff(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
    gcd_gcd_int: fun_int_fun_int_int ).

tff(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
    gcd_gcd_nat: fun_nat_fun_nat_nat ).

tff(sy_c_GCD_Ogcd__class_Olcm_001t__Code____Numeral__Ointeger,type,
    gcd_lcm_Code_integer: fun_Co1981900195452921617nteger ).

tff(sy_c_GCD_Ogcd__class_Olcm_001t__Int__Oint,type,
    gcd_lcm_int: fun_int_fun_int_int ).

tff(sy_c_GCD_Ogcd__class_Olcm_001t__Nat__Onat,type,
    gcd_lcm_nat: fun_nat_fun_nat_nat ).

tff(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
    abs_abs_Code_integer: fun_Co8499238673658658103nteger ).

tff(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
    abs_abs_int: fun_int_int ).

tff(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
    abs_abs_rat: rat > rat ).

tff(sy_c_Groups_Ocomm__monoid_001t__Nat__Onat,type,
    comm_monoid_nat: fun_nat_fun_nat_nat > fun_nat_o ).

tff(sy_c_Groups_Ogroup_001t__Code____Numeral__Ointeger,type,
    group_Code_integer: ( fun_Co1981900195452921617nteger * code_integer * fun_Co8499238673658658103nteger ) > $o ).

tff(sy_c_Groups_Ogroup_001t__Int__Oint,type,
    group_int: ( fun_int_fun_int_int * int * fun_int_int ) > $o ).

tff(sy_c_Groups_Ogroup_001t__Rat__Orat,type,
    group_rat: ( fun_rat_fun_rat_rat * rat * fun_rat_rat ) > $o ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
    minus_8373710615458151222nteger: fun_Co1981900195452921617nteger ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Onatural,type,
    minus_7197305767214868737atural: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
    minus_minus_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
    minus_minus_rat: ( rat * rat ) > rat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
    minus_minus_set_int: ( set_int * set_int ) > set_int ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    minus_7954133019191499631st_nat: ( set_list_nat * set_list_nat ) > set_list_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: ( set_nat * set_nat ) > set_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    minus_1356011639430497352at_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    minus_3314409938677909166at_nat: ( set_Pr4329608150637261639at_nat * set_Pr4329608150637261639at_nat ) > set_Pr4329608150637261639at_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    minus_6452836326544984404t_unit: ( set_Product_unit * set_Product_unit ) > set_Product_unit ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    minus_2163939370556025621et_nat: ( set_set_nat * set_set_nat ) > set_set_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    minus_2612819937483484256nt_int: ( set_se6260736226359567993nt_int * set_se6260736226359567993nt_int ) > set_se6260736226359567993nt_int ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    minus_4207664762107033000at_nat: ( set_se7855581050983116737at_nat * set_se7855581050983116737at_nat ) > set_se7855581050983116737at_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__fun_It__Int__Oint_M_Eo_J,type,
    minus_7151557278095657260_int_o: ( fun_int_o * fun_int_o ) > fun_int_o ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__fun_It__Nat__Onat_M_Eo_J,type,
    minus_8972599623528345224_nat_o: ( fun_nat_o * fun_nat_o ) > fun_nat_o ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    minus_6087510627544308050_nat_o: ( fun_set_nat_o * fun_set_nat_o ) > fun_set_nat_o ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
    minus_3219807633925007451_int_o: ( fun_se7249947736802449410_int_o * fun_se7249947736802449410_int_o ) > fun_se7249947736802449410_int_o ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    minus_752066780107176211_nat_o: ( fun_se4782206882984618170_nat_o * fun_se4782206882984618170_nat_o ) > fun_se4782206882984618170_nat_o ).

tff(sy_c_Groups_Omonoid_001t__Nat__Onat,type,
    monoid_nat: fun_nat_fun_nat_nat > fun_nat_o ).

tff(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
    one_one_Code_integer: code_integer ).

tff(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Onatural,type,
    one_one_Code_natural: code_natural ).

tff(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

tff(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

tff(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
    one_one_rat: rat ).

tff(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
    plus_p5714425477246183910nteger: fun_Co1981900195452921617nteger ).

tff(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Onatural,type,
    plus_p4538020629002901425atural: fun_Co6813818151677833714atural ).

tff(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
    plus_plus_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
    plus_plus_num: num > fun_num_num ).

tff(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
    plus_plus_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Groups_Oplus__class_Oplus_001t__String__Oliteral,type,
    plus_plus_literal: ( literal * literal ) > literal ).

tff(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
    sgn_sgn_Code_integer: fun_Co8499238673658658103nteger ).

tff(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
    sgn_sgn_int: fun_int_int ).

tff(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
    sgn_sgn_rat: rat > rat ).

tff(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
    times_3573771949741848930nteger: fun_Co1981900195452921617nteger ).

tff(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Onatural,type,
    times_2397367101498566445atural: code_natural > fun_Co7398788559141407565atural ).

tff(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
    times_times_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
    times_times_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
    times_times_num: num > fun_num_num ).

tff(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
    times_times_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
    uminus1351360451143612070nteger: fun_Co8499238673658658103nteger ).

tff(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
    uminus_uminus_int: fun_int_int ).

tff(sy_c_Groups_Ouminus__class_Ouminus_001t__Product____Type__Ounit,type,
    uminus2952777764628376836t_unit: product_unit > product_unit ).

tff(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
    uminus_uminus_rat: fun_rat_rat ).

tff(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Ointeger,type,
    zero_z3403309356797280102nteger: code_integer ).

tff(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Onatural,type,
    zero_z2226904508553997617atural: code_natural ).

tff(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
    zero_zero_int: int ).

tff(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

tff(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
    zero_zero_rat: rat ).

tff(sy_c_Groups_Ozero__class_Ozero_001t__String__Oliteral,type,
    zero_zero_literal: literal ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    groups7873554091576472773nteger: ( fun_int_Code_integer * set_int ) > code_integer ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Code____Numeral__Onatural,type,
    groups6697149243333190288atural: ( fun_int_Code_natural * set_int ) > code_natural ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
    groups4538972089207619220nt_int: ( fun_int_int * set_int ) > int ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Nat__Onat,type,
    groups4541462559716669496nt_nat: ( fun_int_nat * set_int ) > nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Rat__Orat,type,
    groups3906332499630173760nt_rat: ( fun_int_rat * set_int ) > rat ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    groups4396056296759096172at_nat: ( fun_list_nat_nat * set_list_nat ) > nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    groups7501900531339628137nteger: fun_nat_Code_integer > fun_se7586849273513198688nteger ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
    groups6325495683096345652atural: ( fun_nat_Code_natural * set_nat ) > code_natural ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
    groups3539618377306564664at_int: fun_nat_int > fun_set_nat_int ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
    groups3542108847815614940at_nat: fun_nat_nat > fun_set_nat_nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Rat__Orat,type,
    groups2906978787729119204at_rat: fun_nat_rat > fun_set_nat_rat ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    groups975429370522433651at_int: fun_Pr7647638413817135296at_int > fun_se9142667792015812128at_int ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    groups977919841031483927at_nat: fun_Pr2602117396471556196at_nat > fun_se4097146774670233028at_nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Rat__Orat,type,
    groups342789780944988191at_rat: fun_Pr8317260874932935276at_rat > fun_se588918216276836300at_rat ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Ounit_001t__Code____Numeral__Ointeger,type,
    groups5393439102872367408nteger: ( fun_Pr1820225556744648189nteger * set_Product_unit ) > code_integer ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Ounit_001t__Int__Oint,type,
    groups8764306707065906943it_int: ( fun_Product_unit_int * set_Product_unit ) > int ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Ounit_001t__Nat__Onat,type,
    groups8766797177574957219it_nat: ( fun_Product_unit_nat * set_Product_unit ) > nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Ounit_001t__Rat__Orat,type,
    groups8131667117488461483it_rat: ( fun_Product_unit_rat * set_Product_unit ) > rat ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
    groups9190459664516455967nteger: ( fun_se7586849273513198688nteger * set_set_nat ) > code_integer ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Int__Oint,type,
    groups8292507037921071086at_int: ( fun_set_nat_int * set_set_nat ) > int ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    groups8294997508430121362at_nat: ( fun_set_nat_nat * set_set_nat ) > nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Rat__Orat,type,
    groups7659867448343625626at_rat: ( fun_set_nat_rat * set_set_nat ) > rat ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Int__Oint,type,
    groups182716853052024971nt_int: ( fun_se9146809000320981208nt_int * set_se6260736226359567993nt_int ) > int ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Nat__Onat,type,
    groups185207323561075247nt_nat: ( fun_se4101287982975402108nt_nat * set_se6260736226359567993nt_int ) > nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Code____Numeral__Ointeger,type,
    groups5114918151688220676nteger: ( fun_se2775971902884351953nteger * set_se7855581050983116737at_nat ) > code_integer ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Int__Oint,type,
    groups178575644746855891at_int: ( fun_se9142667792015812128at_int * set_se7855581050983116737at_nat ) > int ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    groups181066115255906167at_nat: ( fun_se4097146774670233028at_nat * set_se7855581050983116737at_nat ) > nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Rat__Orat,type,
    groups8769308092024186239at_rat: ( fun_se588918216276836300at_rat * set_se7855581050983116737at_nat ) > rat ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    groups3827104343326376752nteger: ( fun_int_Code_integer * set_int ) > code_integer ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Code____Numeral__Onatural,type,
    groups2650699495083094267atural: ( fun_int_Code_natural * set_int ) > code_natural ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Int__Oint,type,
    groups1705073143266064639nt_int: ( fun_int_int * set_int ) > int ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Nat__Onat,type,
    groups1707563613775114915nt_nat: ( fun_int_nat * set_int ) > nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Rat__Orat,type,
    groups1072433553688619179nt_rat: ( fun_int_rat * set_int ) > rat ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint,type,
    groups2905156660866384563at_int: ( fun_list_nat_int * set_list_nat ) > int ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    groups2907647131375434839at_nat: ( fun_list_nat_nat * set_list_nat ) > nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    groups3455450783089532116nteger: fun_nat_Code_integer > fun_se7586849273513198688nteger ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
    groups2279045934846249631atural: ( fun_nat_Code_natural * set_nat ) > code_natural ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Int__Oint,type,
    groups705719431365010083at_int: fun_nat_int > fun_set_nat_int ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
    groups708209901874060359at_nat: fun_nat_nat > fun_set_nat_nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Rat__Orat,type,
    groups73079841787564623at_rat: fun_nat_rat > fun_set_nat_rat ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
    groups1230400874837758585nteger: fun_Pr6026251430685297nteger > fun_se2775971902884351953nteger ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    groups4075276357253098568at_int: fun_Pr7647638413817135296at_int > fun_se9142667792015812128at_int ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    groups4077766827762148844at_nat: fun_Pr2602117396471556196at_nat > fun_se4097146774670233028at_nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Ounit_001t__Code____Numeral__Ointeger,type,
    groups8185359811944078981nteger: ( fun_Pr1820225556744648189nteger * set_Product_unit ) > code_integer ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Ounit_001t__Code____Numeral__Onatural,type,
    groups7008954963700796496atural: ( fun_Pr6278675377255837256atural * set_Product_unit ) > code_natural ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Ounit_001t__Int__Oint,type,
    groups393930778517121108it_int: ( fun_Product_unit_int * set_Product_unit ) > int ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Ounit_001t__Nat__Onat,type,
    groups396421249026171384it_nat: ( fun_Product_unit_nat * set_Product_unit ) > nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Ounit_001t__Rat__Orat,type,
    groups8984663225794451456it_rat: ( fun_Product_unit_rat * set_Product_unit ) > rat ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
    groups2761809935038513290nteger: ( fun_se7586849273513198688nteger * set_set_nat ) > code_integer ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Set__Oset_It__Nat__Onat_J_001t__Int__Oint,type,
    groups4246057289670975065at_int: ( fun_set_nat_int * set_set_nat ) > int ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    groups4248547760180025341at_nat: ( fun_set_nat_nat * set_set_nat ) > nat ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Set__Oset_It__Nat__Onat_J_001t__Rat__Orat,type,
    groups3613417700093529605at_rat: ( fun_set_nat_rat * set_set_nat ) > rat ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Int__Oint,type,
    groups1216284892592621664nt_int: ( fun_se9146809000320981208nt_int * set_se6260736226359567993nt_int ) > int ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Int__Oint,type,
    groups1212143684287452584at_int: ( fun_se9142667792015812128at_int * set_se7855581050983116737at_nat ) > int ).

tff(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
    groups4561878855575611511st_nat: fun_list_nat_nat ).

tff(sy_c_HOL_ONO__MATCH_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    nO_MAT8252062027627875367nteger: fun_Co2081214437782444048eger_o ).

tff(sy_c_HOL_ONO__MATCH_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    nO_MAT8427913294028938742er_int: code_integer > fun_int_o ).

tff(sy_c_HOL_ONO__MATCH_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    nO_MAT8430403764537989018er_nat: code_integer > fun_nat_o ).

tff(sy_c_HOL_ONO__MATCH_001t__Code____Numeral__Ointeger_001t__Rat__Orat,type,
    nO_MAT7795273704451493282er_rat: code_integer > fun_rat_o ).

tff(sy_c_HOL_ONO__MATCH_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    nO_MAT9066612773553876470nteger: int > fun_Code_integer_o ).

tff(sy_c_HOL_ONO__MATCH_001t__Int__Oint_001t__Int__Oint,type,
    nO_MATCH_int_int: fun_int_fun_int_o ).

tff(sy_c_HOL_ONO__MATCH_001t__Int__Oint_001t__Nat__Onat,type,
    nO_MATCH_int_nat: fun_int_fun_nat_o ).

tff(sy_c_HOL_ONO__MATCH_001t__Int__Oint_001t__Rat__Orat,type,
    nO_MATCH_int_rat: int > fun_rat_o ).

tff(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    nO_MAT8694959213317031834nteger: nat > fun_Code_integer_o ).

tff(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Int__Oint,type,
    nO_MATCH_nat_int: fun_nat_fun_int_o ).

tff(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Nat__Onat,type,
    nO_MATCH_nat_nat: fun_nat_fun_nat_o ).

tff(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Rat__Orat,type,
    nO_MATCH_nat_rat: nat > fun_rat_o ).

tff(sy_c_HOL_ONO__MATCH_001t__Rat__Orat_001t__Int__Oint,type,
    nO_MATCH_rat_int: rat > fun_int_o ).

tff(sy_c_HOL_ONO__MATCH_001t__Rat__Orat_001t__Rat__Orat,type,
    nO_MATCH_rat_rat: fun_rat_fun_rat_o ).

tff(sy_c_HOL_OThe_001t__Int__Oint,type,
    the_int: fun_int_o > int ).

tff(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    the_Pr4378521158711661632nt_int: fun_Pr3238374608701176162_int_o > product_prod_int_int ).

tff(sy_c_HOL_Odefault__class_Odefault_001t__Product____Type__Ounit,type,
    defaul566961228789861419t_unit: product_unit ).

tff(sy_c_Heap__Time__Monad_OHeap_OHeap_001t__Product____Type__Ounit,type,
    heap_T6183433275982383450t_unit: fun_he5234484606919320760it_nat > heap_T5738788834812785303t_unit ).

tff(sy_c_Heap__Time__Monad_OHeap_OHeap_001tf__a,type,
    heap_Time_Heap_a2: fun_he1921015881043195817it_nat > heap_Time_Heap_a ).

tff(sy_c_Heap__Time__Monad_OHeap_Osize__Heap_001t__Product____Type__Ounit,type,
    heap_T4142866422068808505t_unit: ( fun_Product_unit_nat * heap_T5738788834812785303t_unit ) > nat ).

tff(sy_c_Heap__Time__Monad_Oassert_001t__Product____Type__Ounit,type,
    heap_T4208721593536448476t_unit: fun_Product_unit_o > fun_Pr292408699144966967t_unit ).

tff(sy_c_Heap__Time__Monad_Oassert_001tf__a,type,
    heap_Time_assert_a: fun_a_o > fun_a_6093291656547882009Heap_a ).

tff(sy_c_Heap__Time__Monad_Obind_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
    heap_T2633723481742716231t_unit: ( heap_T5738788834812785303t_unit * fun_Pr292408699144966967t_unit ) > heap_T5738788834812785303t_unit ).

tff(sy_c_Heap__Time__Monad_Obind_001t__Product____Type__Ounit_001tf__a,type,
    heap_T757603679106148408unit_a: ( heap_T5738788834812785303t_unit * fun_Pr5215307506330550056Heap_a ) > heap_Time_Heap_a ).

tff(sy_c_Heap__Time__Monad_Obind_001tf__a_001t__Product____Type__Ounit,type,
    heap_T8685611227969916822t_unit: ( heap_Time_Heap_a * fun_a_2990225006182978344t_unit ) > heap_T5738788834812785303t_unit ).

tff(sy_c_Heap__Time__Monad_Obind_001tf__a_001tf__a,type,
    heap_Time_bind_a_a: ( heap_Time_Heap_a * fun_a_6093291656547882009Heap_a ) > heap_Time_Heap_a ).

tff(sy_c_Heap__Time__Monad_Oeffect_001t__Product____Type__Ounit,type,
    heap_T6553295506729943825t_unit: ( heap_T5738788834812785303t_unit * heap_e7401611519738050253t_unit * heap_e7401611519738050253t_unit ) > fun_Pr4402817529748954391_nat_o ).

tff(sy_c_Heap__Time__Monad_Oeffect_001tf__a,type,
    heap_Time_effect_a: ( heap_Time_Heap_a * heap_e7401611519738050253t_unit * heap_e7401611519738050253t_unit * a ) > fun_nat_o ).

tff(sy_c_Heap__Time__Monad_Oexecute_001t__Product____Type__Ounit,type,
    heap_T875086893843062177t_unit: heap_T5738788834812785303t_unit > fun_he5234484606919320760it_nat ).

tff(sy_c_Heap__Time__Monad_Oexecute_001tf__a,type,
    heap_Time_execute_a: heap_Time_Heap_a > fun_he1921015881043195817it_nat ).

tff(sy_c_Heap__Time__Monad_Oguard_001t__Product____Type__Ounit,type,
    heap_T8440541562793052209t_unit: ( fun_he8413527710729349838unit_o * fun_he4944555133437423336it_nat ) > heap_T5738788834812785303t_unit ).

tff(sy_c_Heap__Time__Monad_Oguard_001tf__a,type,
    heap_Time_guard_a: ( fun_he8413527710729349838unit_o * fun_he8889679731534893401it_nat ) > heap_Time_Heap_a ).

tff(sy_c_Heap__Time__Monad_Oheap_001t__Product____Type__Ounit,type,
    heap_T6927113302350381334t_unit: fun_he4944555133437423336it_nat > heap_T5738788834812785303t_unit ).

tff(sy_c_Heap__Time__Monad_Oheap_001tf__a,type,
    heap_Time_heap_a: fun_he8889679731534893401it_nat > heap_Time_Heap_a ).

tff(sy_c_Heap__Time__Monad_Oraise_001t__Product____Type__Ounit,type,
    heap_T2927564422264180874t_unit: list_char > heap_T5738788834812785303t_unit ).

tff(sy_c_Heap__Time__Monad_Oraise_001tf__a,type,
    heap_Time_raise_a: list_char > heap_Time_Heap_a ).

tff(sy_c_Heap__Time__Monad_Oreturn_001t__Product____Type__Ounit,type,
    heap_T7507251653302230130t_unit: fun_Pr292408699144966967t_unit ).

tff(sy_c_Heap__Time__Monad_Oreturn_001tf__a,type,
    heap_Time_return_a: fun_a_6093291656547882009Heap_a ).

tff(sy_c_Heap__Time__Monad_Osuccess_001t__Product____Type__Ounit,type,
    heap_T584514906347983379t_unit: heap_T5738788834812785303t_unit > fun_he8413527710729349838unit_o ).

tff(sy_c_Heap__Time__Monad_Osuccess_001tf__a,type,
    heap_Time_success_a: heap_Time_Heap_a > fun_he8413527710729349838unit_o ).

tff(sy_c_Heap__Time__Monad_Otap_001t__Product____Type__Ounit,type,
    heap_T560649228465745139t_unit: fun_he5123621388056232515t_unit > heap_T5738788834812785303t_unit ).

tff(sy_c_Heap__Time__Monad_Otap_001tf__a,type,
    heap_Time_tap_a: fun_he6342020308413958580unit_a > heap_Time_Heap_a ).

tff(sy_c_Heap__Time__Monad_OtimeFrame_001t__Product____Type__Ounit,type,
    heap_T3616969660504097270t_unit: ( nat * option8956607266484857688it_nat ) > option8956607266484857688it_nat ).

tff(sy_c_Heap__Time__Monad_OtimeFrame_001tf__a,type,
    heap_T7616092557645711335rame_a: ( nat * option3562590408128118217it_nat ) > option3562590408128118217it_nat ).

tff(sy_c_Heap__Time__Monad_OtimeFrame__rel_001t__Product____Type__Ounit,type,
    heap_T996182799752388649t_unit: fun_Pr6341106346053136302_nat_o ).

tff(sy_c_Heap__Time__Monad_OtimeFrame__rel_001tf__a,type,
    heap_T8132184524487034138_rel_a: fun_Pr782666833218042862_nat_o ).

tff(sy_c_Heap__Time__Monad_Oureturn_001t__Product____Type__Ounit,type,
    heap_T4284346855313245393t_unit: fun_Pr292408699144966967t_unit ).

tff(sy_c_Heap__Time__Monad_Oureturn_001tf__a,type,
    heap_Time_ureturn_a: fun_a_6093291656547882009Heap_a ).

tff(sy_c_Heap__Time__Monad_Owait,type,
    heap_Time_wait: nat > heap_T5738788834812785303t_unit ).

tff(sy_c_Int_OAbs__Integ,type,
    abs_Integ: fun_Pr7647638413817135296at_int ).

tff(sy_c_Int_ONeg,type,
    neg: fun_num_int ).

tff(sy_c_Int_OPos,type,
    pos: fun_num_int ).

tff(sy_c_Int_ORep__Integ,type,
    rep_Integ: fun_in4514472496968388910at_nat ).

tff(sy_c_Int_Ocr__int,type,
    cr_int: fun_Pr4738963024209204295_int_o ).

tff(sy_c_Int_Odup,type,
    dup: fun_int_int ).

tff(sy_c_Int_Oint_OAbs__int,type,
    abs_int: fun_se9142667792015812128at_int ).

tff(sy_c_Int_Oint_ORep__int,type,
    rep_int: fun_in8804710076054006414at_nat ).

tff(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set_Pr958786334691620121nt_int ).

tff(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set_Pr958786334691620121nt_int ).

tff(sy_c_Int_Ointrel,type,
    intrel: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Int_Onat,type,
    nat2: fun_int_nat ).

tff(sy_c_Int_Opcr__int,type,
    pcr_int: fun_Pr4738963024209204295_int_o ).

tff(sy_c_Int_Oring__1__class_OInts_001t__Code____Numeral__Ointeger,type,
    ring_11222124179247155820nteger: set_Code_integer ).

tff(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
    ring_1_Ints_int: set_int ).

tff(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
    ring_1_Ints_rat: set_rat ).

tff(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
    ring_18347121197199848620nteger: fun_int_Code_integer ).

tff(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
    ring_1_of_int_int: fun_int_int ).

tff(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
    ring_1_of_int_rat: fun_int_rat ).

tff(sy_c_Int_Osub,type,
    sub: num > fun_num_int ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
    inf_inf_int: fun_int_fun_int_int ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
    inf_inf_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__Product____Type__Ounit,type,
    inf_inf_Product_unit: ( product_unit * product_unit ) > product_unit ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    inf_in2572325071724192079at_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
    inf_in94156054473216830char_o: ( fun_list_char_o * fun_list_char_o ) > fun_list_char_o ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    inf_in1782598092406300404_int_o: ( fun_Pr3238374608701176162_int_o * fun_Pr3238374608701176162_int_o ) > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Lattices_Osemilattice__neutr_001t__Nat__Onat,type,
    semila9081495762789891438tr_nat: fun_nat_fun_nat_nat > fun_nat_o ).

tff(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
    semila1623282765462674594er_nat: ( fun_nat_fun_nat_nat * nat * fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > $o ).

tff(sy_c_Lattices_Osemilattice__order_001t__Nat__Onat,type,
    semila1248733672344298208er_nat: ( fun_nat_fun_nat_nat * fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > $o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
    sup_sup_int: fun_int_fun_int_int ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
    sup_sup_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Ounit,type,
    sup_sup_Product_unit: ( product_unit * product_unit ) > product_unit ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: ( set_nat * set_nat ) > set_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    sup_su5525570899277871387at_nat: ( set_Pr4329608150637261639at_nat * set_Pr4329608150637261639at_nat ) > set_Pr4329608150637261639at_nat ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
    lattic8263393255366662781ax_int: set_int > int ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
    lattic8265883725875713057ax_nat: fun_set_nat_nat ).

tff(sy_c_Lattices__Big_Osemilattice__neutr__set_OF_001t__Nat__Onat,type,
    lattic7826324295020591184_F_nat: fun_nat_fun_nat_nat > fun_na106483476154090400at_nat ).

tff(sy_c_Lifting_OQuotient_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral,type,
    quotie6109894551476798619iteral: ( fun_li5260340055336554888char_o * fun_li5857714827575272844iteral * fun_li7473841323753414984t_char * fun_li9104157641623487077eral_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    quotie1194848508323700631at_int: ( fun_Pr2028404973800837934_nat_o * fun_Pr7647638413817135296at_int * fun_in4514472496968388910at_nat * fun_Pr4738963024209204295_int_o ) > $o ).

tff(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: ( list_nat * list_nat ) > list_nat ).

tff(sy_c_List_Oappend_001t__String__Ochar,type,
    append_char: fun_li7251496822767205277t_char ).

tff(sy_c_List_Odistinct_001t__Int__Oint,type,
    distinct_int: list_int > $o ).

tff(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
    distinct_list_nat: list_list_nat > $o ).

tff(sy_c_List_Odistinct_001t__Nat__Onat,type,
    distinct_nat: list_nat > $o ).

tff(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    distin6923225563576452346at_nat: list_P6011104703257516679at_nat > $o ).

tff(sy_c_List_Odistinct_001t__Product____Type__Ounit,type,
    distin4812975378522102150t_unit: list_Product_unit > $o ).

tff(sy_c_List_Odistinct_001t__Set__Oset_It__Nat__Onat_J,type,
    distinct_set_nat: list_set_nat > $o ).

tff(sy_c_List_Odistinct_001t__String__Ochar,type,
    distinct_char: fun_list_char_o ).

tff(sy_c_List_Odrop_001t__Nat__Onat,type,
    drop_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Ofold_001t__Int__Oint_001t__Int__Oint,type,
    fold_int_int: ( fun_int_fun_int_int * list_int ) > fun_int_int ).

tff(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
    fold_nat_nat: ( fun_nat_fun_nat_nat * list_nat ) > fun_nat_nat ).

tff(sy_c_List_Olast_001t__Nat__Onat,type,
    last_nat: fun_list_nat_nat ).

tff(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
    linord1735203802627413978nt_int: ( fun_int_int * list_int ) > list_int ).

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

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

tff(sy_c_List_Olist_OCons_001t__Code____Numeral__Ointeger,type,
    cons_Code_integer: ( code_integer * list_Code_integer ) > list_Code_integer ).

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

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

tff(sy_c_List_Olist_OCons_001t__String__Ochar,type,
    cons_char: char > fun_li3177621261580724543t_char ).

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_ONil_001t__String__Ochar,type,
    nil_char: list_char ).

tff(sy_c_List_Olist_Olist__all2_001t__String__Ochar_001t__String__Ochar,type,
    list_all2_char_char: fun_char_fun_char_o > fun_li5260340055336554888char_o ).

tff(sy_c_List_Olist_Olist__all_001t__String__Ochar,type,
    list_all_char: fun_char_o > fun_list_char_o ).

tff(sy_c_List_Olist_Omap_001t__Code____Numeral__Ointeger_001t__String__Ochar,type,
    map_Co843364623960961780r_char: fun_Co8487158583234504619r_char > fun_li4211551766102051787t_char ).

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_Omap_001t__String__Ochar_001t__Code____Numeral__Ointeger,type,
    map_ch5843194146513361140nteger: fun_ch2863552307440546731nteger > fun_li1966387620041874891nteger ).

tff(sy_c_List_Olist_Omap_001t__String__Ochar_001t__String__Ochar,type,
    map_char_char: fun_char_char > fun_li3177621261580724543t_char ).

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

tff(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
    set_list_nat2: list_list_nat > set_list_nat ).

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

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Ounit,type,
    set_Product_unit2: list_Product_unit > set_Product_unit ).

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__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    set_se4746441305218329646nt_int: list_s8839374986383574687nt_int > set_se6260736226359567993nt_int ).

tff(sy_c_List_Olist_Oset_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_se5049602875457034614at_nat: list_s1210847774152347623at_nat > set_se7855581050983116737at_nat ).

tff(sy_c_List_Olist_Oset_001t__String__Ochar,type,
    set_char2: list_char > set_char ).

tff(sy_c_List_Olist_Osize__list_001t__Typerep__Otyperep,type,
    size_list_typerep: ( fun_typerep_nat * list_typerep ) > nat ).

tff(sy_c_List_Onth_001t__Nat__Onat,type,
    nth_nat: list_nat > fun_nat_nat ).

tff(sy_c_List_Oord_Olexordp_001t__String__Ochar,type,
    lexordp_char: fun_char_fun_char_o > fun_li5260340055336554888char_o ).

tff(sy_c_List_Oord_Olexordp__eq_001t__String__Ochar,type,
    lexordp_eq_char: fun_char_fun_char_o > fun_li5260340055336554888char_o ).

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__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

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

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

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

tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: ( option_num * fun_nat_option_num ) > fun_nat_option_num ).

tff(sy_c_Nat_Onat_Opred,type,
    pred: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Code____Numeral__Ointeger,type,
    semiri2551628265256775642nteger: set_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Code____Numeral__Onatural,type,
    semiri1375223417013493157atural: set_Code_natural ).

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_ONats_001t__Nat__Onat,type,
    semiring_1_Nats_nat: set_nat ).

tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Rat__Orat,type,
    semiring_1_Nats_rat: set_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Onatural,type,
    semiri3763490453095760265atural: fun_nat_Code_natural ).

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__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Code____Numeral__Ointeger,type,
    semiri4055485073559036834nteger: ( fun_Co8499238673658658103nteger * nat ) > fun_Co8499238673658658103nteger ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Code____Numeral__Onatural,type,
    semiri2879080225315754349atural: ( fun_Co7398788559141407565atural * nat * code_natural ) > code_natural ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
    semiri8420488043553186161ux_int: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
    semiri8422978514062236437ux_nat: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
    semiri7787848453975740701ux_rat: ( fun_rat_rat * nat * rat ) > rat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J,type,
    size_s1564113455978345259t_unit: heap_T5738788834812785303t_unit > nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    size_s3023201423986296836st_nat: list_list_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: fun_list_nat_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Ounit_J,type,
    size_s245203480648594047t_unit: list_Product_unit > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    size_s3254054031482475050et_nat: list_set_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    size_s7141307186832570123nt_int: list_s8839374986383574687nt_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    size_s8736152011456118867at_nat: list_s1210847774152347623at_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__String__Ochar_J,type,
    size_size_list_char: fun_list_char_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: fun_num_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
    size_size_option_num: fun_option_num_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s8766407808098229740it_nat: option8956607266484857688it_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s2363601347547812957it_nat: option3562590408128118217it_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
    size_size_char: char > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__String__Oliteral,type,
    size_size_literal: literal > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Typerep__Otyperep,type,
    size_size_typerep: fun_typerep_nat ).

tff(sy_c_Nat__Bijection_Oint__decode,type,
    nat_int_decode: fun_nat_int ).

tff(sy_c_Nat__Bijection_Olist__decode,type,
    nat_list_decode: nat > list_nat ).

tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: fun_nat_fun_nat_o ).

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: fun_list_nat_nat ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun_li1025265955262268170_nat_o ).

tff(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: fun_na1476630412504255114at_nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > fun_na1476630412504255114at_nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: fun_nat_set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Osum__decode,type,
    nat_sum_decode: nat > sum_sum_nat_nat ).

tff(sy_c_Nat__Bijection_Osum__encode,type,
    nat_sum_encode: sum_sum_nat_nat > nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_Num_OBitM,type,
    bitM: fun_num_num ).

tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

tff(sy_c_Num_Onat__of__num,type,
    nat_of_num: fun_num_nat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
    neg_nu8804712462038260780nteger: fun_Co8499238673658658103nteger ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
    neg_numeral_dbl_rat: rat > rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
    neg_nu7757733837767384882nteger: fun_Co8499238673658658103nteger ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
    neg_nu3179335615603231917ec_rat: rat > rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
    neg_nu5831290666863070958nteger: fun_Co8499238673658658103nteger ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
    neg_nu5219082963157363817nc_rat: rat > rat ).

tff(sy_c_Num_Oneg__numeral__class_Ois__num_001t__Code____Numeral__Ointeger,type,
    neg_nu2435721316971906650nteger: fun_Code_integer_o ).

tff(sy_c_Num_Oneg__numeral__class_Ois__num_001t__Int__Oint,type,
    neg_nu7441568979088483369um_int: fun_int_o ).

tff(sy_c_Num_Oneg__numeral__class_Ois__num_001t__Rat__Orat,type,
    neg_nu6808929389511037909um_rat: fun_rat_o ).

tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Code____Numeral__Ointeger,type,
    neg_nu5755505904847501662nteger: ( num * num ) > code_integer ).

tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
    neg_numeral_sub_int: num > fun_num_int ).

tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Rat__Orat,type,
    neg_numeral_sub_rat: ( num * num ) > rat ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

tff(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
    case_num_option_num: ( option_num * fun_num_option_num * fun_num_option_num ) > fun_num_option_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: fun_nat_num ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: fun_num_Code_integer ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Onatural,type,
    numera5444537566228673987atural: fun_num_Code_natural ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: fun_num_int ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: fun_num_nat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: num > rat ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P9117596204409417319it_nat: option8956607266484857688it_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P2651198173097904984it_nat: option3562590408128118217it_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
    some_num: fun_num_option_num ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P1914260805536162275it_nat: fun_Pr7644814098969791837it_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P7913643980934408916it_nat: fun_Pr2403869735082759103it_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Num__Onum,type,
    case_option_o_num: ( $o * fun_num_o ) > fun_option_num_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o2686588417244861013it_nat: ( $o * fun_Pr8032424337255632745_nat_o ) > fun_op479301348043730329_nat_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o3267218206291580230it_nat: ( $o * fun_Pr5746006151868627512_nat_o ) > fun_op9133723461703387112_nat_o ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    case_o356765784539232260er_num: ( code_integer * fun_num_Code_integer * option_num ) > code_integer ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Code____Numeral__Onatural_001t__Num__Onum,type,
    case_o5621594795226839503al_num: ( code_natural * fun_num_Code_natural * option_num ) > code_natural ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
    case_option_int_num: ( int * fun_num_int ) > fun_option_num_int ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Nat__Onat_001t__Num__Onum,type,
    case_option_nat_num: ( nat * fun_num_nat ) > fun_option_num_nat ).

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

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
    case_o6005452278849405969um_num: ( option_num * fun_num_option_num ) > fun_op8989173174992673105on_num ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o7727800614283616222it_nat: ( option8956607266484857688it_nat * fun_Pr8191437870481986254it_nat * option3562590408128118217it_nat ) > option8956607266484857688it_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o6061097939050036047it_nat: ( option3562590408128118217it_nat * fun_Pr2403869735082759103it_nat * option3562590408128118217it_nat ) > option3562590408128118217it_nat ).

tff(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
    size_option_num: fun_num_nat > fun_option_num_nat ).

tff(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    size_o29782932136985253it_nat: ( fun_Pr367382672041210069at_nat * option8956607266484857688it_nat ) > nat ).

tff(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    size_o1731029420616209814it_nat: ( fun_Pr4403862266400233414at_nat * option3562590408128118217it_nat ) > nat ).

tff(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
    the_num: fun_option_num_num ).

tff(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    the_Pr7776125533777466484it_nat: option8956607266484857688it_nat > produc8664842809031399944it_nat ).

tff(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    the_Pr3616839707675578981it_nat: option3562590408128118217it_nat > produc3260487557148687353it_nat ).

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

tff(sy_c_Order__Relation_Owell__order__on_001t__Nat__Onat,type,
    order_2888998067076097458on_nat: set_nat > fun_se4782206882984618170_nat_o ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Predicate__Opred_It__Product____Type__Ounit_J,type,
    bot_bo2717538794563056311t_unit: pred_Product_unit ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Ounit,type,
    bot_bot_Product_unit: product_unit ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    bot_bo3957492148770167129t_unit: set_Product_unit ).

tff(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
    ord_Least_nat: fun_nat_o > nat ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
    ord_le6747313008572928689nteger: fun_Co2081214437782444048eger_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Onatural,type,
    ord_le5570908160329646204atural: fun_Co7302518231560758906ural_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: num > fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le1307284697595431911nteger: ( set_Code_integer * set_Code_integer ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_set_int: ( set_int * set_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > fun_set_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    ord_le7563427860532173253nt_int: set_Pr958786334691620121nt_int > fun_se7249947736802449410_int_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > fun_se4782206882984618170_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_less_set_set_nat: ( set_set_nat * set_set_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    ord_le1924305788584680229nt_int: ( set_se6260736226359567993nt_int * set_se6260736226359567993nt_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    ord_le3519150613208228973at_nat: ( set_se7855581050983116737at_nat * set_se7855581050983116737at_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__String__Oliteral,type,
    ord_less_literal: fun_li4177005666941401710eral_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Int__Oint_M_Eo_J,type,
    ord_less_fun_int_o: ( fun_int_o * fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Nat__Onat_M_Eo_J,type,
    ord_less_fun_nat_o: ( fun_nat_o * fun_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_le921202756137246487_nat_o: ( fun_set_nat_o * fun_set_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
    ord_le8970291702602496086_int_o: ( fun_se7249947736802449410_int_o * fun_se7249947736802449410_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    ord_le6502550848784664846_nat_o: ( fun_se4782206882984618170_nat_o * fun_se4782206882984618170_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: fun_Co2081214437782444048eger_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Onatural,type,
    ord_le1926595141338095240atural: fun_Co7302518231560758906ural_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: ( filter_nat * filter_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: num > fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le7084787975880047091nteger: ( set_Code_integer * set_Code_integer ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_eq_set_int: ( set_int * set_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    ord_le6045566169113846134st_nat: ( set_list_nat * set_list_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > fun_set_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > fun_se7249947736802449410_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > fun_se4782206882984618170_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    ord_le2645283422117218614nteger: ( set_Pr1200682188532975830nteger * set_Pr1200682188532975830nteger ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    ord_le3197784583289274309nt_int: ( set_Pr4166680992137330533nt_int * set_Pr4166680992137330533nt_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    ord_le2932375828722672556nteger: ( set_Pr3486312109254157132nteger * set_Pr3486312109254157132nteger ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J,type,
    ord_le66118011446560909atural: ( set_Pr620054291978045485atural * set_Pr620054291978045485atural ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    ord_le3990897393281686169nt_int: ( set_Pr6700895561065273913nt_int * set_Pr6700895561065273913nt_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    ord_le3507040750410214029t_unit: ( set_Product_unit * set_Product_unit ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: ( set_set_nat * set_set_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    ord_le483042692224249369nt_int: ( set_se6260736226359567993nt_int * set_se6260736226359567993nt_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    ord_le2077887516847798113at_nat: ( set_se7855581050983116737at_nat * set_se7855581050983116737at_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__String__Ochar_J,type,
    ord_less_eq_set_char: ( set_char * set_char ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__String__Oliteral,type,
    ord_less_eq_literal: fun_li4177005666941401710eral_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_M_Eo_J,type,
    ord_le5418975059635765733_int_o: ( fun_int_o * fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    ord_le77980682512674322_int_o: ( fun_int_fun_int_o * fun_int_fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_M_Eo_J,type,
    ord_le7240017405068453697_nat_o: ( fun_nat_o * fun_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_le6468474429079055883_nat_o: ( fun_set_nat_o * fun_set_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
    ord_le2758740137681283170_int_o: ( fun_se7249947736802449410_int_o * fun_se7249947736802449410_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    ord_le290999283863451930_nat_o: ( fun_se4782206882984618170_nat_o * fun_se4782206882984618170_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    ord_le7899862361585801933eger_o: ( fun_fu3422731660242565741eger_o * fun_fu3422731660242565741eger_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    ord_le2876595303158420734_int_o: ( fun_fu2978539155105838238_int_o * fun_fu2978539155105838238_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    ord_le4825817444062779351eger_o: ( fun_fu7695273699059480439eger_o * fun_fu7695273699059480439eger_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_M_Eo_J_J,type,
    ord_le825093278501044790ural_o: ( fun_fu3694549533497745878ural_o * fun_fu3694549533497745878ural_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    ord_le4806730579060245290_int_o: ( fun_fu6865642474282033866_int_o * fun_fu6865642474282033866_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
    ord_max_Code_integer: fun_Co1981900195452921617nteger ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Onatural,type,
    ord_max_Code_natural: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: fun_int_fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: num > fun_num_num ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
    ord_max_rat: ( rat * rat ) > rat ).

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

tff(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
    ord_min_int: fun_int_fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

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

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

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

tff(sy_c_Orderings_Oordering_001t__Nat__Onat,type,
    ordering_nat: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > $o ).

tff(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
    ordering_top_nat: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > fun_nat_o ).

tff(sy_c_Orderings_Opreordering_001t__Nat__Onat,type,
    preordering_nat: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > $o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Ounit,type,
    top_top_Product_unit: product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: code_integer > fun_nat_Code_integer ).

tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Onatural,type,
    power_7079662738309270450atural: code_natural > fun_nat_Code_natural ).

tff(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: fun_int_fun_nat_int ).

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Predicate_Opred_Oeval_001t__Product____Type__Ounit,type,
    eval_Product_unit: pred_Product_unit > fun_Product_unit_o ).

tff(sy_c_Predicate_Osingle_001t__Product____Type__Ounit,type,
    single_Product_unit: product_unit > pred_Product_unit ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
    produc6677183202524767010eger_o: ( code_integer * $o ) > produc6271795597528267376eger_o ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc1086072967326762835nteger: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Onatural_001t__Code____Numeral__Onatural,type,
    produc3574140220909816553atural: ( code_natural * code_natural ) > produc7822875418678951345atural ).

tff(sy_c_Product__Type_OPair_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
    produc584006145561248582it_nat: ( heap_e7401611519738050253t_unit * nat ) > produc6653097349344004940it_nat ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: fun_in1696471567231710611nt_int ).

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

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
    product_Pair_nat_num: ( nat * num ) > product_prod_nat_num ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    produc638857205735767105it_nat: ( nat * option8956607266484857688it_nat ) > produc3911288613690379145it_nat ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    produc4111269673004989362it_nat: ( nat * option3562590408128118217it_nat ) > produc4453839368661128058it_nat ).

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

tff(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: ( num * num ) > product_prod_num_num ).

tff(sy_c_Product__Type_OPair_001t__Predicate__Opred_It__Product____Type__Ounit_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc5069803637994805237atural: pred_Product_unit > fun_Pr160617923510306311atural ).

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

tff(sy_c_Product__Type_OPair_001t__Product____Type__Ounit_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    produc7133225469290080770it_nat: ( product_unit * produc6653097349344004940it_nat ) > produc8664842809031399944it_nat ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
    produc6363374080413544029et_int: ( set_int * set_int ) > produc2115011035271226405et_int ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4532415448927165861et_nat: ( set_nat * set_nat ) > produc7819656566062154093et_nat ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    produc763777882069021527od_o_o: ( set_Product_prod_o_o * set_Product_prod_o_o ) > produc2934264451710624999od_o_o ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc8517790099723286449at_nat: ( set_Product_prod_o_o * set_Pr1261947904930325089at_nat ) > produc732395585841259969at_nat ).

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

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc7264323995540650962nteger: fun_fu3460006408547937207nteger ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Code____Numeral__Onatural_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc3379161622255323613atural: ( fun_Co6386927884524458043e_term * produc7822875418678951345atural ) > produc7476732711998919211atural ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4729913460834754785nt_int: fun_fu465145433024125141nt_int ).

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc565448451787809096nteger: fun_fu8946524575271555235nteger ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc3399137497101226409atural: fun_fu2502439290952149605atural ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc7320969046608631221nt_int: fun_fu9160953773549289085nt_int ).

tff(sy_c_Product__Type_OPair_001tf__a_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    produc9178034014595674355it_nat: ( a * produc6653097349344004940it_nat ) > produc3260487557148687353it_nat ).

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

tff(sy_c_Product__Type_OUnity,type,
    product_Unity: product_unit ).

tff(sy_c_Product__Type_Oapfst_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc3195571438000033749nteger: fun_fu4800041728187187127nteger ).

tff(sy_c_Product__Type_Oapfst_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc2355237562570658114nt_int: fun_int_int > fun_Pr2478283820680522243nt_int ).

tff(sy_c_Product__Type_Oapfst_001t__Int__Oint_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Int__Oint,type,
    produc6151042875033135238rm_int: ( fun_in4421688504851875865e_term * product_prod_int_int ) > produc8857309559938776829rm_int ).

tff(sy_c_Product__Type_Oapfst_001t__Int__Oint_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc3354640145386789057nteger: ( fun_in4421688504851875865e_term * produc6162884466328367972nteger ) > produc7509114972854644512nteger ).

tff(sy_c_Product__Type_Oapfst_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Int__Oint,type,
    produc6429934853825515733rm_int: ( fun_in94812177438132456e_term * product_prod_int_int ) > produc1164942017926883148rm_int ).

tff(sy_c_Product__Type_Oapfst_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc1852316473262731728nt_int: ( fun_in94812177438132456e_term * produc759720530913461378nt_int ) > produc4788131091073775919nt_int ).

tff(sy_c_Product__Type_Oapfst_001t__Int__Oint_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Int__Oint,type,
    produc3213193131064493712rm_int: ( fun_in7158797356850311459e_term * product_prod_int_int ) > produc8574895079301897479rm_int ).

tff(sy_c_Product__Type_Oapfst_001t__Int__Oint_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc8552504676914255415nteger: ( fun_in7158797356850311459e_term * produc6162884466328367972nteger ) > produc2453742332425399446nteger ).

tff(sy_c_Product__Type_Oapfst_001t__Int__Oint_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Int__Oint,type,
    produc7982337911579641349rm_int: ( fun_in1403664741593307672e_term * product_prod_int_int ) > produc4120667822962269308rm_int ).

tff(sy_c_Product__Type_Oapfst_001t__Int__Oint_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc2162821685372896920atural: ( fun_in1403664741593307672e_term * produc8915214464240719482atural ) > produc9140211490168418807atural ).

tff(sy_c_Product__Type_Oapfst_001t__Int__Oint_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Int__Oint,type,
    produc1502976015219776769rm_int: ( fun_in237891044757944084e_term * product_prod_int_int ) > produc6693577919049233272rm_int ).

tff(sy_c_Product__Type_Oapfst_001t__Int__Oint_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc7648640027116863140nt_int: ( fun_in237891044757944084e_term * produc759720530913461378nt_int ) > produc3043004577601930755nt_int ).

tff(sy_c_Product__Type_Oapfst_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    produc986720760941809198at_nat: fun_nat_nat > fun_Pr2118491707810114563at_nat ).

tff(sy_c_Product__Type_Oapfst_001t__Nat__Onat_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Nat__Onat,type,
    produc9061699220754655694rm_nat: ( fun_na1605103330871484349e_term * product_prod_nat_nat ) > produc3811788542593197729rm_nat ).

tff(sy_c_Product__Type_Oapfst_001t__Nat__Onat_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc4795910711687975965nteger: ( fun_na1605103330871484349e_term * produc5240993858131070144nteger ) > produc7509114972854644512nteger ).

tff(sy_c_Product__Type_Oapfst_001t__Nat__Onat_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Nat__Onat,type,
    produc3615840150354174493rm_nat: ( fun_na3603975746771071116e_term * product_prod_nat_nat ) > produc5342793037436079856rm_nat ).

tff(sy_c_Product__Type_Oapfst_001t__Nat__Onat_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc1749517359979953452nt_int: ( fun_na3603975746771071116e_term * produc6945250483304103390nt_int ) > produc4788131091073775919nt_int ).

tff(sy_c_Product__Type_Oapfst_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Nat__Onat,type,
    produc6823882730714392536rm_nat: ( fun_na2562132801372711111e_term * product_prod_nat_nat ) > produc3529374061956318379rm_nat ).

tff(sy_c_Product__Type_Oapfst_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc5927030739694416275nteger: ( fun_na2562132801372711111e_term * produc5240993858131070144nteger ) > produc2453742332425399446nteger ).

tff(sy_c_Product__Type_Oapfst_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Nat__Onat,type,
    produc2369655474374764365rm_nat: ( fun_na6030372222970483132e_term * product_prod_nat_nat ) > produc8298518842471466016rm_nat ).

tff(sy_c_Product__Type_Oapfst_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc8760719785007833588atural: ( fun_na6030372222970483132e_term * produc7993323856043421654atural ) > produc9140211490168418807atural ).

tff(sy_c_Product__Type_Oapfst_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Nat__Onat,type,
    produc6132173967106002505rm_nat: ( fun_na6101041082579861176e_term * product_prod_nat_nat ) > produc1648056901703654172rm_nat ).

tff(sy_c_Product__Type_Oapfst_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc2739701771563428352nt_int: ( fun_na6101041082579861176e_term * produc6945250483304103390nt_int ) > produc3043004577601930755nt_int ).

tff(sy_c_Product__Type_Oapfst_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc5151272107987982333nteger: ( fun_fu9036519777977489245e_term * produc7509114972854644512nteger ) > produc7509114972854644512nteger ).

tff(sy_c_Product__Type_Oapfst_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc2260500861290198387nteger: ( fun_fu3093988009420988519e_term * produc7509114972854644512nteger ) > produc2453742332425399446nteger ).

tff(sy_c_Product__Type_Oapfst_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc5901819666759714173nt_int: ( fun_fu4810317505167811835e_term * produc4788131091073775919nt_int ) > produc4788131091073775919nt_int ).

tff(sy_c_Product__Type_Oapfst_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc479425830829021777nt_int: ( fun_fu1176768540439120679e_term * produc4788131091073775919nt_int ) > produc3043004577601930755nt_int ).

tff(sy_c_Product__Type_Oapfst_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc177741965484698227nteger: ( fun_fu1456393871774969959e_term * produc2453742332425399446nteger ) > produc7509114972854644512nteger ).

tff(sy_c_Product__Type_Oapfst_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc183566614307960809nteger: ( fun_fu2505754153367914865e_term * produc2453742332425399446nteger ) > produc2453742332425399446nteger ).

tff(sy_c_Product__Type_Oapfst_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc1921288387873046805atural: ( fun_fu7248590935147425627e_term * produc9140211490168418807atural ) > produc9140211490168418807atural ).

tff(sy_c_Product__Type_Oapfst_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc7965835600623382609nt_int: ( fun_fu4835546211937200935e_term * produc3043004577601930755nt_int ) > produc4788131091073775919nt_int ).

tff(sy_c_Product__Type_Oapfst_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc7857123342865048869nt_int: ( fun_fu836511555612062035e_term * produc3043004577601930755nt_int ) > produc3043004577601930755nt_int ).

tff(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc6499014454317279255nteger: fun_fu4800041728187187127nteger ).

tff(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc4463282112584876420nt_int: fun_int_int > fun_Pr2478283820680522243nt_int ).

tff(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Int__Oint,type,
    produc2935421404040714269er_int: ( fun_in4744367027289715400nteger * product_prod_int_int ) > produc6162884466328367972nteger ).

tff(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc7159730696577248097e_term: ( fun_in4744367027289715400nteger * produc8857309559938776829rm_int ) > produc7509114972854644512nteger ).

tff(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc5655117854151192939e_term: ( fun_in4744367027289715400nteger * produc8574895079301897479rm_int ) > produc2453742332425399446nteger ).

tff(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Int__Oint,type,
    produc1633769443012300339al_int: ( fun_in7496697025202066910atural * product_prod_int_int ) > produc8915214464240719482atural ).

tff(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc4003887134565203958e_term: ( fun_in7496697025202066910atural * produc4120667822962269308rm_int ) > produc9140211490168418807atural ).

tff(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
    produc6100211334659168315nt_int: ( fun_in4211310926729683942nt_int * product_prod_int_int ) > produc759720530913461378nt_int ).

tff(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc4251495004083222734e_term: ( fun_in4211310926729683942nt_int * produc1164942017926883148rm_int ) > produc4788131091073775919nt_int ).

tff(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc5266752770087298298e_term: ( fun_in4211310926729683942nt_int * produc6693577919049233272rm_int ) > produc3043004577601930755nt_int ).

tff(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    produc3094765310956027504at_nat: fun_nat_nat > fun_Pr2118491707810114563at_nat ).

tff(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Nat__Onat,type,
    produc2688509591584439069er_nat: ( fun_na3822476419092417572nteger * product_prod_nat_nat ) > produc5240993858131070144nteger ).

tff(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc8601001262878435005e_term: ( fun_na3822476419092417572nteger * produc3811788542593197729rm_nat ) > produc7509114972854644512nteger ).

tff(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc3029643916931353799e_term: ( fun_na3822476419092417572nteger * produc3529374061956318379rm_nat ) > produc2453742332425399446nteger ).

tff(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Nat__Onat,type,
    produc1386857630556025139al_nat: ( fun_na6574806417004769082atural * product_prod_nat_nat ) > produc7993323856043421654atural ).

tff(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc1378413197345364818e_term: ( fun_na6574806417004769082atural * produc8298518842471466016rm_nat ) > produc9140211490168418807atural ).

tff(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Nat__Onat,type,
    produc4342665141415107387nt_nat: ( fun_na1173468842265550146nt_int * product_prod_nat_nat ) > produc6945250483304103390nt_int ).

tff(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc4148695890800444458e_term: ( fun_na1173468842265550146nt_int * produc5342793037436079856rm_nat ) > produc4788131091073775919nt_int ).

tff(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc357814514533863510e_term: ( fun_na1173468842265550146nt_int * produc1648056901703654172rm_nat ) > produc3043004577601930755nt_int ).

tff(sy_c_Product__Type_Oapsnd_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc4532778850088682670e_term: ( fun_Pr5876161306168176259nteger * produc7509114972854644512nteger ) > produc7509114972854644512nteger ).

tff(sy_c_Product__Type_Oapsnd_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc2265439103302406840e_term: ( fun_Pr5876161306168176259nteger * produc2453742332425399446nteger ) > produc2453742332425399446nteger ).

tff(sy_c_Product__Type_Oapsnd_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc1166422551560059565e_term: ( fun_Pr74676805955934211atural * produc9140211490168418807atural ) > produc9140211490168418807atural ).

tff(sy_c_Product__Type_Oapsnd_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc2254893824690362237e_term: ( fun_Pr2478283820680522243nt_int * produc4788131091073775919nt_int ) > produc4788131091073775919nt_int ).

tff(sy_c_Product__Type_Oapsnd_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc8061333830752525225e_term: ( fun_Pr2478283820680522243nt_int * produc3043004577601930755nt_int ) > produc3043004577601930755nt_int ).

tff(sy_c_Product__Type_Ocurry_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc175634133007206835_int_o: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).

tff(sy_c_Product__Type_Ocurry_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc1016772743285680337nt_int: fun_Pr982841511892225912nt_int > fun_int_fun_int_int ).

tff(sy_c_Product__Type_Ocurry_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc8249235968001453780nt_int: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_Product__Type_Ocurry_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    produc2062876521443793799_nat_o: fun_Pr2028404973800837934_nat_o > fun_na5914473681949881652_nat_o ).

tff(sy_c_Product__Type_Ocurry_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc4701859392678344816at_nat: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_Product__Type_Ocurry_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
    produc4448170916228617336eger_o: fun_Pr1347628893830317649eger_o > fun_fu3422731660242565741eger_o ).

tff(sy_c_Product__Type_Ocurry_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
    produc2168430993391648169_int_o: fun_Pr1196100989263358594_int_o > fun_fu2978539155105838238_int_o ).

tff(sy_c_Product__Type_Ocurry_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
    produc177509963858859394eger_o: fun_Pr7015759342434529115eger_o > fun_fu7695273699059480439eger_o ).

tff(sy_c_Product__Type_Ocurry_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001_Eo,type,
    produc5480829236514930401ural_o: fun_Pr3733920118687433658ural_o > fun_fu3694549533497745878ural_o ).

tff(sy_c_Product__Type_Ocurry_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
    produc443495763101479893_int_o: fun_Pr3540825087398332590_int_o > fun_fu6865642474282033866_int_o ).

tff(sy_c_Product__Type_Ointernal__case__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc8005341501107743676_int_o: fun_fu1345883967705819239_int_o ).

tff(sy_c_Product__Type_Ointernal__case__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc7926200574084438792nt_int: fun_fu5207212825597307859nt_int ).

tff(sy_c_Product__Type_Ointernal__case__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc297006045350968285nt_int: fun_fu6741006549690681129nt_int ).

tff(sy_c_Product__Type_Ointernal__case__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    produc9189008374811844670_nat_o: fun_fu2847332191311216467_nat_o ).

tff(sy_c_Product__Type_Ointernal__case__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc8104577556401574439at_nat: fun_fu2716251949043459987at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    produc1553301316500091796er_int: ( fun_Co1092983284983208416er_int * produc8923325533196201883nteger ) > int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    produc1555791787009142072er_nat: ( fun_Co3583066429914935172er_nat * produc8923325533196201883nteger ) > nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    produc7336495610019696514er_num: ( fun_Co8521658056209057742er_num * produc8923325533196201883nteger ) > num ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    produc9125791028180074456eger_o: ( fun_Co3544761417380347684eger_o * produc8923325533196201883nteger ) > produc6271795597528267376eger_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: fun_Co2470855966376689427nteger > fun_Pr5876161306168176259nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc4281629032016041417nteger: ( fun_Co1561183069634336733nteger * produc3346328345660746739nteger ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    produc7488178964372371019it_nat: fun_he8369595706569872389it_nat > fun_Pr5651933991979716257it_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    produc5645566021302314940it_nat: fun_he8849654208132081782it_nat > fun_Pr7273243183861550738it_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc4947309494688390418_int_o: fun_fu1345883967705819239_int_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc8211389475949308722nt_int: fun_fu5207212825597307859nt_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: fun_fu6741006549690681129nt_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    produc6081775807080527818_nat_o: fun_nat_fun_nat_o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    produc1830744345554046123nteger: fun_na8135612474337863287nteger > fun_Pr6026251430685297nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
    produc6840382203811409530at_int: fun_nat_fun_nat_int > fun_Pr7647638413817135296at_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    produc2761476792215241774st_nat: ( fun_na819858167508000762st_nat * product_prod_nat_nat ) > list_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    produc6842872674320459806at_nat: fun_nat_fun_nat_nat > fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: fun_na4138050610385705875at_nat > fun_Pr2118491707810114563at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
    produc6207742614233964070at_rat: fun_nat_fun_nat_rat > fun_Pr8317260874932935276at_rat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    produc6395135316500328296_nat_o: fun_fu2847332191311216467_nat_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc4528391901023682385at_nat: fun_fu2716251949043459987at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
    produc478579273971653890on_num: ( fun_na7595719129879163854on_num * product_prod_nat_num ) > option_num ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Predicate__Opred_It__Product____Type__Ounit_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    produc6665183775751917029atural: ( fun_pr6291093682560481277atural * produc4675096598859438275atural ) > produc4675096598859438275atural ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    produc2742967963346599297nteger: fun_Pr5283128996861011131nteger > fun_Pr2541390607678853719nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    produc1137639668546780417nteger: fun_Pr8371818785129130299nteger > fun_Pr6966570558394395095nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    produc5705453634168500973atural: fun_Pr6582898042194347943atural > fun_Pr5219672437278659779atural ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    produc8624921108698535677nt_int: fun_Pr4353654455642104759nt_int > fun_Pr3284083897292419283nt_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    produc4605791130040086525nt_int: fun_Pr8826142978676617911nt_int > fun_Pr5010838083610845651nt_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J_001_Eo,type,
    produc4109468873575309990_int_o: fun_se2051630309975736070_int_o > fun_Pr4126283446944868854_int_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
    produc6247414631856714078_nat_o: fun_se2170385732985038782_nat_o > fun_Pr3947133745332865710_nat_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
    produc410239310623530412_nat_o: fun_se5639961202347202350_nat_o > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
    produc7733317944277263833eger_o: fun_fu3422731660242565741eger_o > fun_Pr1347628893830317649eger_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    produc6766542753387345183nteger: ( fun_fu3460006408547937207nteger * produc7509114972854644512nteger ) > produc7509114972854644512nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Int__Oint_J,type,
    produc3783258407033660385et_int: ( fun_fu1569576118606662777et_int * produc7509114972854644512nteger ) > set_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc7961109426542857093et_nat: ( fun_fu6383124537394581533et_nat * produc7509114972854644512nteger ) > set_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    produc3793265978605622331et_nat: ( fun_fu963894927305104083et_nat * produc7509114972854644512nteger ) > set_set_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    produc852157747730529210nt_int: ( fun_fu6439581347498178894nt_int * produc7509114972854644512nteger ) > set_se6260736226359567993nt_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc2447002572354077954at_nat: ( fun_fu5924095598813543062at_nat * produc7509114972854644512nteger ) > set_se7855581050983116737at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
    produc7271678899836001034_int_o: fun_fu2978539155105838238_int_o > fun_Pr1196100989263358594_int_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    produc7102243260366339645nt_int: ( fun_fu465145433024125141nt_int * produc4788131091073775919nt_int ) > produc4788131091073775919nt_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
    produc833192924009095408et_int: ( fun_fu4266222366127506568et_int * produc4788131091073775919nt_int ) > set_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc5011043943518292116et_nat: ( fun_fu9079770784915425324et_nat * produc4788131091073775919nt_int ) > set_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    produc4945675248400274762et_nat: ( fun_fu5458086908000294626et_nat * produc4788131091073775919nt_int ) > set_set_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    produc6653901401015897067nt_int: ( fun_fu719511775830429311nt_int * produc4788131091073775919nt_int ) > set_se6260736226359567993nt_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc8248746225639445811at_nat: ( fun_fu204026027145793479at_nat * produc4788131091073775919nt_int ) > set_se7855581050983116737at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
    produc7373287791920631523eger_o: fun_fu7695273699059480439eger_o > fun_Pr7015759342434529115eger_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    produc697581972998607883nteger: ( fun_fu8946524575271555235nteger * produc2453742332425399446nteger ) > produc2453742332425399446nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001_Eo,type,
    produc3453235027721926722ural_o: fun_fu3694549533497745878ural_o > fun_Pr3733920118687433658ural_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    produc237422109222252493atural: ( fun_fu2502439290952149605atural * produc9140211490168418807atural ) > produc9140211490168418807atural ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
    produc2207054017359051062_int_o: fun_fu6865642474282033866_int_o > fun_Pr3540825087398332590_int_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    produc6823905543215200229nt_int: ( fun_fu9160953773549289085nt_int * produc3043004577601930755nt_int ) > produc3043004577601930755nt_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    produc5648861294512138360it_nat: fun_a_4942204877778174258it_nat > fun_Pr8191437870481986254it_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    produc7850007024774191849it_nat: fun_a_3426355041879316131it_nat > fun_Pr2403869735082759103it_nat ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc8508995932063986495nteger: fun_Pr6596356882939945227nteger ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc1861665112190112313nteger: produc3346328345660746739nteger > code_integer ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
    produc2838810277069100634it_nat: produc6653097349344004940it_nat > heap_e7401611519738050253t_unit ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: fun_Pr982841511892225912nt_int ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc4337129453304281928e_term: produc3611471421753121338e_term > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc3658391496133417298e_term: produc592931078481472324e_term > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc2007160776547428317e_term: produc7223384021016661839e_term > produc7822875418678951345atural ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc7504962776628944053e_term: produc5464280974217902119e_term > product_prod_int_int ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc8655982770948792545e_term: produc4155239895104388691e_term > product_prod_int_int ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc3547674943397867238nteger: fun_Pr4759914407354907917e_term ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4463008262350210037nt_int: fun_Pr7653582204007996523e_term ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc2481150572478215260nteger: fun_Pr4557384415951816845e_term ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc5314839617791632573atural: fun_Pr944821188406467171e_term ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc3130796691802200265nt_int: fun_Pr5872836680101629291e_term ).

tff(sy_c_Product__Type_Oprod_Ofst_001tf__a_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    produc4481111470393856007it_nat: produc3260487557148687353it_nat > a ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc6174133586879617921nteger: fun_Pr6596356882939945227nteger ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc7955729907432593271nteger: produc3346328345660746739nteger > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
    produc2594868675584459928it_nat: produc6653097349344004940it_nat > nat ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: fun_Pr982841511892225912nt_int ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc648986795810732422e_term: produc3611471421753121338e_term > fun_Co3460408610320954864e_term ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc7766732615400916880e_term: produc592931078481472324e_term > fun_Pr5098669352780829690e_term ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc6115501895814927899e_term: produc7223384021016661839e_term > fun_Pr285225112257423215e_term ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc1448379345012327155e_term: produc5464280974217902119e_term > fun_in5730379987311621695e_term ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    produc7082206506123769631e_term: produc4155239895104388691e_term > fun_Pr7763680713840462443e_term ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc9082904322759093540nteger: fun_Pr2054267358141571796nteger ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc7629796867588368947nt_int: fun_Pr1552678306225191715nt_int ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6589491691745714842nteger: fun_Pr6160931176198666718nteger ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc199808700204356347atural: fun_Pr8015448952463279699atural ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc1557020426977177351nt_int: fun_Pr1987159835869338959nt_int ).

tff(sy_c_Product__Type_Oprod_Osnd_001tf__a_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    produc6461318759014699077it_nat: produc3260487557148687353it_nat > produc6653097349344004940it_nat ).

tff(sy_c_Quotient_OQuotient3_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    quotie6776551016481293500at_int: ( fun_Pr2028404973800837934_nat_o * fun_Pr7647638413817135296at_int * fun_in4514472496968388910at_nat ) > $o ).

tff(sy_c_Quotient_Oquot__type_Oabs_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    quot_a8334514980957697477at_int: ( fun_Pr2028404973800837934_nat_o * fun_se9142667792015812128at_int ) > fun_Pr7647638413817135296at_int ).

tff(sy_c_Quotient_Oquot__type_Orep_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    quot_r1512469584000298464at_nat: fun_in8804710076054006414at_nat > fun_in4514472496968388910at_nat ).

tff(sy_c_Random_Olog,type,
    log: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Olog__rel,type,
    log_rel: fun_Pr3592279944642289454ural_o ).

tff(sy_c_Random__Pred_Onot__randompred,type,
    random6974930770145893639ompred: fun_Pr160617923510306311atural > fun_Pr160617923510306311atural ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: fun_Pr1652463973008025892nt_rat ).

tff(sy_c_Rat_OFract,type,
    fract: fun_int_fun_int_rat ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod_int_int > rat ).

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun_ra4503298312962071866nt_int ).

tff(sy_c_Rat_Ocr__rat,type,
    cr_rat: product_prod_int_int > fun_rat_o ).

tff(sy_c_Rat_Onormalize,type,
    normalize: fun_Pr2478283820680522243nt_int ).

tff(sy_c_Rat_Oof__int,type,
    of_int: fun_int_rat ).

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun_Pr7537129237977221859_rat_o ).

tff(sy_c_Rat_Opositive,type,
    positive: fun_rat_o ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod_int_int ).

tff(sy_c_Rat_Orat_OAbs__rat,type,
    abs_rat: fun_se593059424582005380nt_rat ).

tff(sy_c_Rat_Orat_ORep__rat,type,
    rep_rat: fun_ra7314870381713115802nt_int ).

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Relation_ODomainp_001t__Int__Oint_001t__Int__Oint,type,
    domainp_int_int: fun_int_fun_int_o > fun_int_o ).

tff(sy_c_Relation_ODomainp_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral,type,
    domain7888411570377768909iteral: fun_li9104157641623487077eral_o > fun_list_char_o ).

tff(sy_c_Relation_ODomainp_001t__Nat__Onat_001t__Nat__Onat,type,
    domainp_nat_nat: fun_nat_fun_nat_o > fun_nat_o ).

tff(sy_c_Relation_ODomainp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    domain9213661015745956397nt_rat: fun_Pr7537129237977221859_rat_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Relation_ODomainp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    domain7287725470393535433at_int: fun_Pr4738963024209204295_int_o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_Relation_ODomainp_001t__String__Ochar_001t__String__Ochar,type,
    domainp_char_char: fun_char_fun_char_o > fun_char_o ).

tff(sy_c_Relation_OField_001_Eo,type,
    field_o: set_Product_prod_o_o > set_o ).

tff(sy_c_Relation_OField_001t__Nat__Onat,type,
    field_nat: set_Pr1261947904930325089at_nat > set_nat ).

tff(sy_c_Relation_Orelcompp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    relcom8911704206367525917at_int: ( fun_Pr2028404973800837934_nat_o * fun_Pr4738963024209204295_int_o ) > fun_Pr4738963024209204295_int_o ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
    algebr932160517623751201me_int: fun_int_fun_int_o ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
    algebr934650988132801477me_nat: fun_nat_fun_nat_o ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: fun_Co1981900195452921617nteger ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Onatural,type,
    divide5121882707175180666atural: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: ( rat * rat ) > rat ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
    dvd_dvd_Code_integer: fun_Co2081214437782444048eger_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Onatural,type,
    dvd_dvd_Code_natural: fun_Co7302518231560758906ural_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: fun_rat_fun_rat_o ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
    modulo364778990260209775nteger: fun_Co1981900195452921617nteger ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Onatural,type,
    modulo8411746178871703098atural: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Ounit__factor__class_Ounit__factor_001t__Nat__Onat,type,
    unit_f2748546683901255202or_nat: fun_nat_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
    zero_n356916108424825756nteger: $o > code_integer ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Onatural,type,
    zero_n8403883297036319079atural: $o > code_natural ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_o_int ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: fun_o_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
    zero_n2052037380579107095ol_rat: $o > rat ).

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

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

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

tff(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    collec5989764272469232197st_nat: fun_list_list_nat_o > set_list_list_nat ).

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

tff(sy_c_Set_OCollect_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    collec3343600615725829874at_nat: fun_li7366011709773453460_nat_o > set_li5450038453877631591at_nat ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Product____Type__Ounit_J,type,
    collec4736972532740828926t_unit: fun_li6269451478509088136unit_o > set_li873732205721079667t_unit ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    collect_list_set_nat: fun_list_set_nat_o > set_list_set_nat ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__String__Ochar_J,type,
    collect_list_char: fun_list_char_o > set_list_char ).

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

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

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

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

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    collec957716948307931664et_int: fun_Pr4126283446944868854_int_o > set_Pr2522554150109002629et_int ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    collec6662362479098859352et_nat: fun_Pr3947133745332865710_nat_o > set_Pr5488025237498180813et_nat ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    collec6321179662152712658at_nat: fun_Pr3059359313727237556_nat_o > set_Pr4329608150637261639at_nat ).

tff(sy_c_Set_OCollect_001t__Product____Type__Ounit,type,
    collect_Product_unit: fun_Product_unit_o > set_Product_unit ).

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

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

tff(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    collect_set_list_nat: fun_set_list_nat_o > set_set_list_nat ).

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

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    collec5210948495886036740nt_int: fun_se7249947736802449410_int_o > set_se6260736226359567993nt_int ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    collec5514110066124741708at_nat: fun_se4782206882984618170_nat_o > set_se7855581050983116737at_nat ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    collec7787489603430924120t_unit: fun_se4342725320470837934unit_o > set_set_Product_unit ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    collect_set_set_nat: fun_set_set_nat_o > set_set_set_nat ).

tff(sy_c_Set_OCollect_001t__String__Oliteral,type,
    collect_literal: fun_literal_o > set_literal ).

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

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

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

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

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

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

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
    image_nat_char: ( fun_nat_char * set_nat ) > set_char ).

tff(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( fun_char_nat * set_char ) > set_nat ).

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

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

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

tff(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    insert8211810215607154385at_nat: ( product_prod_nat_nat * set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    insert9069300056098147895at_nat: ( produc3843707927480180839at_nat * set_Pr4329608150637261639at_nat ) > set_Pr4329608150637261639at_nat ).

tff(sy_c_Set_Oinsert_001t__Product____Type__Ounit,type,
    insert_Product_unit: ( product_unit * set_Product_unit ) > set_Product_unit ).

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

tff(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    insert8897473484851387113nt_int: ( set_Pr958786334691620121nt_int * set_se6260736226359567993nt_int ) > set_se6260736226359567993nt_int ).

tff(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    insert9200635055090092081at_nat: ( set_Pr1261947904930325089at_nat * set_se7855581050983116737at_nat ) > set_se7855581050983116737at_nat ).

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

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Code____Numeral__Ointeger,type,
    set_fo1084959871951514735nteger: ( fun_na7078387940644931716nteger * nat * nat ) > fun_Co8499238673658658103nteger ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Code____Numeral__Onatural,type,
    set_fo9131927060563008058atural: ( fun_na607345901702507418atural * nat * nat * code_natural ) > code_natural ).

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

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

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

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Nat__Onat,type,
    set_fo3699595496184130361el_nat: fun_Pr8463018485500633518_nat_o ).

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

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: nat > fun_nat_set_nat ).

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

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: fun_nat_set_nat ).

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

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: fun_nat_set_nat ).

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

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

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4236626031148496127et_nat: set_nat > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    set_or2459421552957432928nt_int: set_Pr958786334691620121nt_int > set_se6260736226359567993nt_int ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_or2762583123196137896at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
    set_or5834768355832116004an_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: fun_nat_set_nat ).

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

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

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

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

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

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    set_or4940836740269066044nt_int: set_Pr958786334691620121nt_int > set_se6260736226359567993nt_int ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_or5243998310507771012at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__String__Oliteral,type,
    set_or2436086697161274615iteral: literal > set_literal ).

tff(sy_c_String_OLiteral,type,
    literal2: fun_o_8412685193225684464iteral ).

tff(sy_c_String_Oascii__of,type,
    ascii_of: fun_char_char ).

tff(sy_c_String_Oasciis__of__literal,type,
    asciis_of_literal: literal > list_Code_integer ).

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

tff(sy_c_String_Ochar_Ocase__char_001_Eo,type,
    case_char_o: fun_o_754185029415132912un_o_o > fun_char_o ).

tff(sy_c_String_Ochar_Ocase__char_001t__String__Oliteral,type,
    case_char_literal: ( fun_o_4077924883752292140iteral * char ) > literal ).

tff(sy_c_String_Ochar_Odigit0,type,
    digit0: fun_char_o ).

tff(sy_c_String_Ochar_Odigit1,type,
    digit1: fun_char_o ).

tff(sy_c_String_Ochar_Odigit2,type,
    digit2: fun_char_o ).

tff(sy_c_String_Ochar_Odigit3,type,
    digit3: fun_char_o ).

tff(sy_c_String_Ochar_Odigit4,type,
    digit4: fun_char_o ).

tff(sy_c_String_Ochar_Odigit5,type,
    digit5: fun_char_o ).

tff(sy_c_String_Ochar_Odigit6,type,
    digit6: fun_char_o ).

tff(sy_c_String_Ochar_Odigit7,type,
    digit7: fun_char_o ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Code____Numeral__Ointeger,type,
    comm_s4049598766089966537nteger: fun_ch2863552307440546731nteger ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

tff(sy_c_String_Ocr__literal,type,
    cr_literal: fun_li9104157641623487077eral_o ).

tff(sy_c_String_Oliteral_OAbs__literal,type,
    abs_literal: fun_li5857714827575272844iteral ).

tff(sy_c_String_Oliteral_Oexplode,type,
    explode: fun_li7473841323753414984t_char ).

tff(sy_c_String_Oliteral__of__asciis,type,
    literal_of_asciis: fun_li9041563744671036928iteral ).

tff(sy_c_String_Opcr__literal,type,
    pcr_literal: fun_li9104157641623487077eral_o ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Code____Numeral__Ointeger,type,
    unique1708501630192214702nteger: fun_Co8487158583234504619r_char ).

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_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
    sum_ca7763040182479039464nt_nat: ( fun_nat_int * fun_nat_int * sum_sum_nat_nat ) > int ).

tff(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_ca6763686470577984908at_nat: ( fun_nat_nat * fun_nat_nat * sum_sum_nat_nat ) > nat ).

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

tff(sy_c_Transfer_Oleft__total_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
    left_t8440299596084406506t_char: fun_li5260340055336554888char_o > $o ).

tff(sy_c_Transfer_Oleft__total_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    left_t3131394472396969446nt_int: fun_Pr9018117161326028078_int_o > $o ).

tff(sy_c_Transfer_Oleft__total_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    left_t1861001511420785031at_int: fun_Pr4738963024209204295_int_o > $o ).

tff(sy_c_Transfer_Oright__total_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    right_1157779049177982926at_int: fun_Pr4738963024209204295_int_o > $o ).

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

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

tff(sy_c_Transitive__Closure_Otranclp_001t__Nat__Onat,type,
    transi2163837189807498211lp_nat: fun_nat_fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_Typedef_Otype__definition_001t__Int__Oint_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    type_d8752208705193531015at_nat: ( fun_in8804710076054006414at_nat * fun_se9142667792015812128at_int * set_se7855581050983116737at_nat ) > $o ).

tff(sy_c_Typedef_Otype__definition_001t__Rat__Orat_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    type_d8554052265237484179nt_int: ( fun_ra7314870381713115802nt_int * fun_se593059424582005380nt_rat * set_se6260736226359567993nt_int ) > $o ).

tff(sy_c_Typedef_Otype__definition_001t__String__Oliteral_001t__List__Olist_It__String__Ochar_J,type,
    type_d4752411451802217481t_char: ( fun_li7473841323753414984t_char * fun_li5857714827575272844iteral * set_list_char ) > $o ).

tff(sy_c_Typerep_Otyperep_OTyperep,type,
    typerep2: ( literal * list_typerep ) > typerep ).

tff(sy_c_Typerep_Otyperep_Osize__typerep,type,
    size_typerep: fun_typerep_nat ).

tff(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: ( fun_li1025265955262268170_nat_o * list_nat ) > $o ).

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

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    accp_P8126237942716283194atural: fun_Pr3592279944642289454ural_o > fun_Pr8359263388657976682ural_o ).

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

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

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    accp_P414730952086964626it_nat: ( fun_Pr6341106346053136302_nat_o * produc3911288613690379145it_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    accp_P6263092265436569219it_nat: ( fun_Pr782666833218042862_nat_o * produc4453839368661128058it_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    accp_P4694664897847628410at_nat: ( fun_Pr8463018485500633518_nat_o * produc9217583074051984497at_nat ) > $o ).

tff(sy_c_Wellfounded_Ofinite__psubset_001t__Int__Oint,type,
    finite_psubset_int: set_Pr2522554150109002629et_int ).

tff(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
    finite_psubset_nat: set_Pr5488025237498180813et_nat ).

tff(sy_c_Wellfounded_Ofinite__psubset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite469560695537375940at_nat: set_Pr4329608150637261639at_nat ).

tff(sy_c_Wellfounded_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

tff(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Owf_001t__Int__Oint,type,
    wf_int: fun_se7249947736802449410_int_o ).

tff(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
    wf_nat: fun_se4782206882984618170_nat_o ).

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

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

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

tff(sy_c_aa_001_Eo_001t__Option__Ooption_It__Num__Onum_J,type,
    aa_o_option_num: ( fun_o_option_num * $o ) > option_num ).

tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_o_P3134438092330336605nt_int: ( fun_o_8988054265482332260nt_int * $o ) > product_prod_int_int ).

tff(sy_c_aa_001_Eo_001t__String__Oliteral,type,
    aa_o_literal: ( fun_o_literal * $o ) > literal ).

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

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__String__Oliteral_J,type,
    aa_o_fun_o_literal: ( fun_o_fun_o_literal * $o ) > fun_o_literal ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J,type,
    aa_o_fun_o_fun_o_o: ( fun_o_fun_o_fun_o_o * $o ) > fun_o_fun_o_o ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__String__Oliteral_J_J,type,
    aa_o_f2313209444760553806iteral: ( fun_o_4805661730443325909iteral * $o ) > fun_o_fun_o_literal ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J_J,type,
    aa_o_f2738860948074597109un_o_o: ( fun_o_8391042909487769924un_o_o * $o ) > fun_o_fun_o_fun_o_o ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__String__Oliteral_J_J_J,type,
    aa_o_f1022258164736565553iteral: ( fun_o_7984170265123672448iteral * $o ) > fun_o_4805661730443325909iteral ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J_J_J,type,
    aa_o_f1377459968498492136un_o_o: ( fun_o_4499574304626203759un_o_o * $o ) > fun_o_8391042909487769924un_o_o ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__String__Oliteral_J_J_J_J,type,
    aa_o_f139249553318509348iteral: ( fun_o_2312947153046382251iteral * $o ) > fun_o_7984170265123672448iteral ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J_J_J_J,type,
    aa_o_f3273824232735246795un_o_o: ( fun_o_9201541227772858138un_o_o * $o ) > fun_o_4499574304626203759un_o_o ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__String__Oliteral_J_J_J_J_J,type,
    aa_o_f1662653615792776199iteral: ( fun_o_3780185944647023446iteral * $o ) > fun_o_2312947153046382251iteral ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J_J_J_J_J,type,
    aa_o_f213386970520222398un_o_o: ( fun_o_2727025911028299589un_o_o * $o ) > fun_o_9201541227772858138un_o_o ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__String__Oliteral_J_J_J_J_J_J,type,
    aa_o_f557580081657435898iteral: ( fun_o_7929707447824500097iteral * $o ) > fun_o_3780185944647023446iteral ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J_J_J_J_J_J,type,
    aa_o_f3238628652476363425un_o_o: ( fun_o_754185029415132912un_o_o * $o ) > fun_o_2727025911028299589un_o_o ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__String__Oliteral_J_J_J_J_J_J_J,type,
    aa_o_f7455844346370101469iteral: ( fun_o_4077924883752292140iteral * $o ) > fun_o_7929707447824500097iteral ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J,type,
    aa_o_f4853929186051764509t_char: ( fun_o_5935606648521830764t_char * $o ) > fun_o_8622092458504374209t_char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J_J,type,
    aa_o_f6593176798326225313iteral: ( fun_o_8412685193225684464iteral * $o ) > fun_o_564418488415493701iteral ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J,type,
    aa_o_f5510610115356171066t_char: ( fun_o_8622092458504374209t_char * $o ) > fun_o_8831979434090536342t_char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J,type,
    aa_o_f293512076901925310iteral: ( fun_o_564418488415493701iteral * $o ) > fun_o_2475340183403050522iteral ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J,type,
    aa_o_f1772367729240163399t_char: ( fun_o_8831979434090536342t_char * $o ) > fun_o_4966981456184207595t_char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J,type,
    aa_o_f2755856924670595275iteral: ( fun_o_2475340183403050522iteral * $o ) > fun_o_2790778467162637167iteral ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J,type,
    aa_o_f2419597222644285284t_char: ( fun_o_4966981456184207595t_char * $o ) > fun_o_4601425329145410496t_char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J,type,
    aa_o_f3445383346798733800iteral: ( fun_o_2790778467162637167iteral * $o ) > fun_o_3955463192764026948iteral ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
    aa_o_f2285770652090220913t_char: ( fun_o_4601425329145410496t_char * $o ) > fun_o_2320692194582262805t_char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J_J,type,
    aa_o_f8819669424537506293iteral: ( fun_o_3955463192764026948iteral * $o ) > fun_o_507274782360720025iteral ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    aa_o_f5688925798504779150t_char: ( fun_o_2320692194582262805t_char * $o ) > fun_o_6199005644470072298t_char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_It__String__Oliteral_Mt__String__Oliteral_J_J,type,
    aa_o_f837559538564922386iteral: ( fun_o_507274782360720025iteral * $o ) > fun_o_5833298648030073966iteral ).

tff(sy_c_aa_001_Eo_001t__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
    aa_o_f2266643866902735003t_char: ( fun_o_6199005644470072298t_char * $o ) > fun_li3177621261580724543t_char ).

tff(sy_c_aa_001_Eo_001t__fun_It__String__Oliteral_Mt__String__Oliteral_J,type,
    aa_o_f1412681823949618463iteral: ( fun_o_5833298648030073966iteral * $o ) > fun_literal_literal ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001_Eo,type,
    aa_Code_integer_o: ( fun_Code_integer_o * code_integer ) > $o ).

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

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

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    aa_Code_integer_num: ( fun_Code_integer_num * code_integer ) > num ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    aa_Cod3168588076859124107eger_o: ( fun_Co1766037875893669450eger_o * code_integer ) > produc6271795597528267376eger_o ).

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__String__Ochar,type,
    aa_Code_integer_char: ( fun_Co8487158583234504619r_char * code_integer ) > char ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
    aa_Cod326357343816142457eger_o: ( fun_Co2081214437782444048eger_o * code_integer ) > fun_Code_integer_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod4161879620478718930nteger: ( fun_Co1981900195452921617nteger * code_integer ) > fun_Co8499238673658658103nteger ).

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J,type,
    aa_Cod5420483313623307717er_nat: ( fun_Co3583066429914935172er_nat * code_integer ) > fun_Code_integer_nat ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Num__Onum_J,type,
    aa_Cod6128727250572145423er_num: ( fun_Co8521658056209057742er_num * code_integer ) > fun_Code_integer_num ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    aa_Cod1096751537565085541eger_o: ( fun_Co3544761417380347684eger_o * code_integer ) > fun_Co1766037875893669450eger_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_Cod8380929501330563836nteger: ( fun_Co2470855966376689427nteger * code_integer ) > fun_Co6597186368970283863nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod1949805079358391713nteger: ( fun_Co3011819262099091808nteger * code_integer ) > fun_int_Code_integer ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod1578151519121547077nteger: ( fun_Co6226381647958292740nteger * code_integer ) > fun_nat_Code_integer ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_Cod3140087502949435154nteger: ( fun_Co5080488584548667985nteger * code_integer ) > fun_na8135612474337863287nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_Cod1978467581589928094nteger: ( fun_Co1561183069634336733nteger * code_integer ) > fun_Pr5876161306168176259nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Product____Type__Ounit_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod3669976074529508130nteger: ( fun_Co6473412210723208761nteger * code_integer ) > fun_Pr1820225556744648189nteger ).

tff(sy_c_aa_001t__Code____Numeral__Onatural_001_Eo,type,
    aa_Code_natural_o: ( fun_Code_natural_o * code_natural ) > $o ).

tff(sy_c_aa_001t__Code____Numeral__Onatural_001t__Code____Numeral__Onatural,type,
    aa_Cod4272041906116450446atural: ( fun_Co7398788559141407565atural * code_natural ) > code_natural ).

tff(sy_c_aa_001t__Code____Numeral__Onatural_001t__Nat__Onat,type,
    aa_Code_natural_nat: ( fun_Code_natural_nat * code_natural ) > nat ).

tff(sy_c_aa_001t__Code____Numeral__Onatural_001t__fun_It__Code____Numeral__Onatural_M_Eo_J,type,
    aa_Cod1485054954640246627ural_o: ( fun_Co7302518231560758906ural_o * code_natural ) > fun_Code_natural_o ).

tff(sy_c_aa_001t__Code____Numeral__Onatural_001t__fun_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    aa_Cod5141017464274028979atural: ( fun_Co6813818151677833714atural * code_natural ) > fun_Co7398788559141407565atural ).

tff(sy_c_aa_001t__Code____Numeral__Onatural_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J,type,
    aa_Cod5320533616963873627atural: ( fun_Co740390889742897050atural * code_natural ) > fun_nat_Code_natural ).

tff(sy_c_aa_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001_Eo,type,
    aa_hea7994211679605476167unit_o: ( fun_he8413527710729349838unit_o * heap_e7401611519738050253t_unit ) > $o ).

tff(sy_c_aa_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    aa_hea684056619475940329it_nat: ( fun_he5234484606919320760it_nat * heap_e7401611519738050253t_unit ) > option8956607266484857688it_nat ).

tff(sy_c_aa_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    aa_hea4724376117875161946it_nat: ( fun_he1921015881043195817it_nat * heap_e7401611519738050253t_unit ) > option3562590408128118217it_nat ).

tff(sy_c_aa_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    aa_hea256619718444519449it_nat: ( fun_he4944555133437423336it_nat * heap_e7401611519738050253t_unit ) > produc8664842809031399944it_nat ).

tff(sy_c_aa_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    aa_hea5645563107516622602it_nat: ( fun_he8889679731534893401it_nat * heap_e7401611519738050253t_unit ) > produc3260487557148687353it_nat ).

tff(sy_c_aa_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Product____Type__Ounit,type,
    aa_hea4146273921685161788t_unit: ( fun_he5123621388056232515t_unit * heap_e7401611519738050253t_unit ) > product_unit ).

tff(sy_c_aa_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__fun_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    aa_hea5083147195637387702it_nat: ( fun_he8369595706569872389it_nat * heap_e7401611519738050253t_unit ) > fun_na5667031169817407269it_nat ).

tff(sy_c_aa_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__fun_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    aa_hea414212936809421479it_nat: ( fun_he8849654208132081782it_nat * heap_e7401611519738050253t_unit ) > fun_na5926011011515597590it_nat ).

tff(sy_c_aa_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001tf__a,type,
    aa_hea5100639427662539565unit_a: ( fun_he6342020308413958580unit_a * heap_e7401611519738050253t_unit ) > a ).

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

tff(sy_c_aa_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    aa_int_Code_integer: ( fun_int_Code_integer * int ) > code_integer ).

tff(sy_c_aa_001t__Int__Oint_001t__Code____Numeral__Onatural,type,
    aa_int_Code_natural: ( fun_int_Code_natural * int ) > code_natural ).

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__Num__Onum,type,
    aa_int_num: ( fun_int_num * int ) > num ).

tff(sy_c_aa_001t__Int__Oint_001t__Option__Ooption_It__Num__Onum_J,type,
    aa_int_option_num: ( fun_int_option_num * int ) > option_num ).

tff(sy_c_aa_001t__Int__Oint_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_int77912060771590321nteger: ( fun_in4744367027289715400nteger * int ) > produc8923325533196201883nteger ).

tff(sy_c_aa_001t__Int__Oint_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    aa_int8200833983109115591atural: ( fun_in7496697025202066910atural * int ) > produc7822875418678951345atural ).

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__Rat__Orat,type,
    aa_int_rat: ( fun_int_rat * int ) > rat ).

tff(sy_c_aa_001t__Int__Oint_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_int8822742982810151799at_nat: ( fun_in8804710076054006414at_nat * int ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_int5322781614173852506e_term: ( fun_in4421688504851875865e_term * int ) > fun_Co3460408610320954864e_term ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_int_fun_int_o: ( fun_int_fun_int_o * int ) > fun_int_o ).

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__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_int408404503994301353e_term: ( fun_in94812177438132456e_term * int ) > fun_in5730379987311621695e_term ).

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__Rat__Orat_J,type,
    aa_int_fun_int_rat: ( fun_int_fun_int_rat * int ) > fun_int_rat ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_int2464330468366073224_int_o: ( fun_in5570076352459579423_int_o * int ) > fun_int_fun_int_o ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    aa_int1966403025003302569nt_int: ( fun_in2204242111582549184nt_int * int ) > fun_in1696471567231710611nt_int ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_int_fun_nat_o: ( fun_int_fun_nat_o * int ) > fun_nat_o ).

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 ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    aa_int431785320988601264at_int: ( fun_in6793278964266839919at_int * int ) > fun_nat_fun_nat_int ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_int1137397288593391972e_term: ( fun_in7158797356850311459e_term * int ) > fun_Pr5098669352780829690e_term ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_int5547325084924761305e_term: ( fun_in1403664741593307672e_term * int ) > fun_Pr285225112257423215e_term ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_int8219552307768696789e_term: ( fun_in237891044757944084e_term * int ) > fun_Pr7763680713840462443e_term ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    aa_int6101145382336329604_nat_o: ( fun_in2375804543815556931_nat_o * int ) > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Product____Type__Ounit_M_Eo_J,type,
    aa_int8968792772080532472unit_o: ( fun_in8984369272479023031unit_o * int ) > fun_Product_unit_o ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Product____Type__Ounit_Mt__Int__Oint_J,type,
    aa_int1856962249220300386it_int: ( fun_in5522822332083694073it_int * int ) > fun_Product_unit_int ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    aa_int_fun_set_nat_o: ( fun_in7136481858371106712_nat_o * int ) > fun_set_nat_o ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J,type,
    aa_int2448287375063835929at_int: ( fun_in6555721293707029592at_int * int ) > fun_set_nat_int ).

tff(sy_c_aa_001t__List__Olist_It__Code____Numeral__Ointeger_J_001_Eo,type,
    aa_lis2813719629158071341eger_o: ( fun_li1325434182413301700eger_o * list_Code_integer ) > $o ).

tff(sy_c_aa_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__List__Olist_It__String__Ochar_J,type,
    aa_lis1268021600136961932t_char: ( fun_li4211551766102051787t_char * list_Code_integer ) > list_char ).

tff(sy_c_aa_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__String__Oliteral,type,
    aa_lis5633869780665501801iteral: ( fun_li9041563744671036928iteral * list_Code_integer ) > literal ).

tff(sy_c_aa_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__fun_It__List__Olist_It__Code____Numeral__Ointeger_J_M_Eo_J,type,
    aa_lis7224287298985949145eger_o: ( fun_li279074556804398960eger_o * list_Code_integer ) > fun_li1325434182413301700eger_o ).

tff(sy_c_aa_001t__List__Olist_It__Int__Oint_J_001_Eo,type,
    aa_list_int_o: ( fun_list_int_o * list_int ) > $o ).

tff(sy_c_aa_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001_Eo,type,
    aa_list_list_nat_o: ( fun_list_list_nat_o * list_list_nat ) > $o ).

tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
    aa_list_nat_o: ( fun_list_nat_o * list_nat ) > $o ).

tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint,type,
    aa_list_nat_int: ( fun_list_nat_int * list_nat ) > int ).

tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    aa_list_nat_nat: ( fun_list_nat_nat * list_nat ) > nat ).

tff(sy_c_aa_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
    aa_lis2793599718760257805_nat_o: ( fun_li7366011709773453460_nat_o * list_P6011104703257516679at_nat ) > $o ).

tff(sy_c_aa_001t__List__Olist_It__Product____Type__Ounit_J_001_Eo,type,
    aa_lis2302375577916549633unit_o: ( fun_li6269451478509088136unit_o * list_Product_unit ) > $o ).

tff(sy_c_aa_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_001_Eo,type,
    aa_list_set_nat_o: ( fun_list_set_nat_o * list_set_nat ) > $o ).

tff(sy_c_aa_001t__List__Olist_It__String__Ochar_J_001_Eo,type,
    aa_list_char_o: ( fun_list_char_o * list_char ) > $o ).

tff(sy_c_aa_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    aa_lis2940536026700532620nteger: ( fun_li1966387620041874891nteger * list_char ) > list_Code_integer ).

tff(sy_c_aa_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
    aa_lis7614194356200183040t_char: ( fun_li3177621261580724543t_char * list_char ) > list_char ).

tff(sy_c_aa_001t__List__Olist_It__String__Ochar_J_001t__Nat__Onat,type,
    aa_list_char_nat: ( fun_list_char_nat * list_char ) > nat ).

tff(sy_c_aa_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral,type,
    aa_list_char_literal: ( fun_li5857714827575272844iteral * list_char ) > literal ).

tff(sy_c_aa_001t__List__Olist_It__String__Ochar_J_001t__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
    aa_lis7814493268959572977char_o: ( fun_li5260340055336554888char_o * list_char ) > fun_list_char_o ).

tff(sy_c_aa_001t__List__Olist_It__String__Ochar_J_001t__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
    aa_lis2542250943915203678t_char: ( fun_li7251496822767205277t_char * list_char ) > fun_li3177621261580724543t_char ).

tff(sy_c_aa_001t__List__Olist_It__String__Ochar_J_001t__fun_It__String__Oliteral_M_Eo_J,type,
    aa_lis3557855670089359654eral_o: ( fun_li9104157641623487077eral_o * list_char ) > fun_literal_o ).

tff(sy_c_aa_001t__Nat__Onat_001_Eo,type,
    aa_nat_o: ( fun_nat_o * nat ) > $o ).

tff(sy_c_aa_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    aa_nat_Code_integer: ( fun_nat_Code_integer * nat ) > code_integer ).

tff(sy_c_aa_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
    aa_nat_Code_natural: ( fun_nat_Code_natural * nat ) > code_natural ).

tff(sy_c_aa_001t__Nat__Onat_001t__Int__Oint,type,
    aa_nat_int: ( fun_nat_int * nat ) > int ).

tff(sy_c_aa_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    aa_nat_list_nat: ( fun_nat_list_nat * nat ) > list_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__Nat__Onat,type,
    aa_nat_nat: ( fun_nat_nat * nat ) > nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__Num__Onum,type,
    aa_nat_num: ( fun_nat_num * nat ) > num ).

tff(sy_c_aa_001t__Nat__Onat_001t__Option__Ooption_It__Num__Onum_J,type,
    aa_nat_option_num: ( fun_nat_option_num * nat ) > option_num ).

tff(sy_c_aa_001t__Nat__Onat_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    aa_nat3804989708481012326it_nat: ( fun_na5667031169817407269it_nat * nat ) > option8956607266484857688it_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    aa_nat3190776082673941207it_nat: ( fun_na5926011011515597590it_nat * nat ) > option3562590408128118217it_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_nat8379393489429068301nteger: ( fun_na3822476419092417572nteger * nat ) > produc8923325533196201883nteger ).

tff(sy_c_aa_001t__Nat__Onat_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    aa_nat7278943374911817763atural: ( fun_na6574806417004769082atural * nat ) > produc7822875418678951345atural ).

tff(sy_c_aa_001t__Nat__Onat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_nat4424723855478970923nt_int: ( fun_na1173468842265550146nt_int * nat ) > product_prod_int_int ).

tff(sy_c_aa_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_nat7603221163087113075at_nat: ( fun_na1476630412504255114at_nat * nat ) > product_prod_nat_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__Rat__Orat,type,
    aa_nat_rat: ( fun_nat_rat * nat ) > rat ).

tff(sy_c_aa_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    aa_nat_set_nat: ( fun_nat_set_nat * nat ) > set_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__String__Oliteral,type,
    aa_nat_literal: ( fun_nat_literal * nat ) > literal ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_nat6383147911832638789nteger: ( fun_na7078387940644931716nteger * nat ) > fun_Co8499238673658658103nteger ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_nat2506196440193460990e_term: ( fun_na1605103330871484349e_term * nat ) > fun_Co3460408610320954864e_term ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_nat2351529209232473362nteger: ( fun_na8856381646659805649nteger * nat ) > fun_Co6226381647958292740nteger ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    aa_nat5282697797315388251atural: ( fun_na607345901702507418atural * nat ) > fun_Co7398788559141407565atural ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_nat_fun_int_o: ( fun_nat_fun_int_o * nat ) > fun_int_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_nat_fun_int_int: ( fun_nat_fun_int_int * nat ) > fun_int_int ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
    aa_nat_fun_int_nat: ( fun_nat_fun_int_nat * nat ) > fun_int_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_nat3917568073327240013e_term: ( fun_na3603975746771071116e_term * nat ) > fun_in5730379987311621695e_term ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    aa_nat6704147566042551216at_int: ( fun_na2382631619178260847at_int * nat ) > fun_int_fun_nat_int ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__List__Olist_It__Int__Oint_J_M_Eo_J,type,
    aa_nat3638697132715757351_int_o: ( fun_na7306406234920378942_int_o * nat ) > fun_list_int_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_M_Eo_J,type,
    aa_nat158606821896287091_nat_o: ( fun_na4138138649119996554_nat_o * nat ) > fun_list_list_nat_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    aa_nat5883842100386203267_nat_o: ( fun_na7234455742697417626_nat_o * nat ) > fun_list_nat_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    aa_nat1970911183850433186_nat_o: ( fun_na5060536216179792737_nat_o * nat ) > fun_li7366011709773453460_nat_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__List__Olist_It__Product____Type__Ounit_J_M_Eo_J,type,
    aa_nat382646585846990998unit_o: ( fun_na2792118505502381397unit_o * nat ) > fun_li6269451478509088136unit_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    aa_nat2247512698059773005_nat_o: ( fun_na6948449159361463396_nat_o * nat ) > fun_list_set_nat_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
    aa_nat3000828340120075458char_o: ( fun_na6185503771353564761char_o * nat ) > fun_list_char_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_nat_fun_nat_o: ( fun_nat_fun_nat_o * nat ) > fun_nat_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J,type,
    aa_nat2010205248571845816nteger: ( fun_na8135612474337863287nteger * nat ) > fun_nat_Code_integer ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_nat_fun_nat_int: ( fun_nat_fun_nat_int * nat ) > fun_nat_int ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    aa_nat6157064778417537467st_nat: ( fun_na819858167508000762st_nat * nat ) > fun_nat_list_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_nat_fun_nat_nat: ( fun_nat_fun_nat_nat * nat ) > fun_nat_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_nat8803788312118491388at_nat: ( fun_na4138050610385705875at_nat * nat ) > fun_na1476630412504255114at_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
    aa_nat_fun_nat_rat: ( fun_nat_fun_nat_rat * nat ) > fun_nat_rat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_nat8314798964848609180_nat_o: ( fun_na7928631014926962227_nat_o * nat ) > fun_nat_fun_nat_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    aa_nat1293065384339088724at_int: ( fun_na7654559027617327379at_int * nat ) > fun_nat_fun_nat_int ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_nat3783148529270815480at_nat: ( fun_na3244735409550470327at_nat * nat ) > fun_nat_fun_nat_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    aa_nat4158579785191972357at_nat: ( fun_na4143055713577223196at_nat * nat ) > fun_na4138050610385705875at_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    aa_nat4259715169212259317_nat_o: ( fun_na5914473681949881652_nat_o * nat ) > fun_na3970208099681795815_nat_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    aa_nat5682243440470347102at_nat: ( fun_na1668100129766719005at_nat * nat ) > fun_na8713714802964096592at_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    aa_nat3713591252005393295on_num: ( fun_na7595719129879163854on_num * nat ) > fun_num_option_num ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_nat5764104769970567432e_term: ( fun_na2562132801372711111e_term * nat ) > fun_Pr5098669352780829690e_term ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_nat950660529447160957e_term: ( fun_na6030372222970483132e_term * nat ) > fun_Pr285225112257423215e_term ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_nat4859330308735838073e_term: ( fun_na6101041082579861176e_term * nat ) > fun_Pr7763680713840462443e_term ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    aa_nat7695548938202568488_nat_o: ( fun_na3970208099681795815_nat_o * nat ) > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_nat7997280740862404369at_nat: ( fun_na8713714802964096592at_nat * nat ) > fun_Pr2118491707810114563at_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Product____Type__Ounit_M_Eo_J,type,
    aa_nat5955485856412068252unit_o: ( fun_na5971062356810558811unit_o * nat ) > fun_Product_unit_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
    aa_nat7626694396248770914it_nat: ( fun_na381414567679918841it_nat * nat ) > fun_Product_unit_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
    aa_nat_fun_rat_rat: ( fun_nat_fun_rat_rat * nat ) > fun_rat_rat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Rat__Orat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    aa_nat298952458717073160at_rat: ( fun_na1148497095348402375at_rat * nat ) > fun_rat_fun_nat_rat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
    aa_nat_fun_set_int_o: ( fun_na8635581178960477080_int_o * nat ) > fun_set_int_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_M_Eo_J,type,
    aa_nat7721107121239158477_nat_o: ( fun_na1108963418173300452_nat_o * nat ) > fun_set_list_nat_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    aa_nat_fun_set_nat_o: ( fun_na8563630686737515764_nat_o * nat ) > fun_set_nat_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J,type,
    aa_nat2732338449433945789at_int: ( fun_na6839772368077139452at_int * nat ) > fun_set_nat_int ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    aa_nat6910189468943142497at_nat: ( fun_na106483476154090400at_nat * nat ) > fun_set_nat_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
    aa_nat4194932087953772816_int_o: ( fun_na6120508140265747663_int_o * nat ) > fun_se7249947736802449410_int_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    aa_nat1727191234135941576_nat_o: ( fun_na4518284863240432007_nat_o * nat ) > fun_se4782206882984618170_nat_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Set__Oset_It__Product____Type__Ounit_J_M_Eo_J,type,
    aa_nat5385523842031975612unit_o: ( fun_na7876804994713067643unit_o * nat ) > fun_se4342725320470837934unit_o ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    aa_nat5160323483289836711_nat_o: ( fun_na3711450577076729278_nat_o * nat ) > fun_set_set_nat_o ).

tff(sy_c_aa_001t__Num__Onum_001_Eo,type,
    aa_num_o: ( fun_num_o * num ) > $o ).

tff(sy_c_aa_001t__Num__Onum_001t__Code____Numeral__Ointeger,type,
    aa_num_Code_integer: ( fun_num_Code_integer * num ) > code_integer ).

tff(sy_c_aa_001t__Num__Onum_001t__Code____Numeral__Onatural,type,
    aa_num_Code_natural: ( fun_num_Code_natural * num ) > code_natural ).

tff(sy_c_aa_001t__Num__Onum_001t__Int__Oint,type,
    aa_num_int: ( fun_num_int * num ) > int ).

tff(sy_c_aa_001t__Num__Onum_001t__Nat__Onat,type,
    aa_num_nat: ( fun_num_nat * num ) > nat ).

tff(sy_c_aa_001t__Num__Onum_001t__Num__Onum,type,
    aa_num_num: ( fun_num_num * num ) > num ).

tff(sy_c_aa_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
    aa_num_option_num: ( fun_num_option_num * num ) > option_num ).

tff(sy_c_aa_001t__Num__Onum_001t__fun_It__Num__Onum_M_Eo_J,type,
    aa_num_fun_num_o: ( fun_num_fun_num_o * num ) > fun_num_o ).

tff(sy_c_aa_001t__Num__Onum_001t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
    aa_num_fun_num_int: ( fun_num_fun_num_int * num ) > fun_num_int ).

tff(sy_c_aa_001t__Option__Ooption_It__Num__Onum_J_001_Eo,type,
    aa_option_num_o: ( fun_option_num_o * option_num ) > $o ).

tff(sy_c_aa_001t__Option__Ooption_It__Num__Onum_J_001t__Int__Oint,type,
    aa_option_num_int: ( fun_option_num_int * option_num ) > int ).

tff(sy_c_aa_001t__Option__Ooption_It__Num__Onum_J_001t__Nat__Onat,type,
    aa_option_num_nat: ( fun_option_num_nat * option_num ) > nat ).

tff(sy_c_aa_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
    aa_option_num_num: ( fun_option_num_num * option_num ) > num ).

tff(sy_c_aa_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
    aa_opt5369159329878751634on_num: ( fun_op8989173174992673105on_num * option_num ) > option_num ).

tff(sy_c_aa_001t__Option__Ooption_It__Num__Onum_J_001t__fun_It__Option__Ooption_It__Num__Onum_J_M_Eo_J,type,
    aa_opt8674010985164783839_num_o: ( fun_op493924057549976310_num_o * option_num ) > fun_option_num_o ).

tff(sy_c_aa_001t__Option__Ooption_It__Num__Onum_J_001t__fun_It__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    aa_opt6324340006354107505_nat_o: ( fun_op298641349230941576_nat_o * option_num ) > fun_op479301348043730329_nat_o ).

tff(sy_c_aa_001t__Option__Ooption_It__Num__Onum_J_001t__fun_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    aa_opt3534265616007204544_nat_o: ( fun_op7041624247167716439_nat_o * option_num ) > fun_op9133723461703387112_nat_o ).

tff(sy_c_aa_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001_Eo,type,
    aa_opt8360315729766332930_nat_o: ( fun_op479301348043730329_nat_o * option8956607266484857688it_nat ) > $o ).

tff(sy_c_aa_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__fun_It__Option__Ooption_It__Num__Onum_J_M_Eo_J,type,
    aa_opt3964967581558639985_num_o: ( fun_op6018817604785332104_num_o * option8956607266484857688it_nat ) > fun_option_num_o ).

tff(sy_c_aa_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__fun_It__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    aa_opt6948326687743049987_nat_o: ( fun_op1633239139951409690_nat_o * option8956607266484857688it_nat ) > fun_op479301348043730329_nat_o ).

tff(sy_c_aa_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__fun_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    aa_opt4896439242367626578_nat_o: ( fun_op2843162390950497001_nat_o * option8956607266484857688it_nat ) > fun_op9133723461703387112_nat_o ).

tff(sy_c_aa_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001_Eo,type,
    aa_opt9064029888611036241_nat_o: ( fun_op9133723461703387112_nat_o * option3562590408128118217it_nat ) > $o ).

tff(sy_c_aa_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__fun_It__Option__Ooption_It__Num__Onum_J_M_Eo_J,type,
    aa_opt1040307220371850048_num_o: ( fun_op8016730479996091735_num_o * option3562590408128118217it_nat ) > fun_option_num_o ).

tff(sy_c_aa_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__fun_It__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    aa_opt7300124240507411154_nat_o: ( fun_op636363337204153321_nat_o * option3562590408128118217it_nat ) > fun_op479301348043730329_nat_o ).

tff(sy_c_aa_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__fun_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    aa_opt9017216189638898209_nat_o: ( fun_op5061235789045561272_nat_o * option3562590408128118217it_nat ) > fun_op9133723461703387112_nat_o ).

tff(sy_c_aa_001t__Predicate__Opred_It__Product____Type__Ounit_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J,type,
    aa_pre2304647123621358054atural: ( fun_pr6291093682560481277atural * pred_Product_unit ) > fun_Pr160617923510306311atural ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
    aa_Pro4926521453222654329eger_o: ( fun_Pr7211993375558241536eger_o * produc8923325533196201883nteger ) > $o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
    aa_Pro5245656335456409788nteger: ( fun_Pr6596356882939945227nteger * produc8923325533196201883nteger ) > code_integer ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Pro4693618923521437820nteger: ( fun_Pr5876161306168176259nteger * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_Pro4709850991492583807nteger: ( fun_Pr8197042229416165454nteger * produc8923325533196201883nteger ) > produc7509114972854644512nteger ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_Pro4291442876242880245nteger: ( fun_Pr2985126522952997572nteger * produc8923325533196201883nteger ) > produc2453742332425399446nteger ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Int__Oint_J,type,
    aa_Pro5426537197693075521et_int: ( fun_Pr3914994171019630096et_int * produc8923325533196201883nteger ) > set_int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Nat__Onat_J,type,
    aa_Pro381016180347496421et_nat: ( fun_Pr6405077315951356852et_nat * produc8923325533196201883nteger ) > set_nat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    aa_Pro3787439844975407259et_nat: ( fun_Pr1826760651301808490et_nat * produc8923325533196201883nteger ) > set_set_nat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    aa_Pro2031029616684803930nt_int: ( fun_Pr4436041713351579233nt_int * produc8923325533196201883nteger ) > set_se6260736226359567993nt_int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    aa_Pro3625874441308352674at_nat: ( fun_Pr6222699239247526825at_nat * produc8923325533196201883nteger ) > set_se7855581050983116737at_nat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J,type,
    aa_Pro4353979721973784418term_o: ( fun_Pr8608829099060101225term_o * produc8923325533196201883nteger ) > fun_fu7717421471775461377term_o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    aa_Pro6864914976855246444nteger: ( fun_Pr5283128996861011131nteger * produc8923325533196201883nteger ) > fun_fu2822266754163309965nteger ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J,type,
    aa_Pro9035772811636298456term_o: ( fun_Pr4680327687137491167term_o * produc8923325533196201883nteger ) > fun_fu4736795817735658359term_o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    aa_Pro5774345223241026284nteger: ( fun_Pr8371818785129130299nteger * produc8923325533196201883nteger ) > fun_fu8902681562042749453nteger ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001_Eo,type,
    aa_Pro6073791466322389475ural_o: ( fun_Pr8359263388657976682ural_o * produc7822875418678951345atural ) > $o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    aa_Pro4262726347734369532atural: ( fun_Pr74676805955934211atural * produc7822875418678951345atural ) > produc7822875418678951345atural ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    aa_Pro6676232280479689400atural: ( fun_Pr160617923510306311atural * produc7822875418678951345atural ) > produc4675096598859438275atural ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    aa_Pro3608106868210220652atural: ( fun_Pr1972435576755982779atural * produc7822875418678951345atural ) > produc9140211490168418807atural ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J,type,
    aa_Pro4093371377458885517term_o: ( fun_Pr8801881646629708820term_o * produc7822875418678951345atural ) > fun_fu7188715730271225026term_o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__fun_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J,type,
    aa_Pro4167682999780799576atural: ( fun_Pr6582898042194347943atural * produc7822875418678951345atural ) > fun_fu7873099233167337571atural ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
    aa_Pro4592068669816069595_int_o: ( fun_Pr3238374608701176162_int_o * product_prod_int_int ) > $o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
    aa_Pro6799258434465214121nt_int: ( fun_Pr982841511892225912nt_int * product_prod_int_int ) > int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_Pro6970504127699319548nt_int: ( fun_Pr2478283820680522243nt_int * product_prod_int_int ) > product_prod_int_int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_Pro8800328608558162860nt_int: ( fun_Pr8696298990344678139nt_int * product_prod_int_int ) > produc4788131091073775919nt_int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_Pro5748361610964203648nt_int: ( fun_Pr4781553364444837327nt_int * product_prod_int_int ) > produc3043004577601930755nt_int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    aa_Pro6166618844887768661nt_rat: ( fun_Pr1652463973008025892nt_rat * product_prod_int_int ) > rat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
    aa_Pro8644714012989290591et_int: ( fun_Pr1023655258412994094et_int * product_prod_int_int ) > set_int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Nat__Onat_J,type,
    aa_Pro3599192995643711491et_nat: ( fun_Pr3513738403344720850et_nat * product_prod_int_int ) > set_nat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    aa_Pro1963545105414287545et_nat: ( fun_Pr27309308411849096et_nat * product_prod_int_int ) > set_set_nat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    aa_Pro7039973274325895612nt_int: ( fun_Pr2465684420796299459nt_int * product_prod_int_int ) > set_se6260736226359567993nt_int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    aa_Pro8634818098949444356at_nat: ( fun_Pr4252341946692247051at_nat * product_prod_int_int ) > set_se7855581050983116737at_nat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    aa_Pro3952998155787061599_int_o: ( fun_Pr9018117161326028078_int_o * product_prod_int_int ) > fun_Pr3238374608701176162_int_o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_Pro4124234152028750080nt_int: ( fun_Pr1308109350235310159nt_int * product_prod_int_int ) > fun_Pr2478283820680522243nt_int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Rat__Orat_M_Eo_J,type,
    aa_Pro2298929076809959388_rat_o: ( fun_Pr7537129237977221859_rat_o * product_prod_int_int ) > fun_rat_o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J,type,
    aa_Pro1235490124264145077term_o: ( fun_Pr7145817153658775868term_o * product_prod_int_int ) > fun_fu3265955482920998386term_o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    aa_Pro3263785050932909160nt_int: ( fun_Pr4353654455642104759nt_int * product_prod_int_int ) > fun_fu4545303443005895275nt_int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_Eo_J,type,
    aa_Pro1023942307601852809term_o: ( fun_Pr5526332658708858896term_o * product_prod_int_int ) > fun_fu6378075912484669126term_o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    aa_Pro7187055436517876584nt_int: ( fun_Pr8826142978676617911nt_int * product_prod_int_int ) > fun_fu5813366174458671467nt_int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
    aa_Pro6150637604445964947_nat_o: ( fun_Pr4796943543331071514_nat_o * product_prod_nat_nat ) > $o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
    aa_Pro4764841431850000930nteger: ( fun_Pr6026251430685297nteger * product_prod_nat_nat ) > code_integer ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    aa_Pro4240683299535347697at_int: ( fun_Pr7647638413817135296at_int * product_prod_nat_nat ) > int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    aa_Pro4243173770044397973at_nat: ( fun_Pr2602117396471556196at_nat * product_prod_nat_nat ) > nat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_Pro262675715343573244at_nat: ( fun_Pr2118491707810114563at_nat * product_prod_nat_nat ) > product_prod_nat_nat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Rat__Orat,type,
    aa_Pro3608043709957902237at_rat: ( fun_Pr8317260874932935276at_rat * product_prod_nat_nat ) > rat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_Pro2912763917461432128_int_o: ( fun_Pr4738963024209204295_int_o * product_prod_nat_nat ) > fun_int_o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_Pro4733806262894120092_nat_o: ( fun_Pr6984107991879650211_nat_o * product_prod_nat_nat ) > fun_nat_o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_Pro1957341097241355813_nat_o: ( fun_Pr3455331679158536364_nat_o * product_prod_nat_nat ) > fun_nat_fun_nat_o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    aa_Pro6895555690143131534at_nat: ( fun_Pr557098546508347797at_nat * product_prod_nat_nat ) > fun_na4138050610385705875at_nat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    aa_Pro989595756709598047_nat_o: ( fun_Pr2028404973800837934_nat_o * product_prod_nat_nat ) > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_Pro8023792847356852296at_nat: ( fun_Pr2477239776880934807at_nat * product_prod_nat_nat ) > fun_Pr2118491707810114563at_nat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001_Eo,type,
    aa_Pro4744234731172327378_nat_o: ( fun_Pr8032424337255632745_nat_o * produc8664842809031399944it_nat ) > $o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    aa_Pro5228051289423594006at_nat: ( fun_Pr367382672041210069at_nat * produc8664842809031399944it_nat ) > nat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    aa_Pro8996696869555642526it_nat: ( fun_Pr7644814098969791837it_nat * produc8664842809031399944it_nat ) > option8956607266484857688it_nat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001_Eo,type,
    aa_Pro5947980084102686253_nat_o: ( fun_Pr3059359313727237556_nat_o * produc3843707927480180839at_nat ) > $o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_Eo,type,
    aa_Pro4652312591602923194eger_o: ( fun_Pr1347628893830317649eger_o * produc7509114972854644512nteger ) > $o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Pro5745608113995628733nteger: ( fun_Pr2054267358141571796nteger * produc7509114972854644512nteger ) > produc8923325533196201883nteger ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_Pro1361292240318994510e_term: ( fun_Pr4759914407354907917e_term * produc7509114972854644512nteger ) > fun_Co3460408610320954864e_term ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_Eo,type,
    aa_Pro5451197003417746155_int_o: ( fun_Pr1196100989263358594_int_o * produc4788131091073775919nt_int ) > $o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_Pro4437156595432605708nt_int: ( fun_Pr1552678306225191715nt_int * produc4788131091073775919nt_int ) > product_prod_int_int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_Pro2551989255418117932e_term: ( fun_Pr7653582204007996523e_term * produc4788131091073775919nt_int ) > fun_in5730379987311621695e_term ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_Eo,type,
    aa_Pro4148064003102580676eger_o: ( fun_Pr7015759342434529115eger_o * produc2453742332425399446nteger ) > $o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Pro4625778721341478855nteger: ( fun_Pr6160931176198666718nteger * produc2453742332425399446nteger ) > produc8923325533196201883nteger ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_Pro4276937008533432526e_term: ( fun_Pr4557384415951816845e_term * produc2453742332425399446nteger ) > fun_Pr5098669352780829690e_term ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001_Eo,type,
    aa_Pro866224779355485219ural_o: ( fun_Pr3733920118687433658ural_o * produc9140211490168418807atural ) > $o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    aa_Pro2627516385176489788atural: ( fun_Pr8015448952463279699atural * produc9140211490168418807atural ) > produc7822875418678951345atural ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_Pro1606062155721680164e_term: ( fun_Pr944821188406467171e_term * produc9140211490168418807atural ) > fun_Pr285225112257423215e_term ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_Eo,type,
    aa_Pro2601676399929874711_int_o: ( fun_Pr3540825087398332590_int_o * produc3043004577601930755nt_int ) > $o ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_Pro2675344531184151096nt_int: ( fun_Pr1987159835869338959nt_int * produc3043004577601930755nt_int ) > product_prod_int_int ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_Pro4010435699744810540e_term: ( fun_Pr5872836680101629291e_term * produc3043004577601930755nt_int ) > fun_Pr7763680713840462443e_term ).

tff(sy_c_aa_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001_Eo,type,
    aa_Pro6426884464393220769_nat_o: ( fun_Pr5746006151868627512_nat_o * produc3260487557148687353it_nat ) > $o ).

tff(sy_c_aa_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    aa_Pro5302094286594872192it_nat: ( fun_Pr2403869735082759103it_nat * produc3260487557148687353it_nat ) > option3562590408128118217it_nat ).

tff(sy_c_aa_001t__Product____Type__Ounit_001_Eo,type,
    aa_Product_unit_o: ( fun_Product_unit_o * product_unit ) > $o ).

tff(sy_c_aa_001t__Product____Type__Ounit_001t__Code____Numeral__Ointeger,type,
    aa_Pro1705297278780387758nteger: ( fun_Pr1820225556744648189nteger * product_unit ) > code_integer ).

tff(sy_c_aa_001t__Product____Type__Ounit_001t__Code____Numeral__Onatural,type,
    aa_Pro528892430537105273atural: ( fun_Pr6278675377255837256atural * product_unit ) > code_natural ).

tff(sy_c_aa_001t__Product____Type__Ounit_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J,type,
    aa_Pro1108561329362217960t_unit: ( fun_Pr292408699144966967t_unit * product_unit ) > heap_T5738788834812785303t_unit ).

tff(sy_c_aa_001t__Product____Type__Ounit_001t__Heap____Time____Monad__OHeap_Itf__a_J,type,
    aa_Pro7886978635576167513Heap_a: ( fun_Pr5215307506330550056Heap_a * product_unit ) > heap_Time_Heap_a ).

tff(sy_c_aa_001t__Product____Type__Ounit_001t__Int__Oint,type,
    aa_Product_unit_int: ( fun_Product_unit_int * product_unit ) > int ).

tff(sy_c_aa_001t__Product____Type__Ounit_001t__Nat__Onat,type,
    aa_Product_unit_nat: ( fun_Product_unit_nat * product_unit ) > nat ).

tff(sy_c_aa_001t__Product____Type__Ounit_001t__Rat__Orat,type,
    aa_Product_unit_rat: ( fun_Product_unit_rat * product_unit ) > rat ).

tff(sy_c_aa_001t__Product____Type__Ounit_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_Pro8249700398669690036_int_o: ( fun_Pr2157672562078508475_int_o * product_unit ) > fun_int_o ).

tff(sy_c_aa_001t__Product____Type__Ounit_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_Pro847370707247602192_nat_o: ( fun_Pr4402817529748954391_nat_o * product_unit ) > fun_nat_o ).

tff(sy_c_aa_001t__Product____Type__Ounit_001t__fun_It__Product____Type__Ounit_M_Eo_J,type,
    aa_Pro4895447120693452255unit_o: ( fun_Pr1520749766685617582unit_o * product_unit ) > fun_Product_unit_o ).

tff(sy_c_aa_001t__Product____Type__Ounit_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    aa_Pro5573132868698907866_nat_o: ( fun_Pr1646868699942875873_nat_o * product_unit ) > fun_set_nat_o ).

tff(sy_c_aa_001t__Rat__Orat_001_Eo,type,
    aa_rat_o: ( fun_rat_o * rat ) > $o ).

tff(sy_c_aa_001t__Rat__Orat_001t__Rat__Orat,type,
    aa_rat_rat: ( fun_rat_rat * rat ) > rat ).

tff(sy_c_aa_001t__Rat__Orat_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_rat8624586542854104963nt_int: ( fun_ra7314870381713115802nt_int * rat ) > set_Pr958786334691620121nt_int ).

tff(sy_c_aa_001t__Rat__Orat_001t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
    aa_rat_fun_int_rat: ( fun_rat_fun_int_rat * rat ) > fun_int_rat ).

tff(sy_c_aa_001t__Rat__Orat_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
    aa_rat_fun_nat_rat: ( fun_rat_fun_nat_rat * rat ) > fun_nat_rat ).

tff(sy_c_aa_001t__Rat__Orat_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    aa_rat250263598611652360at_rat: ( fun_ra5513060666784058567at_rat * rat ) > fun_nat_fun_nat_rat ).

tff(sy_c_aa_001t__Rat__Orat_001t__fun_It__Product____Type__Ounit_Mt__Rat__Orat_J,type,
    aa_rat6507435564262174562it_rat: ( fun_ra5613769993463817465it_rat * rat ) > fun_Product_unit_rat ).

tff(sy_c_aa_001t__Rat__Orat_001t__fun_It__Rat__Orat_M_Eo_J,type,
    aa_rat_fun_rat_o: ( fun_rat_fun_rat_o * rat ) > fun_rat_o ).

tff(sy_c_aa_001t__Rat__Orat_001t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
    aa_rat_fun_rat_rat: ( fun_rat_fun_rat_rat * rat ) > fun_rat_rat ).

tff(sy_c_aa_001t__Rat__Orat_001t__fun_It__Rat__Orat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    aa_rat3146053865535587600at_rat: ( fun_ra3995598502166916815at_rat * rat ) > fun_rat_fun_nat_rat ).

tff(sy_c_aa_001t__Rat__Orat_001t__fun_It__Rat__Orat_Mt__fun_It__Rat__Orat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_J,type,
    aa_rat7467449789483521253at_rat: ( fun_ra8505079968215247908at_rat * rat ) > fun_ra3995598502166916815at_rat ).

tff(sy_c_aa_001t__Set__Oset_It__Int__Oint_J_001_Eo,type,
    aa_set_int_o: ( fun_set_int_o * set_int ) > $o ).

tff(sy_c_aa_001t__Set__Oset_It__Int__Oint_J_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_set_int_fun_int_o: ( fun_se867122708194164540_int_o * set_int ) > fun_int_o ).

tff(sy_c_aa_001t__Set__Oset_It__Int__Oint_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_set7310040872927613301nt_int: ( fun_se4827636401813558708nt_int * set_int ) > fun_int_int ).

tff(sy_c_aa_001t__Set__Oset_It__Int__Oint_J_001t__fun_It__List__Olist_It__Int__Oint_J_M_Eo_J,type,
    aa_set1055383481605741461_int_o: ( fun_se4632862981408181932_int_o * set_int ) > fun_list_int_o ).

tff(sy_c_aa_001t__Set__Oset_It__Int__Oint_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_set6310687161026558745at_int: ( fun_se2640714827120536920at_int * set_int ) > fun_nat_int ).

tff(sy_c_aa_001t__Set__Oset_It__Int__Oint_J_001t__fun_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
    aa_set5425576183772390127_int_o: ( fun_se2051630309975736070_int_o * set_int ) > fun_set_int_o ).

tff(sy_c_aa_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001_Eo,type,
    aa_set_list_nat_o: ( fun_set_list_nat_o * set_list_nat ) > $o ).

tff(sy_c_aa_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__fun_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_M_Eo_J,type,
    aa_set5589568368246272301_nat_o: ( fun_se6885878543466214468_nat_o * set_list_nat ) > fun_list_list_nat_o ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
    aa_set_nat_o: ( fun_set_nat_o * set_nat ) > $o ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
    aa_set871585252387937313nteger: ( fun_se7586849273513198688nteger * set_nat ) > code_integer ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001t__Int__Oint,type,
    aa_set_nat_int: ( fun_set_nat_int * set_nat ) > int ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    aa_set_nat_nat: ( fun_set_nat_nat * set_nat ) > nat ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001t__Rat__Orat,type,
    aa_set_nat_rat: ( fun_set_nat_rat * set_nat ) > rat ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_set_nat_fun_int_o: ( fun_se7360890175450123928_int_o * set_nat ) > fun_int_o ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_set3358596244511904025nt_int: ( fun_se876191773397849432nt_int * set_nat ) > fun_int_int ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    aa_set3062951307625612365_nat_o: ( fun_se4323335347534645604_nat_o * set_nat ) > fun_list_nat_o ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_set_nat_fun_nat_o: ( fun_se382663106265794036_nat_o * set_nat ) > fun_nat_o ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_set2359242532610849469at_int: ( fun_se7912642235559603452at_int * set_nat ) > fun_nat_int ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_set6537093552120046177at_nat: ( fun_se1179353343636554400at_nat * set_nat ) > fun_nat_nat ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001t__fun_It__Product____Type__Ounit_M_Eo_J,type,
    aa_set3064922119574097746unit_o: ( fun_se4446312468244607761unit_o * set_nat ) > fun_Product_unit_o ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    aa_set7861427066675100071_nat_o: ( fun_se2170385732985038782_nat_o * set_nat ) > fun_set_nat_o ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_Eo,type,
    aa_set6170482117830564987_int_o: ( fun_se7249947736802449410_int_o * set_Pr958786334691620121nt_int ) > $o ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Int__Oint,type,
    aa_set4750725449604864009nt_int: ( fun_se9146809000320981208nt_int * set_Pr958786334691620121nt_int ) > int ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Nat__Onat,type,
    aa_set4753215920113914285nt_nat: ( fun_se4101287982975402108nt_nat * set_Pr958786334691620121nt_int ) > nat ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Rat__Orat,type,
    aa_set4118085860027418549nt_rat: ( fun_se593059424582005380nt_rat * set_Pr958786334691620121nt_int ) > rat ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
    aa_set3702741264012733747_nat_o: ( fun_se4782206882984618170_nat_o * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Code____Numeral__Ointeger,type,
    aa_set5601480140167866242nteger: ( fun_se2775971902884351953nteger * set_Pr1261947904930325089at_nat ) > code_integer ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Int__Oint,type,
    aa_set4746584241299694929at_int: ( fun_se9142667792015812128at_int * set_Pr1261947904930325089at_nat ) > int ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    aa_set4749074711808745205at_nat: ( fun_se4097146774670233028at_nat * set_Pr1261947904930325089at_nat ) > nat ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Rat__Orat,type,
    aa_set4113944651722249469at_rat: ( fun_se588918216276836300at_rat * set_Pr1261947904930325089at_nat ) > rat ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    aa_set1857767853180004543_nat_o: ( fun_se374060482380105870_nat_o * set_Pr1261947904930325089at_nat ) > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    aa_set7675215768510388063_nat_o: ( fun_se5639961202347202350_nat_o * set_Pr1261947904930325089at_nat ) > fun_se4782206882984618170_nat_o ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    aa_set6353399069419443519_nat_o: ( fun_se5282722391638144526_nat_o * set_Pr4329608150637261639at_nat ) > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Ounit_J_001_Eo,type,
    aa_set8780210009671629607unit_o: ( fun_se4342725320470837934unit_o * set_Product_unit ) > $o ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Ounit_J_001t__fun_It__List__Olist_It__Product____Type__Ounit_J_M_Eo_J,type,
    aa_set1739257761580076089unit_o: ( fun_se4193343538020515080unit_o * set_Product_unit ) > fun_li6269451478509088136unit_o ).

tff(sy_c_aa_001t__Set__Oset_It__Product____Type__Ounit_J_001t__fun_It__Product____Type__Ounit_M_Eo_J,type,
    aa_set2953008695252327103unit_o: ( fun_se8207921111072574094unit_o * set_Product_unit ) > fun_Product_unit_o ).

tff(sy_c_aa_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001_Eo,type,
    aa_set_set_nat_o: ( fun_set_set_nat_o * set_set_nat ) > $o ).

tff(sy_c_aa_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__fun_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    aa_set474910819808044769_nat_o: ( fun_se6503204438271284984_nat_o * set_set_nat ) > fun_list_set_nat_o ).

tff(sy_c_aa_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    aa_set4404905041378927473_nat_o: ( fun_se2389560805064864136_nat_o * set_set_nat ) > fun_set_nat_o ).

tff(sy_c_aa_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
    aa_set4062574822058255039_int_o: ( fun_se3135557758678015630_int_o * set_se6260736226359567993nt_int ) > fun_se7249947736802449410_int_o ).

tff(sy_c_aa_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    aa_set7533644503045286079_nat_o: ( fun_se7472145016457562254_nat_o * set_se7855581050983116737at_nat ) > fun_se4782206882984618170_nat_o ).

tff(sy_c_aa_001t__Set__Oset_It__String__Ochar_J_001t__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
    aa_set3791436276075643467char_o: ( fun_se8116641925208253410char_o * set_char ) > fun_list_char_o ).

tff(sy_c_aa_001t__String__Ochar_001_Eo,type,
    aa_char_o: ( fun_char_o * char ) > $o ).

tff(sy_c_aa_001t__String__Ochar_001t__Nat__Onat,type,
    aa_char_nat: ( fun_char_nat * char ) > nat ).

tff(sy_c_aa_001t__String__Ochar_001t__String__Ochar,type,
    aa_char_char: ( fun_char_char * char ) > char ).

tff(sy_c_aa_001t__String__Ochar_001t__fun_It__String__Ochar_M_Eo_J,type,
    aa_char_fun_char_o: ( fun_char_fun_char_o * char ) > fun_char_o ).

tff(sy_c_aa_001t__String__Oliteral_001_Eo,type,
    aa_literal_o: ( fun_literal_o * literal ) > $o ).

tff(sy_c_aa_001t__String__Oliteral_001t__List__Olist_It__String__Ochar_J,type,
    aa_literal_list_char: ( fun_li7473841323753414984t_char * literal ) > list_char ).

tff(sy_c_aa_001t__String__Oliteral_001t__String__Oliteral,type,
    aa_literal_literal: ( fun_literal_literal * literal ) > literal ).

tff(sy_c_aa_001t__String__Oliteral_001t__fun_It__String__Oliteral_M_Eo_J,type,
    aa_lit5986169069484365087eral_o: ( fun_li4177005666941401710eral_o * literal ) > fun_literal_o ).

tff(sy_c_aa_001t__Typerep__Otyperep_001t__Nat__Onat,type,
    aa_typerep_nat: ( fun_typerep_nat * typerep ) > nat ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
    aa_fun774133186966548707eger_o: ( fun_fu1769128043447206010eger_o * fun_Co8499238673658658103nteger ) > $o ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_fun4435975922079263404nteger: ( fun_fu1763652423599918571nteger * fun_Co8499238673658658103nteger ) > fun_Co8499238673658658103nteger ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_fun2509385170507305080nteger: ( fun_fu4800041728187187127nteger * fun_Co8499238673658658103nteger ) > fun_Pr5876161306168176259nteger ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001_Eo,type,
    aa_fun7659544460261446762term_o: ( fun_fu7717421471775461377term_o * fun_Co3460408610320954864e_term ) > $o ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_fun6004758606051371726nteger: ( fun_fu2822266754163309965nteger * fun_Co3460408610320954864e_term ) > produc7509114972854644512nteger ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_fun8764196472072707742e_term: ( fun_fu9036519777977489245e_term * fun_Co3460408610320954864e_term ) > fun_Co3460408610320954864e_term ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J,type,
    aa_fun5125889547937825198eger_o: ( fun_fu3422731660242565741eger_o * fun_Co3460408610320954864e_term ) > fun_Pr7211993375558241536eger_o ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    aa_fun6437779379678770976nteger: ( fun_fu3460006408547937207nteger * fun_Co3460408610320954864e_term ) > fun_Pr8197042229416165454nteger ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    aa_fun1064201865214304482et_int: ( fun_fu1569576118606662777et_int * fun_Co3460408610320954864e_term ) > fun_Pr3914994171019630096et_int ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    aa_fun3554285010146031238et_nat: ( fun_fu6383124537394581533et_nat * fun_Co3460408610320954864e_term ) > fun_Pr6405077315951356852et_nat ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    aa_fun966729468685823036et_nat: ( fun_fu963894927305104083et_nat * fun_Co3460408610320954864e_term ) > fun_Pr1826760651301808490et_nat ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    aa_fun5905053582222755087nt_int: ( fun_fu6439581347498178894nt_int * fun_Co3460408610320954864e_term ) > fun_Pr4436041713351579233nt_int ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    aa_fun7691711108118702679at_nat: ( fun_fu5924095598813543062at_nat * fun_Co3460408610320954864e_term ) > fun_Pr6222699239247526825at_nat ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_fun1525513021908036776e_term: ( fun_fu3093988009420988519e_term * fun_Co3460408610320954864e_term ) > fun_Pr5098669352780829690e_term ).

tff(sy_c_aa_001t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    aa_fun2986507860694160780it_nat: ( fun_fu2492676192234506515it_nat * fun_he4944555133437423336it_nat ) > fun_he5234484606919320760it_nat ).

tff(sy_c_aa_001t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    aa_fun3370411987892275404it_nat: ( fun_fu9056938625493858259it_nat * fun_he8889679731534893401it_nat ) > fun_he1921015881043195817it_nat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_M_Eo_J_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_fun6243517787477317628_int_o: ( fun_fu3883539188510166275_int_o * fun_int_o ) > fun_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_M_Eo_J_001t__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J,type,
    aa_fun6019004165459918095nteger: ( fun_fu6361890901329191134nteger * fun_int_o ) > fun_int_Code_integer ).

tff(sy_c_aa_001t__fun_It__Int__Oint_M_Eo_J_001t__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J,type,
    aa_fun1254081949116331354atural: ( fun_fu1591967865782657961atural * fun_int_o ) > fun_int_Code_natural ).

tff(sy_c_aa_001t__fun_It__Int__Oint_M_Eo_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_fun8822586851003501406nt_int: ( fun_fu7811933856372049837nt_int * fun_int_o ) > fun_int_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_M_Eo_J_001t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
    aa_fun3777065833657922306nt_nat: ( fun_fu1078644964449000785nt_nat * fun_int_o ) > fun_int_nat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_M_Eo_J_001t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
    aa_fun268837275264525578nt_rat: ( fun_fu3922030663826099033nt_rat * fun_int_o ) > fun_int_rat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_fun2884842898402423460nteger: ( fun_fu6811445973280110947nteger * fun_int_Code_integer ) > fun_Co3011819262099091808nteger ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_fun3336537925265328129_int_o: ( fun_fu5271299750659287320_int_o * fun_int_Code_integer ) > fun_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J,type,
    aa_fun5304498221708291146nteger: ( fun_fu3628599891403802633nteger * fun_int_Code_integer ) > fun_int_Code_integer ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_001t__fun_It__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun8818272713621706452_int_o: ( fun_fu63825462011493995_int_o * fun_int_Code_integer ) > fun_fu5271299750659287320_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_fun296549085072024822_int_o: ( fun_fu2231310910465984013_int_o * fun_int_Code_natural ) > fun_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J_001t__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J,type,
    aa_fun4771645977350521568atural: ( fun_fu3090746827843086623atural * fun_int_Code_natural ) > fun_int_Code_natural ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J_001t__fun_It__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun4460768838848905406_int_o: ( fun_fu619080566302588373_int_o * fun_int_Code_natural ) > fun_fu2231310910465984013_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
    aa_fun_int_int_o: ( fun_fun_int_int_o * fun_int_int ) > $o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_fun8964553180794676938nteger: ( fun_fu6921880696561781257nteger * fun_int_int ) > fun_Co8499238673658658103nteger ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_fun8155382727227203058_int_o: ( fun_fu4534828536049688329_int_o * fun_int_int ) > fun_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_fun6280345759651049192nt_int: ( fun_fu1143798476471108391nt_int * fun_int_int ) > fun_int_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun5911324870756489105nt_int: ( fun_fu923337204662071120nt_int * fun_int_int ) > fun_int_fun_int_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Set__Oset_It__Int__Oint_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun8076508010538293575nt_int: ( fun_fu3739269683040691718nt_int * fun_int_int ) > fun_se4827636401813558708nt_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J,type,
    aa_fun6864876441520360999eger_o: ( fun_fu726873822431485118eger_o * fun_int_int ) > fun_fu1769128043447206010eger_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    aa_fun2545150047610057353_int_o: ( fun_fu4208408993564413984_int_o * fun_int_int ) > fun_fun_int_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun1697523107123627702_int_o: ( fun_fu4125167268176844237_int_o * fun_int_int ) > fun_fu4534828536049688329_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Nat__Onat_J_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_fun5425778157628386638_int_o: ( fun_fu1805223966450871909_int_o * fun_int_nat ) > fun_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Nat__Onat_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_fun2328901131235339916nt_int: ( fun_fu6415725884910174923nt_int * fun_int_nat ) > fun_int_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Nat__Onat_J_001t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
    aa_fun6506752150744536624nt_nat: ( fun_fu8905809029841901679nt_nat * fun_int_nat ) > fun_int_nat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    aa_fun1022895345097303677nt_nat: ( fun_fu9041382543001606716nt_nat * fun_int_nat ) > fun_nat_fun_int_nat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Nat__Onat_J_001t__fun_It__fun_It__Int__Oint_Mt__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun8703205213358249326_int_o: ( fun_fu2760153274138146437_int_o * fun_int_nat ) > fun_fu1805223966450871909_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001_Eo,type,
    aa_fun3020711759848831579term_o: ( fun_fu3265955482920998386term_o * fun_in5730379987311621695e_term ) > $o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun3520829229142255404nt_int: ( fun_fu4545303443005895275nt_int * fun_in5730379987311621695e_term ) > produc4788131091073775919nt_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_fun5947755341433680828e_term: ( fun_fu4810317505167811835e_term * fun_in5730379987311621695e_term ) > fun_in5730379987311621695e_term ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    aa_fun586359091965693151_int_o: ( fun_fu2978539155105838238_int_o * fun_in5730379987311621695e_term ) > fun_Pr3238374608701176162_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    aa_fun260840925927807678nt_int: ( fun_fu465145433024125141nt_int * fun_in5730379987311621695e_term ) > fun_Pr8696298990344678139nt_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    aa_fun5919120857301092337et_int: ( fun_fu4266222366127506568et_int * fun_in5730379987311621695e_term ) > fun_Pr1023655258412994094et_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    aa_fun8409204002232819093et_nat: ( fun_fu9079770784915425324et_nat * fun_in5730379987311621695e_term ) > fun_Pr3513738403344720850et_nat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    aa_fun8341152516354955083et_nat: ( fun_fu5458086908000294626et_nat * fun_in5730379987311621695e_term ) > fun_Pr27309308411849096et_nat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    aa_fun4635030967160187712nt_int: ( fun_fu719511775830429311nt_int * fun_in5730379987311621695e_term ) > fun_Pr2465684420796299459nt_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    aa_fun6421688493056135304at_nat: ( fun_fu204026027145793479at_nat * fun_in5730379987311621695e_term ) > fun_Pr4252341946692247051at_nat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_fun1932935823081541608e_term: ( fun_fu1176768540439120679e_term * fun_in5730379987311621695e_term ) > fun_Pr7763680713840462443e_term ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Rat__Orat_J_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_fun7094158214180239686_int_o: ( fun_fu3473604023002724957_int_o * fun_int_rat ) > fun_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Rat__Orat_J_001t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
    aa_fun1481061427733998144nt_rat: ( fun_fu1008360527747082367nt_rat * fun_int_rat ) > fun_int_rat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Rat__Orat_J_001t__fun_It__Rat__Orat_Mt__fun_It__Int__Oint_Mt__Rat__Orat_J_J,type,
    aa_fun9155288305517341845nt_rat: ( fun_fu114989186063767636nt_rat * fun_int_rat ) > fun_rat_fun_int_rat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__Rat__Orat_J_001t__fun_It__fun_It__Int__Oint_Mt__Rat__Orat_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun1445893531507182942_int_o: ( fun_fu7714913403879666293_int_o * fun_int_rat ) > fun_fu3473604023002724957_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_fun282240425178129291nt_int: ( fun_fu76847565846999002nt_int * fun_int_fun_int_o ) > fun_int_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
    aa_fun4460091444687325999nt_nat: ( fun_fu2566930710778725758nt_nat * fun_int_fun_int_o ) > fun_int_nat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun6133342665515662012_int_o: ( fun_fu138107234227827011_int_o * fun_int_fun_int_o ) > fun_int_fun_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun7323606162024495668nt_int: ( fun_fu6453395667394816515nt_int * fun_int_fun_int_o ) > fun_int_fun_int_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    aa_fun4023131440832735197nt_int: ( fun_fu8427683764667544036nt_int * fun_int_fun_int_o ) > fun_in1696471567231710611nt_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    aa_fun7329327620659528856_int_o: ( fun_fu1345883967705819239_int_o * fun_int_fun_int_o ) > fun_Pr3238374608701176162_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun82482788810740534_int_o: ( fun_fu1481857817119020493_int_o * fun_int_fun_int_int ) > fun_int_fun_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun6119123525526402106nt_int: ( fun_fu6092417073193843705nt_int * fun_int_fun_int_int ) > fun_int_fun_int_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    aa_fun1034830265760468567nt_int: ( fun_fu918106826131887214nt_int * fun_int_fun_int_int ) > fun_in1696471567231710611nt_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
    aa_fun914008879417524284nt_int: ( fun_fu5207212825597307859nt_int * fun_int_fun_int_int ) > fun_Pr982841511892225912nt_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun1486234411616799902nt_int: ( fun_fu6909826129937320757nt_int * fun_int_fun_int_int ) > fun_fu76847565846999002nt_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun4545072666309863515_int_o: ( fun_fu4592178885959468770_int_o * fun_in1696471567231710611nt_int ) > fun_int_fun_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun7461304428735181397nt_int: ( fun_fu717549361983437348nt_int * fun_in1696471567231710611nt_int ) > fun_int_fun_int_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    aa_fun4152872853588681340nt_int: ( fun_fu3898734532474705027nt_int * fun_in1696471567231710611nt_int ) > fun_in1696471567231710611nt_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun8026974734089330650nt_int: ( fun_fu6741006549690681129nt_int * fun_in1696471567231710611nt_int ) > fun_Pr2478283820680522243nt_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_fun472946340410393319nt_int: ( fun_fu267553481079263030nt_int * fun_int_fun_nat_o ) > fun_int_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
    aa_fun4650797359919590027nt_nat: ( fun_fu2757636626010989786nt_nat * fun_int_fun_nat_o ) > fun_int_nat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun16395659681819580_nat_o: ( fun_fu927436805355353155_nat_o * fun_int_fun_nat_o ) > fun_int_fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_fun8696964665364114571at_int: ( fun_fu7304003943241017050at_int * fun_int_fun_nat_o ) > fun_nat_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun3651443648018535471at_nat: ( fun_fu570715051317967998at_nat * fun_int_fun_nat_o ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun5439836533389078972_int_o: ( fun_fu7493154764834273859_int_o * fun_int_fun_nat_o ) > fun_nat_fun_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun6062826860762256214nt_int: ( fun_fu1834763485345162221nt_int * fun_int_fun_nat_int ) > fun_fu267553481079263030nt_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    aa_fun3875905286069234426at_int: ( fun_fu7839431928035568017at_int * fun_int_fun_nat_int ) > fun_fu7304003943241017050at_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    aa_fun8303507722728657494nt_nat: ( fun_fu6398909621167755501nt_nat * fun_int_fun_nat_nat ) > fun_fu2757636626010989786nt_nat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun6116586148035635706at_nat: ( fun_fu3180206027003385489at_nat * fun_int_fun_nat_nat ) > fun_fu570715051317967998at_nat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun8623471109766342856_int_o: ( fun_fu3968995010428139015_int_o * fun_in2375804543815556931_nat_o ) > fun_Pr4738963024209204295_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
    aa_fun5797050646745590254nt_nat: ( fun_fu3043648374899647533nt_nat * fun_in8984369272479023031unit_o ) > fun_int_nat ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    aa_fun5573515973562188204unit_o: ( fun_fu6257966931869675755unit_o * fun_in8984369272479023031unit_o ) > fun_in8984369272479023031unit_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun7970191300016449456_int_o: ( fun_fu4586251570762536175_int_o * fun_in8984369272479023031unit_o ) > fun_Pr2157672562078508475_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_fun1461796484577889457nt_int: ( fun_fu3354657885645732096nt_int * fun_in7136481858371106712_nat_o ) > fun_int_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    aa_fun7992546631809917244_nat_o: ( fun_fu1780879341396661443_nat_o * fun_in7136481858371106712_nat_o ) > fun_in7136481858371106712_nat_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J,type,
    aa_fun5805668902077597195at_int: ( fun_fu4103912687242499418at_int * fun_in7136481858371106712_nat_o ) > fun_set_nat_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun8216954948888934460_int_o: ( fun_fu6598766258096663491_int_o * fun_in7136481858371106712_nat_o ) > fun_se7360890175450123928_int_o ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun1283830802835827434nt_int: ( fun_fu1817862469383530113nt_int * fun_in6555721293707029592at_int ) > fun_fu3354657885645732096nt_int ).

tff(sy_c_aa_001t__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J,type,
    aa_fun5576367408028688196at_int: ( fun_fu6187811966698790363at_int * fun_in6555721293707029592at_int ) > fun_fu4103912687242499418at_int ).

tff(sy_c_aa_001t__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_001_Eo,type,
    aa_fun_list_char_o_o: ( fun_fu2157805571543427467ar_o_o * fun_list_char_o ) > $o ).

tff(sy_c_aa_001t__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_001t__fun_It__fun_It__List__Olist_It__String__Ochar_J_M_Eo_J_M_Eo_J,type,
    aa_fun3501637381074999839ar_o_o: ( fun_fu5789129120561279854ar_o_o * fun_list_char_o ) > fun_fu2157805571543427467ar_o_o ).

tff(sy_c_aa_001t__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_001_Eo,type,
    aa_fun8069410965070236123char_o: ( fun_fu7823346190879031666char_o * fun_li3177621261580724543t_char ) > $o ).

tff(sy_c_aa_001t__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_001t__fun_It__fun_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_M_Eo_J,type,
    aa_fun8853676540986067637char_o: ( fun_fu74290633640755532char_o * fun_li3177621261580724543t_char ) > fun_fu7823346190879031666char_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_M_Eo_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_fun5937193402907943164_nat_o: ( fun_fu4001317426178549763_nat_o * fun_nat_o ) > fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_M_Eo_J_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J,type,
    aa_fun3969837373441324047nteger: ( fun_fu7898940055406642654nteger * fun_nat_o ) > fun_nat_Code_integer ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_M_Eo_J_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J,type,
    aa_fun8428287193952513114atural: ( fun_fu3129017019860109481atural * fun_nat_o ) > fun_nat_Code_natural ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_M_Eo_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_fun27009930614080094at_int: ( fun_fu7052161110045437101at_int * fun_nat_o ) > fun_nat_int ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_M_Eo_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun4204860950123276802at_nat: ( fun_fu318872218122388049at_nat * fun_nat_o ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_M_Eo_J_001t__fun_It__Nat__Onat_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    aa_fun6990899178391151900on_num: ( fun_fu2797749360310127211on_num * fun_nat_o ) > fun_nat_option_num ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_M_Eo_J_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
    aa_fun696632391729880074at_rat: ( fun_fu3162257917499486297at_rat * fun_nat_o ) > fun_nat_rat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_fun8469111845059130348nteger: ( fun_fu7333075202678876331nteger * fun_nat_Code_integer ) > fun_Co6226381647958292740nteger ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_fun3480067038916266681_nat_o: ( fun_fu5838931486547983824_nat_o * fun_nat_Code_integer ) > fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J,type,
    aa_fun589076786913273746nteger: ( fun_fu2499394402704830801nteger * fun_nat_Code_integer ) > fun_nat_Code_integer ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun2745573882733218024_nat_o: ( fun_fu8532066432991188095_nat_o * fun_nat_Code_integer ) > fun_fu5838931486547983824_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_fun3918145391673965625nteger: ( fun_fu6327154928193479288nteger * fun_nat_Code_integer ) > fun_fu2499394402704830801nteger ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_fun440078198722963374_nat_o: ( fun_fu2798942646354680517_nat_o * fun_nat_Code_natural ) > fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J,type,
    aa_fun56224542555504168atural: ( fun_fu1961541339144114791atural * fun_nat_Code_natural ) > fun_nat_Code_natural ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_001t__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun7611442044815192786_nat_o: ( fun_fu9087321537282282473_nat_o * fun_nat_Code_natural ) > fun_fu2798942646354680517_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_001t__fun_It__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J_J,type,
    aa_fun1651958757531339674atural: ( fun_fu8575605362271143641atural * fun_nat_Code_natural ) > fun_fu1961541339144114791atural ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Int__Oint_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_fun2180201864171524266_nat_o: ( fun_fu8207122332086543297_nat_o * fun_nat_int ) > fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Int__Oint_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_fun5565043122120104496at_int: ( fun_fu8464300013002972271at_int * fun_nat_int ) > fun_nat_int ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Int__Oint_J_001t__fun_It__fun_It__Nat__Onat_Mt__Int__Oint_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun8566290670789036234_nat_o: ( fun_fu7440396458814700513_nat_o * fun_nat_int ) > fun_fu8207122332086543297_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Int__Oint_J_001t__fun_It__fun_It__Nat__Onat_Mt__Int__Oint_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    aa_fun5140708168904641446at_int: ( fun_fu960360787657196261at_int * fun_nat_int ) > fun_fu8464300013002972271at_int ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_fun8673969331427483654_nat_o: ( fun_fu5477517762487726877_nat_o * fun_nat_nat ) > fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_fun1613598493704395220at_int: ( fun_fu4512855384587262995at_int * fun_nat_nat ) > fun_nat_int ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun5791449513213591928at_nat: ( fun_fu7002938529518989751at_nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    aa_fun6517009837249254290on_num: ( fun_fu5130958774822480081on_num * fun_nat_nat ) > fun_nat_option_num ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun6650658898577385413at_nat: ( fun_fu4413992040155564420at_nat * fun_nat_nat ) > fun_nat_fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun4561837934398617467at_nat: ( fun_fu2454420574033104442at_nat * fun_nat_nat ) > fun_se1179353343636554400at_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun6348600740168882050_nat_o: ( fun_fu6075382464776002713_nat_o * fun_nat_nat ) > fun_fu5477517762487726877_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun6854695580826382482at_nat: ( fun_fu5935951186182374353at_nat * fun_nat_nat ) > fun_fu7002938529518989751at_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Option__Ooption_It__Num__Onum_J_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_fun1675964276245824108_nat_o: ( fun_fu2897328509465618819_nat_o * fun_nat_option_num ) > fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Option__Ooption_It__Num__Onum_J_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun7387201323300553874at_nat: ( fun_fu1502755112258691281at_nat * fun_nat_option_num ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Option__Ooption_It__Num__Onum_J_J_001t__fun_It__Nat__Onat_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    aa_fun7716400287237121964on_num: ( fun_fu1574286537503582443on_num * fun_nat_option_num ) > fun_nat_option_num ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_fun1118977351124560894_nat_o: ( fun_fu7145897819039579925_nat_o * fun_nat_rat ) > fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
    aa_fun765758790203053448at_rat: ( fun_fu8328862064278946247at_rat * fun_nat_rat ) > fun_nat_rat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__Rat__Orat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    aa_fun5559679822142647773at_rat: ( fun_fu4710970720072501148at_rat * fun_nat_rat ) > fun_rat_fun_nat_rat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun8314661095172591474_nat_o: ( fun_fu1806770557662746761_nat_o * fun_nat_rat ) > fun_fu7145897819039579925_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    aa_fun3638354034882855594at_rat: ( fun_fu6579618355643354601at_rat * fun_nat_rat ) > fun_fu8328862064278946247at_rat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_fun5488476222267017447nt_int: ( fun_fu5283083362935887158nt_int * fun_nat_fun_int_o ) > fun_int_int ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
    aa_fun442955204921438347nt_nat: ( fun_fu7773166507867613914nt_nat * fun_nat_fun_int_o ) > fun_int_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun7393466555336238524_nat_o: ( fun_fu8304507701009772099_nat_o * fun_nat_fun_int_o ) > fun_int_fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_fun4489122510365962891at_int: ( fun_fu3096161788242865370at_int * fun_nat_fun_int_o ) > fun_nat_int ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun8666973529875159599at_nat: ( fun_fu5586244933174592126at_nat * fun_nat_fun_int_o ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun3593535392188722108_int_o: ( fun_fu5646853623633916995_int_o * fun_nat_fun_int_o ) > fun_nat_fun_int_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun3345280138976288598nt_int: ( fun_fu7802804735324696557nt_int * fun_nat_fun_int_int ) > fun_fu5283083362935887158nt_int ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    aa_fun1158358564283266810at_int: ( fun_fu4584101141160326545at_int * fun_nat_fun_int_int ) > fun_fu3096161788242865370at_int ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    aa_fun5585961000942689878nt_nat: ( fun_fu3143578834292514029nt_nat * fun_nat_fun_int_nat ) > fun_fu7773166507867613914nt_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun3399039426249668090at_nat: ( fun_fu9148247276982919825at_nat * fun_nat_fun_int_nat ) > fun_fu5586244933174592126at_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_fun4679828425598226919at_int: ( fun_fu3286867703475129398at_int * fun_nat_fun_nat_o ) > fun_nat_int ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun8857679445107423627at_nat: ( fun_fu5776950848406856154at_nat * fun_nat_fun_nat_o ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun6699960423209655484_nat_o: ( fun_fu6436183194761443139_nat_o * fun_nat_fun_nat_o ) > fun_nat_fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    aa_fun5734951013428723122at_int: ( fun_fu8732410533422943817at_int * fun_nat_fun_nat_int ) > fun_fu3286867703475129398at_int ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun7975631875395124402at_nat: ( fun_fu4073184632390761289at_nat * fun_nat_fun_nat_nat ) > fun_fu5776950848406856154at_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_J,type,
    aa_fun824319262034445606_nat_o: ( fun_fu6384561554008368101_nat_o * fun_na5914473681949881652_nat_o ) > fun_na5914473681949881652_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    aa_fun5912894404062470332_nat_o: ( fun_fu2847332191311216467_nat_o * fun_na5914473681949881652_nat_o ) > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    aa_fun6875833325056119164at_nat: ( fun_fu2716251949043459987at_nat * fun_na1668100129766719005at_nat ) > fun_Pr2477239776880934807at_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun5389978208584994248_nat_o: ( fun_fu7641778686208158983_nat_o * fun_na3970208099681795815_nat_o ) > fun_Pr6984107991879650211_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun8994848686300240182at_nat: ( fun_fu5053878551662330229at_nat * fun_na5971062356810558811unit_o ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    aa_fun8483372159859046644unit_o: ( fun_fu2062424135460034099unit_o * fun_na5971062356810558811unit_o ) > fun_na5971062356810558811unit_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun6915127332797442224_nat_o: ( fun_fu1214092143650121711_nat_o * fun_na5971062356810558811unit_o ) > fun_Pr4402817529748954391_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun7187098123162309461at_nat: ( fun_fu6204623786860714916at_nat * fun_na8563630686737515764_nat_o ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    aa_fun1792963693083279932_nat_o: ( fun_fu3568225526390654915_nat_o * fun_na8563630686737515764_nat_o ) > fun_na8563630686737515764_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    aa_fun255155394271802123at_nat: ( fun_fu6089003341714010202at_nat * fun_na8563630686737515764_nat_o ) > fun_set_nat_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun2835368149466334012_nat_o: ( fun_fu8123456035635431619_nat_o * fun_na8563630686737515764_nat_o ) > fun_se382663106265794036_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun585184524491904070at_nat: ( fun_fu9114370601171615197at_nat * fun_na106483476154090400at_nat ) > fun_fu6204623786860714916at_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    aa_fun6284103964554420476at_nat: ( fun_fu8651063156380458899at_nat * fun_na106483476154090400at_nat ) > fun_fu6089003341714010202at_nat ).

tff(sy_c_aa_001t__fun_It__Num__Onum_Mt__Int__Oint_J_001t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
    aa_fun329774391934724804um_int: ( fun_fu7117095973476296707um_int * fun_num_int ) > fun_num_int ).

tff(sy_c_aa_001t__fun_It__Num__Onum_Mt__Int__Oint_J_001t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
    aa_fun5215869348392759218um_num: ( fun_fu5322398707847370225um_num * fun_num_int ) > fun_num_num ).

tff(sy_c_aa_001t__fun_It__Num__Onum_Mt__Int__Oint_J_001t__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    aa_fun3858197967482829954on_num: ( fun_fu7964671872712412353on_num * fun_num_int ) > fun_num_option_num ).

tff(sy_c_aa_001t__fun_It__Num__Onum_Mt__Num__Onum_J_001t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
    aa_fun3162900327506486194um_int: ( fun_fu726849872193282289um_int * fun_num_num ) > fun_num_int ).

tff(sy_c_aa_001t__fun_It__Num__Onum_Mt__Num__Onum_J_001t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
    aa_fun8048995283964520608um_num: ( fun_fu8155524643419131615um_num * fun_num_num ) > fun_num_num ).

tff(sy_c_aa_001t__fun_It__Num__Onum_Mt__Num__Onum_J_001t__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    aa_fun612943910911948912on_num: ( fun_fu4719417816141531311on_num * fun_num_num ) > fun_num_option_num ).

tff(sy_c_aa_001t__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J_001t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
    aa_fun2256179333612926594um_int: ( fun_fu1947565687807238081um_int * fun_num_option_num ) > fun_num_int ).

tff(sy_c_aa_001t__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J_001t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
    aa_fun7142274290070961008um_num: ( fun_fu152868422178311599um_num * fun_num_option_num ) > fun_num_num ).

tff(sy_c_aa_001t__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J_001t__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    aa_fun3636977833914307392on_num: ( fun_fu2987389051837124479on_num * fun_num_option_num ) > fun_num_option_num ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001_Eo,type,
    aa_fun7993445754732355552term_o: ( fun_fu4736795817735658359term_o * fun_Pr5098669352780829690e_term ) > $o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_fun8425650785172306510nteger: ( fun_fu8902681562042749453nteger * fun_Pr5098669352780829690e_term ) > produc2453742332425399446nteger ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_fun1536964767670826920e_term: ( fun_fu1456393871774969959e_term * fun_Pr5098669352780829690e_term ) > fun_Co3460408610320954864e_term ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J,type,
    aa_fun4505422719058378424eger_o: ( fun_fu7695273699059480439eger_o * fun_Pr5098669352780829690e_term ) > fun_Pr7211993375558241536eger_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    aa_fun8303988899834042124nteger: ( fun_fu8946524575271555235nteger * fun_Pr5098669352780829690e_term ) > fun_Pr2985126522952997572nteger ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_fun1248587000295462322e_term: ( fun_fu2505754153367914865e_term * fun_Pr5098669352780829690e_term ) > fun_Pr5098669352780829690e_term ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001_Eo,type,
    aa_fun1221993630413146411term_o: ( fun_fu7188715730271225026term_o * fun_Pr285225112257423215e_term ) > $o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    aa_fun7725423394461249828atural: ( fun_fu7873099233167337571atural * fun_Pr285225112257423215e_term ) > produc9140211490168418807atural ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_M_Eo_J,type,
    aa_fun9089857049899810327ural_o: ( fun_fu3694549533497745878ural_o * fun_Pr285225112257423215e_term ) > fun_Pr8359263388657976682ural_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J,type,
    aa_fun3413103502972484686atural: ( fun_fu2502439290952149605atural * fun_Pr285225112257423215e_term ) > fun_Pr1972435576755982779atural ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_fun6933112156808570396e_term: ( fun_fu7248590935147425627e_term * fun_Pr285225112257423215e_term ) > fun_Pr285225112257423215e_term ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_Eo,type,
    aa_fun2650603632653508759_int_o: ( fun_fu7503566767892366382_int_o * fun_Pr2478283820680522243nt_int ) > $o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
    aa_fun7041730090766645614at_rat: ( fun_fu1098312133097393581at_rat * fun_Pr2478283820680522243nt_int ) > fun_rat_rat ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
    aa_fun3734617511155415725_int_o: ( fun_fu5323616038969315652_int_o * fun_Pr2478283820680522243nt_int ) > fun_fu7503566767892366382_int_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Rat__Orat_Mt__Rat__Orat_J_M_Eo_J,type,
    aa_fun382765352565706371_rat_o: ( fun_fu4306744791715064858_rat_o * fun_Pr2478283820680522243nt_int ) > fun_fun_rat_rat_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001_Eo,type,
    aa_fun7650857933016935215term_o: ( fun_fu6378075912484669126term_o * fun_Pr7763680713840462443e_term ) > $o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun9040010020517048364nt_int: ( fun_fu5813366174458671467nt_int * fun_Pr7763680713840462443e_term ) > produc3043004577601930755nt_int ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_fun3544567364070095336e_term: ( fun_fu4835546211937200935e_term * fun_Pr7763680713840462443e_term ) > fun_in5730379987311621695e_term ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    aa_fun3538162476766722827_int_o: ( fun_fu6865642474282033866_int_o * fun_Pr7763680713840462443e_term ) > fun_Pr3238374608701176162_int_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    aa_fun3265472288063244902nt_int: ( fun_fu9160953773549289085nt_int * fun_Pr7763680713840462443e_term ) > fun_Pr4781553364444837327nt_int ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    aa_fun49903357033182740e_term: ( fun_fu836511555612062035e_term * fun_Pr7763680713840462443e_term ) > fun_Pr7763680713840462443e_term ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_Eo,type,
    aa_fun8408151826566693824at_o_o: ( fun_fu6891447142836604759at_o_o * fun_Pr4796943543331071514_nat_o ) > $o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    aa_fun359341089595923442_nat_o: ( fun_fu3275390434952296369_nat_o * fun_Pr4796943543331071514_nat_o ) > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
    aa_fun3538214411680374423_nat_o: ( fun_fu8391177546919232046_nat_o * fun_Pr2118491707810114563at_nat ) > $o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_fun8399641135911395862nt_int: ( fun_fu3064074204415222357nt_int * fun_Pr2118491707810114563at_nat ) > fun_int_int ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    aa_fun856177901615811547_int_o: ( fun_fu4638078822479986034_int_o * fun_Pr2118491707810114563at_nat ) > fun_fun_int_int_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    aa_fun6920661532540581549_nat_o: ( fun_fu3373238862511017284_nat_o * fun_Pr2118491707810114563at_nat ) > fun_fu8391177546919232046_nat_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    aa_fun1144309370076579528_nat_o: ( fun_fu2341038929883010567_nat_o * fun_Pr4738963024209204295_int_o ) > fun_in2375804543815556931_nat_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    aa_fun2489310642977492936_nat_o: ( fun_fu3617688454659378439_nat_o * fun_Pr6984107991879650211_nat_o ) > fun_na3970208099681795815_nat_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_Eo_J_001t__fun_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    aa_fun2566793329686429606_nat_o: ( fun_fu1322491050725084261_nat_o * fun_Pr3059359313727237556_nat_o ) > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_M_Eo_J_001t__fun_It__Product____Type__Ounit_M_Eo_J,type,
    aa_fun644178081534259290unit_o: ( fun_fu182299653934541849unit_o * fun_Product_unit_o ) > fun_Product_unit_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_fun3412420425980457528nteger: ( fun_fu6709995119303243583nteger * fun_Pr1820225556744648189nteger ) > fun_Co6473412210723208761nteger ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Product____Type__Ounit_M_Eo_J,type,
    aa_fun7797430150634904527unit_o: ( fun_fu6229700951322188702unit_o * fun_Pr1820225556744648189nteger ) > fun_Product_unit_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__Int__Oint_J_001t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Int__Oint_J_J,type,
    aa_fun4982840383720920169it_int: ( fun_fu8548799023038560624it_int * fun_Product_unit_int ) > fun_in5522822332083694073it_int ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__Int__Oint_J_001t__fun_It__Product____Type__Ounit_M_Eo_J,type,
    aa_fun693351410520992030unit_o: ( fun_fu7766707205226367213unit_o * fun_Product_unit_int ) > fun_Product_unit_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
    aa_fun277559562889681605it_nat: ( fun_fu511840035978624972it_nat * fun_Product_unit_nat ) > fun_na381414567679918841it_nat ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J_001t__fun_It__Product____Type__Ounit_M_Eo_J,type,
    aa_fun9021389585855579330unit_o: ( fun_fu6871373343706178705unit_o * fun_Product_unit_nat ) > fun_Product_unit_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__Rat__Orat_J_001t__fun_It__Rat__Orat_Mt__fun_It__Product____Type__Ounit_Mt__Rat__Orat_J_J,type,
    aa_fun5217320890768598205it_rat: ( fun_fu1171974497074579908it_rat * fun_Product_unit_rat ) > fun_ra5613769993463817465it_rat ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    aa_fun249790601762366384unit_o: ( fun_fu934241560069853935unit_o * fun_Pr2157672562078508475_int_o ) > fun_in8984369272479023031unit_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
    aa_fun4006079477318474295it_nat: ( fun_fu8835698356539894734it_nat * fun_Pr2157672562078508475_int_o ) > fun_Product_unit_nat ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun2646465928216627636_int_o: ( fun_fu8485898235817490163_int_o * fun_Pr2157672562078508475_int_o ) > fun_Pr2157672562078508475_int_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    aa_fun3452303748553993392unit_o: ( fun_fu6254727761009756655unit_o * fun_Pr4402817529748954391_nat_o ) > fun_na5971062356810558811unit_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
    aa_fun1580523165164119515it_nat: ( fun_fu6410142044385539954it_nat * fun_Pr4402817529748954391_nat_o ) > fun_Product_unit_nat ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun1884058921492388972_nat_o: ( fun_fu5406395769199844267_nat_o * fun_Pr4402817529748954391_nat_o ) > fun_Pr4402817529748954391_nat_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_001t__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
    aa_fun2167693721382921342it_nat: ( fun_fu6371730626855558661it_nat * fun_Pr1520749766685617582unit_o ) > fun_Product_unit_nat ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    aa_fun7194487344118582332unit_o: ( fun_fu7877188687433499331unit_o * fun_Pr1520749766685617582unit_o ) > fun_Pr1520749766685617582unit_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_001t__fun_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
    aa_fun6562703090490685585it_nat: ( fun_fu3293275151183473192it_nat * fun_Pr1646868699942875873_nat_o ) > fun_Product_unit_nat ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    aa_fun4072893971397127680_nat_o: ( fun_fu6154516897781741631_nat_o * fun_Pr1646868699942875873_nat_o ) > fun_Pr1646868699942875873_nat_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Ounit_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Product____Type__Ounit_M_Eo_J_J,type,
    aa_fun6872337739698859568unit_o: ( fun_fu27622379790049391unit_o * fun_Pr1646868699942875873_nat_o ) > fun_se4446312468244607761unit_o ).

tff(sy_c_aa_001t__fun_It__Rat__Orat_Mt__Rat__Orat_J_001_Eo,type,
    aa_fun_rat_rat_o: ( fun_fun_rat_rat_o * fun_rat_rat ) > $o ).

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

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Code____Numeral__Ointeger_J,type,
    aa_fun4587287018503165694nteger: ( fun_fu5510336631606277565nteger * fun_se7586849273513198688nteger ) > fun_se7586849273513198688nteger ).

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_M_Eo_J,type,
    aa_fun1141455960422277694_nat_o: ( fun_fu1047064357265694805_nat_o * fun_set_nat_int ) > fun_set_nat_o ).

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,
    aa_fun3600621996379952540at_int: ( fun_fu3492966305048685787at_int * fun_set_nat_int ) > fun_set_nat_int ).

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__Rat__Orat_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Rat__Orat_J,type,
    aa_fun8478617152544252148at_rat: ( fun_fu3811435807551234611at_rat * fun_set_nat_rat ) > fun_set_nat_rat ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_fun7007818992012977073nt_int: ( fun_fu8900680393080819712nt_int * fun_se7360890175450123928_int_o ) > fun_int_int ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    aa_fun9086678093774146108_nat_o: ( fun_fu2875010803360890307_nat_o * fun_se7360890175450123928_int_o ) > fun_in7136481858371106712_nat_o ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J,type,
    aa_fun5327250345464651019at_int: ( fun_fu3625494130629553242at_int * fun_se7360890175450123928_int_o ) > fun_set_nat_int ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun87714373998387516_int_o: ( fun_fu7692897720060892355_int_o * fun_se7360890175450123928_int_o ) > fun_se7360890175450123928_int_o ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun7130641361752856810nt_int: ( fun_fu3776778612278285441nt_int * fun_se876191773397849432nt_int ) > fun_fu8900680393080819712nt_int ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J,type,
    aa_fun3157859007690329412at_int: ( fun_fu2189424204742085595at_int * fun_se876191773397849432nt_int ) > fun_fu3625494130629553242at_int ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_fun6199171195344186545at_int: ( fun_fu6904464733620061952at_int * fun_se382663106265794036_nat_o ) > fun_nat_int ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun1153650177998607445at_nat: ( fun_fu171175841697012900at_nat * fun_se382663106265794036_nat_o ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    aa_fun9105140013458882876_nat_o: ( fun_fu1657029809911482051_nat_o * fun_se382663106265794036_nat_o ) > fun_na8563630686737515764_nat_o ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J,type,
    aa_fun2061028301734939751at_int: ( fun_fu359272086899841974at_int * fun_se382663106265794036_nat_o ) > fun_set_nat_int ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    aa_fun6238879321244136459at_nat: ( fun_fu2849355231831568730at_nat * fun_se382663106265794036_nat_o ) > fun_set_nat_nat ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun924172432987161148_nat_o: ( fun_fu6212260319156258755_nat_o * fun_se382663106265794036_nat_o ) > fun_se382663106265794036_nat_o ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    aa_fun2727794920753907526at_int: ( fun_fu7137239131779924701at_int * fun_se7912642235559603452at_int ) > fun_fu6904464733620061952at_int ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Int__Oint_J_J,type,
    aa_fun8609273137425597436at_int: ( fun_fu2855959522417824915at_int * fun_se7912642235559603452at_int ) > fun_fu359272086899841974at_int ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun5868473082678171206at_nat: ( fun_fu3378010530705604573at_nat * fun_se1179353343636554400at_nat ) > fun_fu171175841697012900at_nat ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    aa_fun1036415571424436988at_nat: ( fun_fu6829939267127632275at_nat * fun_se1179353343636554400at_nat ) > fun_fu2849355231831568730at_nat ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
    aa_fun6889641154785167298_int_o: ( fun_fu5088349107139067777_int_o * fun_se7249947736802449410_int_o ) > fun_se7249947736802449410_int_o ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Nat__Onat_J_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Nat__Onat_J,type,
    aa_fun209864160807355708nt_nat: ( fun_fu8621421401018972867nt_nat * fun_se4101287982975402108nt_nat ) > fun_se4101287982975402108nt_nat ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    aa_fun3315923329236548914_nat_o: ( fun_fu2380148858382964977_nat_o * fun_se4782206882984618170_nat_o ) > fun_se4782206882984618170_nat_o ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Code____Numeral__Ointeger_J,type,
    aa_fun5683560775029648636nteger: ( fun_fu5303860655635331075nteger * fun_se2775971902884351953nteger ) > fun_se2775971902884351953nteger ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
    aa_fun4978281634197224252at_nat: ( fun_fu6666991544514956483at_nat * fun_se4097146774670233028at_nat ) > fun_se4097146774670233028at_nat ).

tff(sy_c_aa_001t__fun_Itf__a_Mt__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_J_001t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    aa_fun372765568861188684it_nat: ( fun_fu6488529940522614227it_nat * fun_a_2990225006182978344t_unit ) > fun_he5234484606919320760it_nat ).

tff(sy_c_aa_001t__fun_Itf__a_Mt__Heap____Time____Monad__OHeap_Itf__a_J_J_001t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    aa_fun6119220767042692876it_nat: ( fun_fu4855298819597708819it_nat * fun_a_6093291656547882009Heap_a ) > fun_he1921015881043195817it_nat ).

tff(sy_c_aa_001tf__a_001_Eo,type,
    aa_a_o: ( fun_a_o * a ) > $o ).

tff(sy_c_aa_001tf__a_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J,type,
    aa_a_H4862568901524997081t_unit: ( fun_a_2990225006182978344t_unit * a ) > heap_T5738788834812785303t_unit ).

tff(sy_c_aa_001tf__a_001t__Heap____Time____Monad__OHeap_Itf__a_J,type,
    aa_a_H8829438773920479562Heap_a: ( fun_a_6093291656547882009Heap_a * a ) > heap_Time_Heap_a ).

tff(sy_c_aa_001tf__a_001t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__fun_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J,type,
    aa_a_f112692477493148739it_nat: ( fun_a_6373226321756256586it_nat * a ) > fun_he8369595706569872389it_nat ).

tff(sy_c_aa_001tf__a_001t__fun_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__fun_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_J,type,
    aa_a_f1761598956578211508it_nat: ( fun_a_3171520138701809467it_nat * a ) > fun_he8849654208132081782it_nat ).

tff(sy_c_aa_001tf__a_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    aa_a_f2724909083865345251it_nat: ( fun_a_4942204877778174258it_nat * a ) > fun_Pr5651933991979716257it_nat ).

tff(sy_c_aa_001tf__a_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    aa_a_f2042851359257671380it_nat: ( fun_a_3426355041879316131it_nat * a ) > fun_Pr7273243183861550738it_nat ).

tff(sy_c_fequal_001_Eo,type,
    fequal_o: fun_o_fun_o_o ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_o ).

tff(sy_c_fequal_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    fequal5224421480335545304nteger: fun_li279074556804398960eger_o ).

tff(sy_c_fequal_001t__List__Olist_It__String__Ochar_J,type,
    fequal_list_char: fun_li5260340055336554888char_o ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_o ).

tff(sy_c_fequal_001t__Num__Onum,type,
    fequal_num: fun_num_fun_num_o ).

tff(sy_c_fequal_001t__String__Ochar,type,
    fequal_char: fun_char_fun_char_o ).

tff(sy_c_member_001t__Code____Numeral__Ointeger,type,
    member_Code_integer: ( code_integer * set_Code_integer ) > $o ).

tff(sy_c_member_001t__Code____Numeral__Onatural,type,
    member_Code_natural: ( code_natural * set_Code_natural ) > $o ).

tff(sy_c_member_001t__Int__Oint,type,
    member_int: ( int * set_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__String__Ochar_J,type,
    member_list_char: ( list_char * set_list_char ) > $o ).

tff(sy_c_member_001t__Nat__Onat,type,
    member_nat: fun_na8563630686737515764_nat_o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: product_prod_int_int > fun_se7249947736802449410_int_o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > fun_se4782206882984618170_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__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    member2572552093476627150et_int: ( produc2115011035271226405et_int * set_Pr2522554150109002629et_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: ( produc7819656566062154093et_nat * set_Pr5488025237498180813et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    member444158400953824016od_o_o: ( produc2934264451710624999od_o_o * set_Pr1932065953672099015od_o_o ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member4095101504841534314at_nat: ( produc732395585841259969at_nat * set_Pr457366540195662369at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: ( produc3843707927480180839at_nat * set_Pr4329608150637261639at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    member6901806533512420791nteger: ( produc7509114972854644512nteger * set_Pr1200682188532975830nteger ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member5890090881338784710nt_int: ( produc4788131091073775919nt_int * set_Pr4166680992137330533nt_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    member8978266922497061677nteger: ( produc2453742332425399446nteger * set_Pr3486312109254157132nteger ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    member6441364043385305230atural: ( produc9140211490168418807atural * set_Pr620054291978045485atural ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member753276645541486746nt_int: ( produc3043004577601930755nt_int * set_Pr6700895561065273913nt_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Ounit,type,
    member_Product_unit: ( product_unit * set_Product_unit ) > $o ).

tff(sy_c_member_001t__Rat__Orat,type,
    member_rat: ( rat * set_rat ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: ( set_nat * set_set_nat ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member2340774599025711042nt_int: ( set_Pr958786334691620121nt_int * set_se6260736226359567993nt_int ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: ( set_Pr1261947904930325089at_nat * set_se7855581050983116737at_nat ) > $o ).

tff(sy_c_member_001t__String__Ochar,type,
    member_char: ( char * set_char ) > $o ).

tff(sy_v_P,type,
    p: fun_a_o ).

tff(sy_v_h,type,
    h: heap_e7401611519738050253t_unit ).

tff(sy_v_x,type,
    x: a ).

% Relevant facts (8939)
tff(fact_0_success__ifI,axiom,
    ! [C: $o,T: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,E: heap_Time_Heap_a] :
      ( ( (C)
       => aa_hea7994211679605476167unit_o(heap_Time_success_a(T),Ha) )
     => ( ( ~ (C)
         => aa_hea7994211679605476167unit_o(heap_Time_success_a(E),Ha) )
       => aa_hea7994211679605476167unit_o(heap_Time_success_a($ite((C),T,E)),Ha) ) ) ).

% success_ifI
tff(fact_1_success__tapI,axiom,
    ! [F: fun_he6342020308413958580unit_a,Ha: heap_e7401611519738050253t_unit] : aa_hea7994211679605476167unit_o(heap_Time_success_a(heap_Time_tap_a(F)),Ha) ).

% success_tapI
tff(fact_2_success__guardI,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,F: fun_he8889679731534893401it_nat] :
      ( aa_hea7994211679605476167unit_o(Pa,Ha)
     => aa_hea7994211679605476167unit_o(heap_Time_success_a(heap_Time_guard_a(Pa,F)),Ha) ) ).

% success_guardI
tff(fact_3_success__ureturnI,axiom,
    ! [Xa: a,Ha: heap_e7401611519738050253t_unit] : aa_hea7994211679605476167unit_o(heap_Time_success_a(aa_a_H8829438773920479562Heap_a(heap_Time_ureturn_a,Xa)),Ha) ).

% success_ureturnI
tff(fact_4_success__heapI,axiom,
    ! [F: fun_he8889679731534893401it_nat,Ha: heap_e7401611519738050253t_unit] : aa_hea7994211679605476167unit_o(heap_Time_success_a(heap_Time_heap_a(F)),Ha) ).

% success_heapI
tff(fact_5_success__returnI,axiom,
    ! [Xa: a,Ha: heap_e7401611519738050253t_unit] : aa_hea7994211679605476167unit_o(heap_Time_success_a(aa_a_H8829438773920479562Heap_a(heap_Time_return_a,Xa)),Ha) ).

% success_returnI
tff(fact_6_effect__success,axiom,
    ! [C: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: a,N: nat] :
      ( aa_nat_o(heap_Time_effect_a(C,Ha,H,R),N)
     => aa_hea7994211679605476167unit_o(heap_Time_success_a(C),Ha) ) ).

% effect_success
tff(fact_7_success__effectE,axiom,
    ! [C: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(heap_Time_success_a(C),Ha)
     => ~ ! [R2: a,H2: heap_e7401611519738050253t_unit,N2: nat] : ~ aa_nat_o(heap_Time_effect_a(C,Ha,H2,R2),N2) ) ).

% success_effectE
tff(fact_8_execute__assert_I2_J,axiom,
    ! [Pa: fun_a_o,Xa: a,Ha: heap_e7401611519738050253t_unit] :
      ( ~ aa_a_o(Pa,Xa)
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(heap_Time_assert_a(Pa),Xa)),Ha) = none_P2651198173097904984it_nat ) ) ).

% execute_assert(2)
tff(fact_9_success__bind__effectI,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,Xa: a,N: nat,G: fun_a_6093291656547882009Heap_a] :
      ( aa_nat_o(heap_Time_effect_a(F,Ha,H,Xa),N)
     => ( aa_hea7994211679605476167unit_o(heap_Time_success_a(aa_a_H8829438773920479562Heap_a(G,Xa)),H)
       => aa_hea7994211679605476167unit_o(heap_Time_success_a(heap_Time_bind_a_a(F,G)),Ha) ) ) ).

% success_bind_effectI
tff(fact_10_successI,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) != none_P2651198173097904984it_nat )
     => aa_hea7994211679605476167unit_o(heap_Time_success_a(F),Ha) ) ).

% successI
tff(fact_11_success__def,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(heap_Time_success_a(F),Ha)
    <=> ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) != none_P2651198173097904984it_nat ) ) ).

% success_def
tff(fact_12_execute__bind_I2_J,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,G: fun_a_6093291656547882009Heap_a] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) = none_P2651198173097904984it_nat )
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_a_a(F,G)),Ha) = none_P2651198173097904984it_nat ) ) ).

% execute_bind(2)
tff(fact_13_execute__guard_I1_J,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,F: fun_he8889679731534893401it_nat] :
      ( ~ aa_hea7994211679605476167unit_o(Pa,Ha)
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_guard_a(Pa,F)),Ha) = none_P2651198173097904984it_nat ) ) ).

% execute_guard(1)
tff(fact_14_effect__deterministic_I3_J,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,A: a,N: nat,H3: heap_e7401611519738050253t_unit,B: a,N3: nat] :
      ( aa_nat_o(heap_Time_effect_a(F,Ha,H,A),N)
     => ( aa_nat_o(heap_Time_effect_a(F,Ha,H3,B),N3)
       => ( N = N3 ) ) ) ).

% effect_deterministic(3)
tff(fact_15_effect__deterministic_I2_J,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,A: a,N: nat,H3: heap_e7401611519738050253t_unit,B: a,N3: nat] :
      ( aa_nat_o(heap_Time_effect_a(F,Ha,H,A),N)
     => ( aa_nat_o(heap_Time_effect_a(F,Ha,H3,B),N3)
       => ( H = H3 ) ) ) ).

% effect_deterministic(2)
tff(fact_16_effect__deterministic_I1_J,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,A: a,N: nat,H3: heap_e7401611519738050253t_unit,B: a,N3: nat] :
      ( aa_nat_o(heap_Time_effect_a(F,Ha,H,A),N)
     => ( aa_nat_o(heap_Time_effect_a(F,Ha,H3,B),N3)
       => ( A = B ) ) ) ).

% effect_deterministic(1)
tff(fact_17_Heap__eqI,axiom,
    ! [F: heap_Time_Heap_a,G: heap_Time_Heap_a] :
      ( ! [H4: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),H4) = aa_hea4724376117875161946it_nat(heap_Time_execute_a(G),H4)
     => ( F = G ) ) ).

% Heap_eqI
tff(fact_18_effect__ifE,axiom,
    ! [C: $o,T: heap_Time_Heap_a,E: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: a,N: nat] :
      ( aa_nat_o(heap_Time_effect_a($ite((C),T,E),Ha,H,R),N)
     => ( ( (C)
         => ~ aa_nat_o(heap_Time_effect_a(T,Ha,H,R),N) )
       => ~ ( ~ (C)
           => ~ aa_nat_o(heap_Time_effect_a(E,Ha,H,R),N) ) ) ) ).

% effect_ifE
tff(fact_19_effect__ifI,axiom,
    ! [C: $o,T: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: a,N: nat,E: heap_Time_Heap_a] :
      ( ( (C)
       => aa_nat_o(heap_Time_effect_a(T,Ha,H,R),N) )
     => ( ( ~ (C)
         => aa_nat_o(heap_Time_effect_a(E,Ha,H,R),N) )
       => aa_nat_o(heap_Time_effect_a($ite((C),T,E),Ha,H,R),N) ) ) ).

% effect_ifI
tff(fact_20_bind__ureturn,axiom,
    ! [F: heap_Time_Heap_a] : heap_Time_bind_a_a(F,heap_Time_ureturn_a) = F ).

% bind_ureturn
tff(fact_21_execute__raise,axiom,
    ! [S: list_char,X: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_raise_a(S)),X) = none_P2651198173097904984it_nat ).

% execute_raise
tff(fact_22_effect__tapE,axiom,
    ! [F: fun_he6342020308413958580unit_a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: a,N: nat] :
      ( aa_nat_o(heap_Time_effect_a(heap_Time_tap_a(F),Ha,H,R),N)
     => ~ ( ( H = Ha )
         => ( ( R = aa_hea5100639427662539565unit_a(F,Ha) )
           => ( N != one_one_nat ) ) ) ) ).

% effect_tapE
tff(fact_23_effect__tapI,axiom,
    ! [H: heap_e7401611519738050253t_unit,Ha: heap_e7401611519738050253t_unit,R: a,F: fun_he6342020308413958580unit_a] :
      ( ( H = Ha )
     => ( ( R = aa_hea5100639427662539565unit_a(F,Ha) )
       => aa_nat_o(heap_Time_effect_a(heap_Time_tap_a(F),Ha,H,R),one_one_nat) ) ) ).

% effect_tapI
tff(fact_24_effect__raiseE,axiom,
    ! [Xa: list_char,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: a,N: nat] : ~ aa_nat_o(heap_Time_effect_a(heap_Time_raise_a(Xa),Ha,H,R),N) ).

% effect_raiseE
tff(fact_25_effect__ureturnE,axiom,
    ! [Xa: a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: a,N: nat] :
      ( aa_nat_o(heap_Time_effect_a(aa_a_H8829438773920479562Heap_a(heap_Time_ureturn_a,Xa),Ha,H,R),N)
     => ~ ( ( R = Xa )
         => ( ( H = Ha )
           => ( N != zero_zero_nat ) ) ) ) ).

% effect_ureturnE
tff(fact_26_effect__ureturnI,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,Xa: a] :
      ( ( Ha = H )
     => aa_nat_o(heap_Time_effect_a(aa_a_H8829438773920479562Heap_a(heap_Time_ureturn_a,Xa),Ha,H,Xa),zero_zero_nat) ) ).

% effect_ureturnI
tff(fact_27_effect__returnE,axiom,
    ! [Xa: a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: a,N: nat] :
      ( aa_nat_o(heap_Time_effect_a(aa_a_H8829438773920479562Heap_a(heap_Time_return_a,Xa),Ha,H,R),N)
     => ~ ( ( R = Xa )
         => ( ( H = Ha )
           => ( N != one_one_nat ) ) ) ) ).

% effect_returnE
tff(fact_28_effect__returnI,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,Xa: a] :
      ( ( Ha = H )
     => aa_nat_o(heap_Time_effect_a(aa_a_H8829438773920479562Heap_a(heap_Time_return_a,Xa),Ha,H,Xa),one_one_nat) ) ).

% effect_returnI
tff(fact_29_execute__guard_I2_J,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,F: fun_he4944555133437423336it_nat] :
      ( aa_hea7994211679605476167unit_o(Pa,Ha)
     => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(heap_T8440541562793052209t_unit(Pa,F)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,aa_hea256619718444519449it_nat(F,Ha)) ) ) ).

% execute_guard(2)
tff(fact_30_execute__guard_I2_J,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,F: fun_he8889679731534893401it_nat] :
      ( aa_hea7994211679605476167unit_o(Pa,Ha)
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_guard_a(Pa,F)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_hea5645563107516622602it_nat(F,Ha)) ) ) ).

% execute_guard(2)
tff(fact_31_effect__bindE,axiom,
    ! [F: heap_Time_Heap_a,G: fun_a_6093291656547882009Heap_a,Ha: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R3: a,N: nat] :
      ( aa_nat_o(heap_Time_effect_a(heap_Time_bind_a_a(F,G),Ha,H3,R3),N)
     => ~ ! [H2: heap_e7401611519738050253t_unit,R2: a,N1: nat] :
            ( aa_nat_o(heap_Time_effect_a(F,Ha,H2,R2),N1)
           => ! [N22: nat] :
                ( aa_nat_o(heap_Time_effect_a(aa_a_H8829438773920479562Heap_a(G,R2),H2,H3,R3),N22)
               => ( N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_32_option_Oinject,axiom,
    ! [X2: produc8664842809031399944it_nat,Y2: produc8664842809031399944it_nat] :
      ( ( aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_33_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( aa_num_option_num(some_num,X2) = aa_num_option_num(some_num,Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_34_not__Some__eq,axiom,
    ! [Xa: option3562590408128118217it_nat] :
      ( ! [Y: produc3260487557148687353it_nat] : Xa != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Y)
    <=> ( Xa = none_P2651198173097904984it_nat ) ) ).

% not_Some_eq
tff(fact_35_not__Some__eq,axiom,
    ! [Xa: option8956607266484857688it_nat] :
      ( ! [Y: produc8664842809031399944it_nat] : Xa != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Y)
    <=> ( Xa = none_P9117596204409417319it_nat ) ) ).

% not_Some_eq
tff(fact_36_not__Some__eq,axiom,
    ! [Xa: option_num] :
      ( ! [Y: num] : Xa != aa_num_option_num(some_num,Y)
    <=> ( Xa = none_num ) ) ).

% not_Some_eq
tff(fact_37_not__None__eq,axiom,
    ! [Xa: option3562590408128118217it_nat] :
      ( ( Xa != none_P2651198173097904984it_nat )
    <=> ? [Y: produc3260487557148687353it_nat] : Xa = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Y) ) ).

% not_None_eq
tff(fact_38_not__None__eq,axiom,
    ! [Xa: option8956607266484857688it_nat] :
      ( ( Xa != none_P9117596204409417319it_nat )
    <=> ? [Y: produc8664842809031399944it_nat] : Xa = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Y) ) ).

% not_None_eq
tff(fact_39_not__None__eq,axiom,
    ! [Xa: option_num] :
      ( ( Xa != none_num )
    <=> ? [Y: num] : Xa = aa_num_option_num(some_num,Y) ) ).

% not_None_eq
tff(fact_40_combine__options__cases,axiom,
    ! [Xa: option3562590408128118217it_nat,Pa: fun_op5061235789045561272_nat_o,Y3: option3562590408128118217it_nat] :
      ( ( ( Xa = none_P2651198173097904984it_nat )
       => aa_opt9064029888611036241_nat_o(aa_opt9017216189638898209_nat_o(Pa,Xa),Y3) )
     => ( ( ( Y3 = none_P2651198173097904984it_nat )
         => aa_opt9064029888611036241_nat_o(aa_opt9017216189638898209_nat_o(Pa,Xa),Y3) )
       => ( ! [A2: produc3260487557148687353it_nat,B2: produc3260487557148687353it_nat] :
              ( ( Xa = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,A2) )
             => ( ( Y3 = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,B2) )
               => aa_opt9064029888611036241_nat_o(aa_opt9017216189638898209_nat_o(Pa,Xa),Y3) ) )
         => aa_opt9064029888611036241_nat_o(aa_opt9017216189638898209_nat_o(Pa,Xa),Y3) ) ) ) ).

% combine_options_cases
tff(fact_41_combine__options__cases,axiom,
    ! [Xa: option3562590408128118217it_nat,Pa: fun_op636363337204153321_nat_o,Y3: option8956607266484857688it_nat] :
      ( ( ( Xa = none_P2651198173097904984it_nat )
       => aa_opt8360315729766332930_nat_o(aa_opt7300124240507411154_nat_o(Pa,Xa),Y3) )
     => ( ( ( Y3 = none_P9117596204409417319it_nat )
         => aa_opt8360315729766332930_nat_o(aa_opt7300124240507411154_nat_o(Pa,Xa),Y3) )
       => ( ! [A2: produc3260487557148687353it_nat,B2: produc8664842809031399944it_nat] :
              ( ( Xa = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,A2) )
             => ( ( Y3 = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,B2) )
               => aa_opt8360315729766332930_nat_o(aa_opt7300124240507411154_nat_o(Pa,Xa),Y3) ) )
         => aa_opt8360315729766332930_nat_o(aa_opt7300124240507411154_nat_o(Pa,Xa),Y3) ) ) ) ).

% combine_options_cases
tff(fact_42_combine__options__cases,axiom,
    ! [Xa: option3562590408128118217it_nat,Pa: fun_op8016730479996091735_num_o,Y3: option_num] :
      ( ( ( Xa = none_P2651198173097904984it_nat )
       => aa_option_num_o(aa_opt1040307220371850048_num_o(Pa,Xa),Y3) )
     => ( ( ( Y3 = none_num )
         => aa_option_num_o(aa_opt1040307220371850048_num_o(Pa,Xa),Y3) )
       => ( ! [A2: produc3260487557148687353it_nat,B2: num] :
              ( ( Xa = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,A2) )
             => ( ( Y3 = aa_num_option_num(some_num,B2) )
               => aa_option_num_o(aa_opt1040307220371850048_num_o(Pa,Xa),Y3) ) )
         => aa_option_num_o(aa_opt1040307220371850048_num_o(Pa,Xa),Y3) ) ) ) ).

% combine_options_cases
tff(fact_43_combine__options__cases,axiom,
    ! [Xa: option8956607266484857688it_nat,Pa: fun_op2843162390950497001_nat_o,Y3: option3562590408128118217it_nat] :
      ( ( ( Xa = none_P9117596204409417319it_nat )
       => aa_opt9064029888611036241_nat_o(aa_opt4896439242367626578_nat_o(Pa,Xa),Y3) )
     => ( ( ( Y3 = none_P2651198173097904984it_nat )
         => aa_opt9064029888611036241_nat_o(aa_opt4896439242367626578_nat_o(Pa,Xa),Y3) )
       => ( ! [A2: produc8664842809031399944it_nat,B2: produc3260487557148687353it_nat] :
              ( ( Xa = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,A2) )
             => ( ( Y3 = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,B2) )
               => aa_opt9064029888611036241_nat_o(aa_opt4896439242367626578_nat_o(Pa,Xa),Y3) ) )
         => aa_opt9064029888611036241_nat_o(aa_opt4896439242367626578_nat_o(Pa,Xa),Y3) ) ) ) ).

% combine_options_cases
tff(fact_44_combine__options__cases,axiom,
    ! [Xa: option8956607266484857688it_nat,Pa: fun_op1633239139951409690_nat_o,Y3: option8956607266484857688it_nat] :
      ( ( ( Xa = none_P9117596204409417319it_nat )
       => aa_opt8360315729766332930_nat_o(aa_opt6948326687743049987_nat_o(Pa,Xa),Y3) )
     => ( ( ( Y3 = none_P9117596204409417319it_nat )
         => aa_opt8360315729766332930_nat_o(aa_opt6948326687743049987_nat_o(Pa,Xa),Y3) )
       => ( ! [A2: produc8664842809031399944it_nat,B2: produc8664842809031399944it_nat] :
              ( ( Xa = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,A2) )
             => ( ( Y3 = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,B2) )
               => aa_opt8360315729766332930_nat_o(aa_opt6948326687743049987_nat_o(Pa,Xa),Y3) ) )
         => aa_opt8360315729766332930_nat_o(aa_opt6948326687743049987_nat_o(Pa,Xa),Y3) ) ) ) ).

% combine_options_cases
tff(fact_45_combine__options__cases,axiom,
    ! [Xa: option8956607266484857688it_nat,Pa: fun_op6018817604785332104_num_o,Y3: option_num] :
      ( ( ( Xa = none_P9117596204409417319it_nat )
       => aa_option_num_o(aa_opt3964967581558639985_num_o(Pa,Xa),Y3) )
     => ( ( ( Y3 = none_num )
         => aa_option_num_o(aa_opt3964967581558639985_num_o(Pa,Xa),Y3) )
       => ( ! [A2: produc8664842809031399944it_nat,B2: num] :
              ( ( Xa = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,A2) )
             => ( ( Y3 = aa_num_option_num(some_num,B2) )
               => aa_option_num_o(aa_opt3964967581558639985_num_o(Pa,Xa),Y3) ) )
         => aa_option_num_o(aa_opt3964967581558639985_num_o(Pa,Xa),Y3) ) ) ) ).

% combine_options_cases
tff(fact_46_combine__options__cases,axiom,
    ! [Xa: option_num,Pa: fun_op7041624247167716439_nat_o,Y3: option3562590408128118217it_nat] :
      ( ( ( Xa = none_num )
       => aa_opt9064029888611036241_nat_o(aa_opt3534265616007204544_nat_o(Pa,Xa),Y3) )
     => ( ( ( Y3 = none_P2651198173097904984it_nat )
         => aa_opt9064029888611036241_nat_o(aa_opt3534265616007204544_nat_o(Pa,Xa),Y3) )
       => ( ! [A2: num,B2: produc3260487557148687353it_nat] :
              ( ( Xa = aa_num_option_num(some_num,A2) )
             => ( ( Y3 = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,B2) )
               => aa_opt9064029888611036241_nat_o(aa_opt3534265616007204544_nat_o(Pa,Xa),Y3) ) )
         => aa_opt9064029888611036241_nat_o(aa_opt3534265616007204544_nat_o(Pa,Xa),Y3) ) ) ) ).

% combine_options_cases
tff(fact_47_combine__options__cases,axiom,
    ! [Xa: option_num,Pa: fun_op298641349230941576_nat_o,Y3: option8956607266484857688it_nat] :
      ( ( ( Xa = none_num )
       => aa_opt8360315729766332930_nat_o(aa_opt6324340006354107505_nat_o(Pa,Xa),Y3) )
     => ( ( ( Y3 = none_P9117596204409417319it_nat )
         => aa_opt8360315729766332930_nat_o(aa_opt6324340006354107505_nat_o(Pa,Xa),Y3) )
       => ( ! [A2: num,B2: produc8664842809031399944it_nat] :
              ( ( Xa = aa_num_option_num(some_num,A2) )
             => ( ( Y3 = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,B2) )
               => aa_opt8360315729766332930_nat_o(aa_opt6324340006354107505_nat_o(Pa,Xa),Y3) ) )
         => aa_opt8360315729766332930_nat_o(aa_opt6324340006354107505_nat_o(Pa,Xa),Y3) ) ) ) ).

% combine_options_cases
tff(fact_48_combine__options__cases,axiom,
    ! [Xa: option_num,Pa: fun_op493924057549976310_num_o,Y3: option_num] :
      ( ( ( Xa = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(Pa,Xa),Y3) )
     => ( ( ( Y3 = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(Pa,Xa),Y3) )
       => ( ! [A2: num,B2: num] :
              ( ( Xa = aa_num_option_num(some_num,A2) )
             => ( ( Y3 = aa_num_option_num(some_num,B2) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(Pa,Xa),Y3) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(Pa,Xa),Y3) ) ) ) ).

% combine_options_cases
tff(fact_49_split__option__all,axiom,
    ! [Pa: fun_op9133723461703387112_nat_o] :
      ( ! [X_1: option3562590408128118217it_nat] : aa_opt9064029888611036241_nat_o(Pa,X_1)
    <=> ( aa_opt9064029888611036241_nat_o(Pa,none_P2651198173097904984it_nat)
        & ! [X3: produc3260487557148687353it_nat] : aa_opt9064029888611036241_nat_o(Pa,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X3)) ) ) ).

% split_option_all
tff(fact_50_split__option__all,axiom,
    ! [Pa: fun_op479301348043730329_nat_o] :
      ( ! [X_1: option8956607266484857688it_nat] : aa_opt8360315729766332930_nat_o(Pa,X_1)
    <=> ( aa_opt8360315729766332930_nat_o(Pa,none_P9117596204409417319it_nat)
        & ! [X3: produc8664842809031399944it_nat] : aa_opt8360315729766332930_nat_o(Pa,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X3)) ) ) ).

% split_option_all
tff(fact_51_split__option__all,axiom,
    ! [Pa: fun_option_num_o] :
      ( ! [X_1: option_num] : aa_option_num_o(Pa,X_1)
    <=> ( aa_option_num_o(Pa,none_num)
        & ! [X3: num] : aa_option_num_o(Pa,aa_num_option_num(some_num,X3)) ) ) ).

% split_option_all
tff(fact_52_split__option__ex,axiom,
    ! [Pa: fun_op9133723461703387112_nat_o] :
      ( ? [X_1: option3562590408128118217it_nat] : aa_opt9064029888611036241_nat_o(Pa,X_1)
    <=> ( aa_opt9064029888611036241_nat_o(Pa,none_P2651198173097904984it_nat)
        | ? [X3: produc3260487557148687353it_nat] : aa_opt9064029888611036241_nat_o(Pa,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X3)) ) ) ).

% split_option_ex
tff(fact_53_split__option__ex,axiom,
    ! [Pa: fun_op479301348043730329_nat_o] :
      ( ? [X_1: option8956607266484857688it_nat] : aa_opt8360315729766332930_nat_o(Pa,X_1)
    <=> ( aa_opt8360315729766332930_nat_o(Pa,none_P9117596204409417319it_nat)
        | ? [X3: produc8664842809031399944it_nat] : aa_opt8360315729766332930_nat_o(Pa,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X3)) ) ) ).

% split_option_ex
tff(fact_54_split__option__ex,axiom,
    ! [Pa: fun_option_num_o] :
      ( ? [X_1: option_num] : aa_option_num_o(Pa,X_1)
    <=> ( aa_option_num_o(Pa,none_num)
        | ? [X3: num] : aa_option_num_o(Pa,aa_num_option_num(some_num,X3)) ) ) ).

% split_option_ex
tff(fact_55_mem__Collect__eq,axiom,
    ! [A: set_Pr1261947904930325089at_nat,Pa: fun_se4782206882984618170_nat_o] :
      ( member2643936169264416010at_nat(A,collec5514110066124741708at_nat(Pa))
    <=> aa_set3702741264012733747_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_56_mem__Collect__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int,Pa: fun_se7249947736802449410_int_o] :
      ( member2340774599025711042nt_int(A,collec5210948495886036740nt_int(Pa))
    <=> aa_set6170482117830564987_int_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_57_mem__Collect__eq,axiom,
    ! [A: set_nat,Pa: fun_set_nat_o] :
      ( member_set_nat(A,collect_set_nat(Pa))
    <=> aa_set_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_58_mem__Collect__eq,axiom,
    ! [A: nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),collect_nat(Pa))
    <=> aa_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_59_mem__Collect__eq,axiom,
    ! [A: int,Pa: fun_int_o] :
      ( member_int(A,collect_int(Pa))
    <=> aa_int_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_60_Collect__mem__eq,axiom,
    ! [A3: set_se7855581050983116737at_nat] : collec5514110066124741708at_nat(aTP_Lamm_a(A3)) = A3 ).

% Collect_mem_eq
tff(fact_61_Collect__mem__eq,axiom,
    ! [A3: set_se6260736226359567993nt_int] : collec5210948495886036740nt_int(aTP_Lamm_aa(A3)) = A3 ).

% Collect_mem_eq
tff(fact_62_Collect__mem__eq,axiom,
    ! [A3: set_set_nat] : collect_set_nat(aTP_Lamm_ab(A3)) = A3 ).

% Collect_mem_eq
tff(fact_63_Collect__mem__eq,axiom,
    ! [A3: set_nat] : collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ac,A3)) = A3 ).

% Collect_mem_eq
tff(fact_64_Collect__mem__eq,axiom,
    ! [A3: set_int] : collect_int(aTP_Lamm_ad(A3)) = A3 ).

% Collect_mem_eq
tff(fact_65_Collect__cong,axiom,
    ! [Pa: fun_se4782206882984618170_nat_o,Q: fun_se4782206882984618170_nat_o] :
      ( ! [X4: set_Pr1261947904930325089at_nat] :
          ( aa_set3702741264012733747_nat_o(Pa,X4)
        <=> aa_set3702741264012733747_nat_o(Q,X4) )
     => ( collec5514110066124741708at_nat(Pa) = collec5514110066124741708at_nat(Q) ) ) ).

% Collect_cong
tff(fact_66_Collect__cong,axiom,
    ! [Pa: fun_se7249947736802449410_int_o,Q: fun_se7249947736802449410_int_o] :
      ( ! [X4: set_Pr958786334691620121nt_int] :
          ( aa_set6170482117830564987_int_o(Pa,X4)
        <=> aa_set6170482117830564987_int_o(Q,X4) )
     => ( collec5210948495886036740nt_int(Pa) = collec5210948495886036740nt_int(Q) ) ) ).

% Collect_cong
tff(fact_67_Collect__cong,axiom,
    ! [Pa: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X4: set_nat] :
          ( aa_set_nat_o(Pa,X4)
        <=> aa_set_nat_o(Q,X4) )
     => ( collect_set_nat(Pa) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_68_Collect__cong,axiom,
    ! [Pa: fun_nat_o,Q: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_nat_o(Pa,X4)
        <=> aa_nat_o(Q,X4) )
     => ( collect_nat(Pa) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_69_Collect__cong,axiom,
    ! [Pa: fun_int_o,Q: fun_int_o] :
      ( ! [X4: int] :
          ( aa_int_o(Pa,X4)
        <=> aa_int_o(Q,X4) )
     => ( collect_int(Pa) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_70_option_Oexhaust,axiom,
    ! [Y3: option3562590408128118217it_nat] :
      ( ( Y3 != none_P2651198173097904984it_nat )
     => ~ ! [X22: produc3260487557148687353it_nat] : Y3 != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X22) ) ).

% option.exhaust
tff(fact_71_option_Oexhaust,axiom,
    ! [Y3: option8956607266484857688it_nat] :
      ( ( Y3 != none_P9117596204409417319it_nat )
     => ~ ! [X22: produc8664842809031399944it_nat] : Y3 != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X22) ) ).

% option.exhaust
tff(fact_72_option_Oexhaust,axiom,
    ! [Y3: option_num] :
      ( ( Y3 != none_num )
     => ~ ! [X22: num] : Y3 != aa_num_option_num(some_num,X22) ) ).

% option.exhaust
tff(fact_73_option_OdiscI,axiom,
    ! [Option: option3562590408128118217it_nat,X2: produc3260487557148687353it_nat] :
      ( ( Option = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X2) )
     => ( Option != none_P2651198173097904984it_nat ) ) ).

% option.discI
tff(fact_74_option_OdiscI,axiom,
    ! [Option: option8956607266484857688it_nat,X2: produc8664842809031399944it_nat] :
      ( ( Option = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2) )
     => ( Option != none_P9117596204409417319it_nat ) ) ).

% option.discI
tff(fact_75_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = aa_num_option_num(some_num,X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_76_option_Odistinct_I1_J,axiom,
    ! [X2: produc3260487557148687353it_nat] : none_P2651198173097904984it_nat != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X2) ).

% option.distinct(1)
tff(fact_77_option_Odistinct_I1_J,axiom,
    ! [X2: produc8664842809031399944it_nat] : none_P9117596204409417319it_nat != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2) ).

% option.distinct(1)
tff(fact_78_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != aa_num_option_num(some_num,X2) ).

% option.distinct(1)
tff(fact_79_effect__bindI,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: a,N: nat,G: fun_a_6093291656547882009Heap_a,H3: heap_e7401611519738050253t_unit,R3: a,N3: nat] :
      ( aa_nat_o(heap_Time_effect_a(F,Ha,H,R),N)
     => ( aa_nat_o(heap_Time_effect_a(aa_a_H8829438773920479562Heap_a(G,R),H,H3,R3),N3)
       => aa_nat_o(heap_Time_effect_a(heap_Time_bind_a_a(F,G),Ha,H3,R3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N3)) ) ) ).

% effect_bindI
tff(fact_80_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_81_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_82_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_83_double__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% double_eq_0_iff
tff(fact_84_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_85_add__0,axiom,
    ! [A: literal] : plus_plus_literal(zero_zero_literal,A) = A ).

% add_0
tff(fact_86_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_87_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_88_add__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).

% add_0
tff(fact_89_add__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add_0
tff(fact_90_add__0,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,zero_z2226904508553997617atural),A) = A ).

% add_0
tff(fact_91_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xa: nat,Y3: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y3) )
    <=> ( ( Xa = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_92_add__eq__0__iff__both__eq__0,axiom,
    ! [Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y3) = zero_zero_nat )
    <=> ( ( Xa = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_93_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_94_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_95_add__cancel__right__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_right
tff(fact_96_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_97_add__cancel__right__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B) )
    <=> ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_right_right
tff(fact_98_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_99_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_100_add__cancel__right__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_left
tff(fact_101_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_102_add__cancel__right__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A) )
    <=> ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_right_left
tff(fact_103_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_104_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_105_add__cancel__left__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = A )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_right
tff(fact_106_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_107_add__cancel__left__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B) = A )
    <=> ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_left_right
tff(fact_108_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_109_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_110_add__cancel__left__left,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = A )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_left
tff(fact_111_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_112_add__cancel__left__left,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A) = A )
    <=> ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_left_left
tff(fact_113_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_114_double__zero__sym,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% double_zero_sym
tff(fact_115_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_116_add_Oright__neutral,axiom,
    ! [A: literal] : plus_plus_literal(A,zero_zero_literal) = A ).

% add.right_neutral
tff(fact_117_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_118_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_119_add_Oright__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ).

% add.right_neutral
tff(fact_120_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_121_add_Oright__neutral,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_z2226904508553997617atural) = A ).

% add.right_neutral
tff(fact_122_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_123_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_124_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_125_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_126_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_127_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_128_zero__reorient,axiom,
    ! [Xa: nat] :
      ( ( zero_zero_nat = Xa )
    <=> ( Xa = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_129_zero__reorient,axiom,
    ! [Xa: int] :
      ( ( zero_zero_int = Xa )
    <=> ( Xa = zero_zero_int ) ) ).

% zero_reorient
tff(fact_130_zero__reorient,axiom,
    ! [Xa: code_integer] :
      ( ( zero_z3403309356797280102nteger = Xa )
    <=> ( Xa = zero_z3403309356797280102nteger ) ) ).

% zero_reorient
tff(fact_131_zero__reorient,axiom,
    ! [Xa: rat] :
      ( ( zero_zero_rat = Xa )
    <=> ( Xa = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_132_zero__reorient,axiom,
    ! [Xa: code_natural] :
      ( ( zero_z2226904508553997617atural = Xa )
    <=> ( Xa = zero_z2226904508553997617atural ) ) ).

% zero_reorient
tff(fact_133_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_134_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_135_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_136_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_137_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_138_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_139_group__cancel_Oadd1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A3),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_140_group__cancel_Oadd1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_141_group__cancel_Oadd1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A3),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_142_group__cancel_Oadd2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_143_group__cancel_Oadd2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_144_group__cancel_Oadd2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_145_add_Oassoc,axiom,
    ! [A: literal,B: literal,C: literal] : plus_plus_literal(plus_plus_literal(A,B),C) = plus_plus_literal(A,plus_plus_literal(B,C)) ).

% add.assoc
tff(fact_146_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.assoc
tff(fact_147_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_148_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.assoc
tff(fact_149_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_150_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_151_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_152_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_153_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ).

% add.commute
tff(fact_154_add_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ).

% add.commute
tff(fact_155_add_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ).

% add.commute
tff(fact_156_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.left_commute
tff(fact_157_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.left_commute
tff(fact_158_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.left_commute
tff(fact_159_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_160_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_161_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_162_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_163_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_164_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_165_one__reorient,axiom,
    ! [Xa: nat] :
      ( ( one_one_nat = Xa )
    <=> ( Xa = one_one_nat ) ) ).

% one_reorient
tff(fact_166_one__reorient,axiom,
    ! [Xa: int] :
      ( ( one_one_int = Xa )
    <=> ( Xa = one_one_int ) ) ).

% one_reorient
tff(fact_167_one__reorient,axiom,
    ! [Xa: code_integer] :
      ( ( one_one_Code_integer = Xa )
    <=> ( Xa = one_one_Code_integer ) ) ).

% one_reorient
tff(fact_168_one__reorient,axiom,
    ! [Xa: rat] :
      ( ( one_one_rat = Xa )
    <=> ( Xa = one_one_rat ) ) ).

% one_reorient
tff(fact_169_one__reorient,axiom,
    ! [Xa: code_natural] :
      ( ( one_one_Code_natural = Xa )
    <=> ( Xa = one_one_Code_natural ) ) ).

% one_reorient
tff(fact_170_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_171_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_172_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_173_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_174_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,zero_z2226904508553997617atural),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_175_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_176_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_177_add_Ocomm__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ).

% add.comm_neutral
tff(fact_178_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_179_add_Ocomm__neutral,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_z2226904508553997617atural) = A ).

% add.comm_neutral
tff(fact_180_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_181_add_Ogroup__left__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).

% add.group_left_neutral
tff(fact_182_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_183_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_184_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_185_bind__return,axiom,
    ! [F: heap_Time_Heap_a] : heap_Time_bind_a_a(F,heap_Time_return_a) = heap_T757603679106148408unit_a(heap_Time_wait(one_one_nat),aTP_Lamm_ae(F)) ).

% bind_return
tff(fact_186_Euclid__induct,axiom,
    ! [Pa: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A2: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(Pa,A2),B2)
        <=> aa_nat_o(aa_nat_fun_nat_o(Pa,B2),A2) )
     => ( ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,A2),zero_zero_nat)
       => ( ! [A2: nat,B2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(Pa,A2),B2)
             => aa_nat_o(aa_nat_fun_nat_o(Pa,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(Pa,A),B) ) ) ) ).

% Euclid_induct
tff(fact_187_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_188_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_189_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
tff(fact_190_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_191_zero__neq__one,axiom,
    zero_z2226904508553997617atural != one_one_Code_natural ).

% zero_neq_one
tff(fact_192_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( A = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_193_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_194_add__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% add_0_iff
tff(fact_195_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_196_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_197_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_198_verit__sum__simplify,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ).

% verit_sum_simplify
tff(fact_199_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_200_verit__sum__simplify,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_z2226904508553997617atural) = A ).

% verit_sum_simplify
tff(fact_201_guard__def,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,F: fun_he8889679731534893401it_nat] : heap_Time_guard_a(Pa,F) = heap_Time_Heap_a2(aa_fun3370411987892275404it_nat(aTP_Lamm_af(Pa),F)) ).

% guard_def
tff(fact_202_guard__def,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,F: fun_he4944555133437423336it_nat] : heap_T8440541562793052209t_unit(Pa,F) = heap_T6183433275982383450t_unit(aa_fun2986507860694160780it_nat(aTP_Lamm_ag(Pa),F)) ).

% guard_def
tff(fact_203_raise__def,axiom,
    ! [S: list_char] : heap_Time_raise_a(S) = heap_Time_Heap_a2(aTP_Lamm_ah) ).

% raise_def
tff(fact_204_raise__def,axiom,
    ! [S: list_char] : heap_T2927564422264180874t_unit(S) = heap_T6183433275982383450t_unit(aTP_Lamm_ai) ).

% raise_def
tff(fact_205_successE,axiom,
    ! [F: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(heap_T584514906347983379t_unit(F),Ha)
     => ~ ! [R2: product_unit,H2: produc6653097349344004940it_nat] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(F),Ha) != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,H2)) ) ).

% successE
tff(fact_206_successE,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(heap_Time_success_a(F),Ha)
     => ~ ! [R2: a,H2: produc6653097349344004940it_nat] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(R2,H2)) ) ).

% successE
tff(fact_207_Heap__cases,axiom,
    ! [F: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit] :
      ( ! [X4: product_unit,H2: produc6653097349344004940it_nat] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(F),Ha) != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X4,H2))
     => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(F),Ha) = none_P9117596204409417319it_nat ) ) ).

% Heap_cases
tff(fact_208_Heap__cases,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit] :
      ( ! [X4: a,H2: produc6653097349344004940it_nat] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(X4,H2))
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) = none_P2651198173097904984it_nat ) ) ).

% Heap_cases
tff(fact_209_execute__heap,axiom,
    ! [F: fun_he4944555133437423336it_nat] : heap_T875086893843062177t_unit(heap_T6927113302350381334t_unit(F)) = comp_P3118722334806803912t_unit(some_P1914260805536162275it_nat,F) ).

% execute_heap
tff(fact_210_execute__heap,axiom,
    ! [F: fun_he8889679731534893401it_nat] : heap_Time_execute_a(heap_Time_heap_a(F)) = comp_P8552900652980440422t_unit(some_P7913643980934408916it_nat,F) ).

% execute_heap
tff(fact_211_Heap_Oinject,axiom,
    ! [Xa: fun_he5234484606919320760it_nat,Ya: fun_he5234484606919320760it_nat] :
      ( ( heap_T6183433275982383450t_unit(Xa) = heap_T6183433275982383450t_unit(Ya) )
    <=> ( Xa = Ya ) ) ).

% Heap.inject
tff(fact_212_Heap__execute,axiom,
    ! [F: heap_Time_Heap_a] : heap_Time_Heap_a2(heap_Time_execute_a(F)) = F ).

% Heap_execute
tff(fact_213_Heap__execute,axiom,
    ! [F: heap_T5738788834812785303t_unit] : heap_T6183433275982383450t_unit(heap_T875086893843062177t_unit(F)) = F ).

% Heap_execute
tff(fact_214_rewriteR__comp__comp2,axiom,
    ! [G: fun_int_int,Ha: fun_nat_int,R1: fun_nat_int,R22: fun_nat_nat,F: fun_int_nat,L: fun_nat_nat] :
      ( ( comp_int_int_nat(G,Ha) = comp_nat_int_nat(R1,R22) )
     => ( ( comp_int_nat_nat(F,R1) = L )
       => ( comp_int_nat_nat(comp_int_nat_int(F,G),Ha) = comp_nat_nat_nat(L,R22) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_215_rewriteR__comp__comp2,axiom,
    ! [G: fun_nat_int,Ha: fun_nat_nat,R1: fun_int_int,R22: fun_nat_int,F: fun_int_nat,L: fun_int_nat] :
      ( ( comp_nat_int_nat(G,Ha) = comp_int_int_nat(R1,R22) )
     => ( ( comp_int_nat_int(F,R1) = L )
       => ( comp_nat_nat_nat(comp_int_nat_nat(F,G),Ha) = comp_int_nat_nat(L,R22) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_216_rewriteR__comp__comp2,axiom,
    ! [G: fun_int_int,Ha: fun_int_int,R1: fun_int_int,R22: fun_int_int,F: fun_int_nat,L: fun_int_nat] :
      ( ( comp_int_int_int(G,Ha) = comp_int_int_int(R1,R22) )
     => ( ( comp_int_nat_int(F,R1) = L )
       => ( comp_int_nat_int(comp_int_nat_int(F,G),Ha) = comp_int_nat_int(L,R22) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_217_rewriteR__comp__comp2,axiom,
    ! [G: fun_Co8487158583234504619r_char,Ha: fun_Co8499238673658658103nteger,R1: fun_char_char,R22: fun_Co8487158583234504619r_char,F: fun_char_char,L: fun_char_char] :
      ( ( comp_C3856975138083468656nteger(G,Ha) = comp_c4562247934908617188nteger(R1,R22) )
     => ( ( comp_char_char_char(F,R1) = L )
       => ( comp_C3856975138083468656nteger(comp_c4562247934908617188nteger(F,G),Ha) = comp_c4562247934908617188nteger(L,R22) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_218_rewriteR__comp__comp2,axiom,
    ! [G: fun_int_num,Ha: fun_int_int,R1: fun_nat_num,R22: fun_int_nat,F: fun_num_nat,L: fun_nat_nat] :
      ( ( comp_int_num_int(G,Ha) = comp_nat_num_int(R1,R22) )
     => ( ( comp_num_nat_nat(F,R1) = L )
       => ( comp_int_nat_int(comp_num_nat_int(F,G),Ha) = comp_nat_nat_int(L,R22) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_219_rewriteR__comp__comp2,axiom,
    ! [G: fun_int_nat,Ha: fun_nat_int,R1: fun_nat_nat,R22: fun_nat_nat,F: fun_nat_num,L: fun_nat_num] :
      ( ( comp_int_nat_nat(G,Ha) = comp_nat_nat_nat(R1,R22) )
     => ( ( comp_nat_num_nat(F,R1) = L )
       => ( comp_int_num_nat(comp_nat_num_int(F,G),Ha) = comp_nat_num_nat(L,R22) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_220_rewriteR__comp__comp2,axiom,
    ! [G: fun_nat_nat,Ha: fun_int_nat,R1: fun_int_nat,R22: fun_int_int,F: fun_nat_nat,L: fun_int_nat] :
      ( ( comp_nat_nat_int(G,Ha) = comp_int_nat_int(R1,R22) )
     => ( ( comp_nat_nat_int(F,R1) = L )
       => ( comp_nat_nat_int(comp_nat_nat_nat(F,G),Ha) = comp_int_nat_int(L,R22) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_221_rewriteR__comp__comp2,axiom,
    ! [G: fun_nat_nat,Ha: fun_int_nat,R1: fun_int_nat,R22: fun_int_int,F: fun_nat_num,L: fun_int_num] :
      ( ( comp_nat_nat_int(G,Ha) = comp_int_nat_int(R1,R22) )
     => ( ( comp_nat_num_int(F,R1) = L )
       => ( comp_nat_num_int(comp_nat_num_nat(F,G),Ha) = comp_int_num_int(L,R22) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_222_rewriteR__comp__comp2,axiom,
    ! [G: fun_char_char,Ha: fun_Co8487158583234504619r_char,R1: fun_Co8487158583234504619r_char,R22: fun_Co8499238673658658103nteger,F: fun_char_char,L: fun_Co8487158583234504619r_char] :
      ( ( comp_c4562247934908617188nteger(G,Ha) = comp_C3856975138083468656nteger(R1,R22) )
     => ( ( comp_c4562247934908617188nteger(F,R1) = L )
       => ( comp_c4562247934908617188nteger(comp_char_char_char(F,G),Ha) = comp_C3856975138083468656nteger(L,R22) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_223_rewriteR__comp__comp2,axiom,
    ! [G: fun_char_char,Ha: fun_Co8487158583234504619r_char,R1: fun_char_char,R22: fun_Co8487158583234504619r_char,F: fun_char_char,L: fun_char_char] :
      ( ( comp_c4562247934908617188nteger(G,Ha) = comp_c4562247934908617188nteger(R1,R22) )
     => ( ( comp_char_char_char(F,R1) = L )
       => ( comp_c4562247934908617188nteger(comp_char_char_char(F,G),Ha) = comp_c4562247934908617188nteger(L,R22) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_224_rewriteL__comp__comp2,axiom,
    ! [F: fun_num_num,G: fun_nat_num,L1: fun_nat_num,L2: fun_nat_nat,Ha: fun_int_nat,R: fun_int_nat] :
      ( ( comp_num_num_nat(F,G) = comp_nat_num_nat(L1,L2) )
     => ( ( comp_nat_nat_int(L2,Ha) = R )
       => ( comp_num_num_int(F,comp_nat_num_int(G,Ha)) = comp_nat_num_int(L1,R) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_225_rewriteL__comp__comp2,axiom,
    ! [F: fun_num_nat,G: fun_nat_num,L1: fun_int_nat,L2: fun_nat_int,Ha: fun_int_nat,R: fun_int_int] :
      ( ( comp_num_nat_nat(F,G) = comp_int_nat_nat(L1,L2) )
     => ( ( comp_nat_int_int(L2,Ha) = R )
       => ( comp_num_nat_int(F,comp_nat_num_int(G,Ha)) = comp_int_nat_int(L1,R) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_226_rewriteL__comp__comp2,axiom,
    ! [F: fun_char_char,G: fun_char_char,L1: fun_char_char,L2: fun_char_char,Ha: fun_Co8487158583234504619r_char,R: fun_Co8487158583234504619r_char] :
      ( ( comp_char_char_char(F,G) = comp_char_char_char(L1,L2) )
     => ( ( comp_c4562247934908617188nteger(L2,Ha) = R )
       => ( comp_c4562247934908617188nteger(F,comp_c4562247934908617188nteger(G,Ha)) = comp_c4562247934908617188nteger(L1,R) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_227_rewriteL__comp__comp2,axiom,
    ! [F: fun_nat_num,G: fun_nat_nat,L1: fun_num_num,L2: fun_nat_num,Ha: fun_int_nat,R: fun_int_num] :
      ( ( comp_nat_num_nat(F,G) = comp_num_num_nat(L1,L2) )
     => ( ( comp_nat_num_int(L2,Ha) = R )
       => ( comp_nat_num_int(F,comp_nat_nat_int(G,Ha)) = comp_num_num_int(L1,R) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_228_rewriteL__comp__comp2,axiom,
    ! [F: fun_int_nat,G: fun_nat_int,L1: fun_num_nat,L2: fun_nat_num,Ha: fun_int_nat,R: fun_int_num] :
      ( ( comp_int_nat_nat(F,G) = comp_num_nat_nat(L1,L2) )
     => ( ( comp_nat_num_int(L2,Ha) = R )
       => ( comp_int_nat_int(F,comp_nat_int_int(G,Ha)) = comp_num_nat_int(L1,R) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_229_rewriteL__comp__comp2,axiom,
    ! [F: fun_num_nat,G: fun_nat_num,L1: fun_nat_nat,L2: fun_nat_nat,Ha: fun_int_nat,R: fun_int_nat] :
      ( ( comp_num_nat_nat(F,G) = comp_nat_nat_nat(L1,L2) )
     => ( ( comp_nat_nat_int(L2,Ha) = R )
       => ( comp_num_nat_int(F,comp_nat_num_int(G,Ha)) = comp_nat_nat_int(L1,R) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_230_rewriteL__comp__comp2,axiom,
    ! [F: fun_int_nat,G: fun_nat_int,L1: fun_nat_nat,L2: fun_nat_nat,Ha: fun_int_nat,R: fun_int_nat] :
      ( ( comp_int_nat_nat(F,G) = comp_nat_nat_nat(L1,L2) )
     => ( ( comp_nat_nat_int(L2,Ha) = R )
       => ( comp_int_nat_int(F,comp_nat_int_int(G,Ha)) = comp_nat_nat_int(L1,R) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_231_rewriteL__comp__comp2,axiom,
    ! [F: fun_nat_nat,G: fun_int_nat,L1: fun_int_nat,L2: fun_int_int,Ha: fun_int_int,R: fun_int_int] :
      ( ( comp_nat_nat_int(F,G) = comp_int_nat_int(L1,L2) )
     => ( ( comp_int_int_int(L2,Ha) = R )
       => ( comp_nat_nat_int(F,comp_int_nat_int(G,Ha)) = comp_int_nat_int(L1,R) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_232_rewriteL__comp__comp2,axiom,
    ! [F: fun_nat_nat,G: fun_int_nat,L1: fun_int_nat,L2: fun_int_int,Ha: fun_nat_int,R: fun_nat_int] :
      ( ( comp_nat_nat_int(F,G) = comp_int_nat_int(L1,L2) )
     => ( ( comp_int_int_nat(L2,Ha) = R )
       => ( comp_nat_nat_nat(F,comp_int_nat_nat(G,Ha)) = comp_int_nat_nat(L1,R) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_233_rewriteL__comp__comp2,axiom,
    ! [F: fun_char_char,G: fun_Co8487158583234504619r_char,L1: fun_char_char,L2: fun_Co8487158583234504619r_char,Ha: fun_Co8499238673658658103nteger,R: fun_Co8487158583234504619r_char] :
      ( ( comp_c4562247934908617188nteger(F,G) = comp_c4562247934908617188nteger(L1,L2) )
     => ( ( comp_C3856975138083468656nteger(L2,Ha) = R )
       => ( comp_c4562247934908617188nteger(F,comp_C3856975138083468656nteger(G,Ha)) = comp_c4562247934908617188nteger(L1,R) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_234_rewriteR__comp__comp,axiom,
    ! [G: fun_Co8487158583234504619r_char,Ha: fun_Co8499238673658658103nteger,R: fun_Co8487158583234504619r_char,F: fun_char_char] :
      ( ( comp_C3856975138083468656nteger(G,Ha) = R )
     => ( comp_C3856975138083468656nteger(comp_c4562247934908617188nteger(F,G),Ha) = comp_c4562247934908617188nteger(F,R) ) ) ).

% rewriteR_comp_comp
tff(fact_235_rewriteR__comp__comp,axiom,
    ! [G: fun_nat_nat,Ha: fun_int_nat,R: fun_int_nat,F: fun_nat_num] :
      ( ( comp_nat_nat_int(G,Ha) = R )
     => ( comp_nat_num_int(comp_nat_num_nat(F,G),Ha) = comp_nat_num_int(F,R) ) ) ).

% rewriteR_comp_comp
tff(fact_236_rewriteR__comp__comp,axiom,
    ! [G: fun_int_int,Ha: fun_int_int,R: fun_int_int,F: fun_int_nat] :
      ( ( comp_int_int_int(G,Ha) = R )
     => ( comp_int_nat_int(comp_int_nat_int(F,G),Ha) = comp_int_nat_int(F,R) ) ) ).

% rewriteR_comp_comp
tff(fact_237_rewriteR__comp__comp,axiom,
    ! [G: fun_char_char,Ha: fun_Co8487158583234504619r_char,R: fun_Co8487158583234504619r_char,F: fun_char_char] :
      ( ( comp_c4562247934908617188nteger(G,Ha) = R )
     => ( comp_c4562247934908617188nteger(comp_char_char_char(F,G),Ha) = comp_c4562247934908617188nteger(F,R) ) ) ).

% rewriteR_comp_comp
tff(fact_238_rewriteR__comp__comp,axiom,
    ! [G: fun_nat_num,Ha: fun_int_nat,R: fun_int_num,F: fun_num_num] :
      ( ( comp_nat_num_int(G,Ha) = R )
     => ( comp_nat_num_int(comp_num_num_nat(F,G),Ha) = comp_num_num_int(F,R) ) ) ).

% rewriteR_comp_comp
tff(fact_239_rewriteR__comp__comp,axiom,
    ! [G: fun_nat_nat,Ha: fun_nat_nat,R: fun_nat_nat,F: fun_nat_nat] :
      ( ( comp_nat_nat_nat(G,Ha) = R )
     => ( comp_nat_nat_nat(comp_nat_nat_nat(F,G),Ha) = comp_nat_nat_nat(F,R) ) ) ).

% rewriteR_comp_comp
tff(fact_240_rewriteR__comp__comp,axiom,
    ! [G: fun_int_nat,Ha: fun_int_int,R: fun_int_nat,F: fun_nat_num] :
      ( ( comp_int_nat_int(G,Ha) = R )
     => ( comp_int_num_int(comp_nat_num_int(F,G),Ha) = comp_nat_num_int(F,R) ) ) ).

% rewriteR_comp_comp
tff(fact_241_rewriteR__comp__comp,axiom,
    ! [G: fun_int_nat,Ha: fun_int_int,R: fun_int_nat,F: fun_nat_nat] :
      ( ( comp_int_nat_int(G,Ha) = R )
     => ( comp_int_nat_int(comp_nat_nat_int(F,G),Ha) = comp_nat_nat_int(F,R) ) ) ).

% rewriteR_comp_comp
tff(fact_242_rewriteR__comp__comp,axiom,
    ! [G: fun_Co1981900195452921617nteger,Ha: fun_Co8499238673658658103nteger,R: fun_Co1981900195452921617nteger,F: fun_fu4800041728187187127nteger] :
      ( ( comp_C7233611567195229654nteger(G,Ha) = R )
     => ( comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(F,G),Ha) = comp_f6739083541385432444nteger(F,R) ) ) ).

% rewriteR_comp_comp
tff(fact_243_rewriteR__comp__comp,axiom,
    ! [G: fun_fu1763652423599918571nteger,Ha: fun_Co1981900195452921617nteger,R: fun_Co1981900195452921617nteger,F: fun_fu4800041728187187127nteger] :
      ( ( comp_f228016600672082096nteger(G,Ha) = R )
     => ( comp_f6739083541385432444nteger(comp_f3848972703483044950nteger(F,G),Ha) = comp_f6739083541385432444nteger(F,R) ) ) ).

% rewriteR_comp_comp
tff(fact_244_rewriteL__comp__comp,axiom,
    ! [F: fun_num_num,G: fun_nat_num,L: fun_nat_num,Ha: fun_int_nat] :
      ( ( comp_num_num_nat(F,G) = L )
     => ( comp_num_num_int(F,comp_nat_num_int(G,Ha)) = comp_nat_num_int(L,Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_245_rewriteL__comp__comp,axiom,
    ! [F: fun_nat_nat,G: fun_int_nat,L: fun_int_nat,Ha: fun_int_int] :
      ( ( comp_nat_nat_int(F,G) = L )
     => ( comp_nat_nat_int(F,comp_int_nat_int(G,Ha)) = comp_int_nat_int(L,Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_246_rewriteL__comp__comp,axiom,
    ! [F: fun_char_char,G: fun_char_char,L: fun_char_char,Ha: fun_Co8487158583234504619r_char] :
      ( ( comp_char_char_char(F,G) = L )
     => ( comp_c4562247934908617188nteger(F,comp_c4562247934908617188nteger(G,Ha)) = comp_c4562247934908617188nteger(L,Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_247_rewriteL__comp__comp,axiom,
    ! [F: fun_nat_num,G: fun_nat_nat,L: fun_nat_num,Ha: fun_int_nat] :
      ( ( comp_nat_num_nat(F,G) = L )
     => ( comp_nat_num_int(F,comp_nat_nat_int(G,Ha)) = comp_nat_num_int(L,Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_248_rewriteL__comp__comp,axiom,
    ! [F: fun_char_char,G: fun_Co8487158583234504619r_char,L: fun_Co8487158583234504619r_char,Ha: fun_Co8499238673658658103nteger] :
      ( ( comp_c4562247934908617188nteger(F,G) = L )
     => ( comp_c4562247934908617188nteger(F,comp_C3856975138083468656nteger(G,Ha)) = comp_C3856975138083468656nteger(L,Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_249_rewriteL__comp__comp,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,L: fun_int_num,Ha: fun_int_int] :
      ( ( comp_nat_num_int(F,G) = L )
     => ( comp_nat_num_int(F,comp_int_nat_int(G,Ha)) = comp_int_num_int(L,Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_250_rewriteL__comp__comp,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,L: fun_nat_nat,Ha: fun_nat_nat] :
      ( ( comp_nat_nat_nat(F,G) = L )
     => ( comp_nat_nat_nat(F,comp_nat_nat_nat(G,Ha)) = comp_nat_nat_nat(L,Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_251_rewriteL__comp__comp,axiom,
    ! [F: fun_int_nat,G: fun_int_int,L: fun_int_nat,Ha: fun_int_int] :
      ( ( comp_int_nat_int(F,G) = L )
     => ( comp_int_nat_int(F,comp_int_int_int(G,Ha)) = comp_int_nat_int(L,Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_252_rewriteL__comp__comp,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,L: fun_Co1561183069634336733nteger,Ha: fun_Co8499238673658658103nteger] :
      ( ( comp_f6739083541385432444nteger(F,G) = L )
     => ( comp_f6739083541385432444nteger(F,comp_C7233611567195229654nteger(G,Ha)) = comp_C7848664941760151970nteger(L,Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_253_rewriteL__comp__comp,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_fu1763652423599918571nteger,L: fun_fu4800041728187187127nteger,Ha: fun_Co1981900195452921617nteger] :
      ( ( comp_f3848972703483044950nteger(F,G) = L )
     => ( comp_f6739083541385432444nteger(F,comp_f228016600672082096nteger(G,Ha)) = comp_f6739083541385432444nteger(L,Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_254_Heap_Oexhaust,axiom,
    ! [Y3: heap_T5738788834812785303t_unit] :
      ~ ! [X4: fun_he5234484606919320760it_nat] : Y3 != heap_T6183433275982383450t_unit(X4) ).

% Heap.exhaust
tff(fact_255_heap__def,axiom,
    ! [F: fun_he8889679731534893401it_nat] : heap_Time_heap_a(F) = heap_Time_Heap_a2(comp_P8552900652980440422t_unit(some_P7913643980934408916it_nat,F)) ).

% heap_def
tff(fact_256_heap__def,axiom,
    ! [F: fun_he4944555133437423336it_nat] : heap_T6927113302350381334t_unit(F) = heap_T6183433275982383450t_unit(comp_P3118722334806803912t_unit(some_P1914260805536162275it_nat,F)) ).

% heap_def
tff(fact_257_execute_Osimps,axiom,
    ! [F: fun_he1921015881043195817it_nat] : heap_Time_execute_a(heap_Time_Heap_a2(F)) = F ).

% execute.simps
tff(fact_258_execute_Osimps,axiom,
    ! [F: fun_he5234484606919320760it_nat] : heap_T875086893843062177t_unit(heap_T6183433275982383450t_unit(F)) = F ).

% execute.simps
tff(fact_259_comp__apply,axiom,
    ! [F: fun_char_char,G: fun_Co8487158583234504619r_char,Xa: code_integer] : aa_Code_integer_char(comp_c4562247934908617188nteger(F,G),Xa) = aa_char_char(F,aa_Code_integer_char(G,Xa)) ).

% comp_apply
tff(fact_260_comp__apply,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,Xa: int] : aa_int_num(comp_nat_num_int(F,G),Xa) = aa_nat_num(F,aa_int_nat(G,Xa)) ).

% comp_apply
tff(fact_261_comp__apply,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Xa: nat] : aa_nat_nat(comp_nat_nat_nat(F,G),Xa) = aa_nat_nat(F,aa_nat_nat(G,Xa)) ).

% comp_apply
tff(fact_262_comp__apply,axiom,
    ! [F: fun_int_nat,G: fun_int_int,Xa: int] : aa_int_nat(comp_int_nat_int(F,G),Xa) = aa_int_nat(F,aa_int_int(G,Xa)) ).

% comp_apply
tff(fact_263_comp__apply,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,Xa: code_integer] : aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(F,G),Xa) = aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,Xa)) ).

% comp_apply
tff(fact_264_old_Oprod_Oinject,axiom,
    ! [A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger,A4: fun_Co3460408610320954864e_term,B4: produc8923325533196201883nteger] :
      ( ( aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_265_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr285225112257423215e_term,B: produc7822875418678951345atural,A4: fun_Pr285225112257423215e_term,B4: produc7822875418678951345atural] :
      ( ( aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A),B) = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_266_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,A4: fun_Pr5098669352780829690e_term,B4: produc8923325533196201883nteger] :
      ( ( aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_267_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A4: fun_Pr7763680713840462443e_term,B4: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_268_old_Oprod_Oinject,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_269_prod_Oinject,axiom,
    ! [X1: fun_Co3460408610320954864e_term,X2: produc8923325533196201883nteger,Y1: fun_Co3460408610320954864e_term,Y2: produc8923325533196201883nteger] :
      ( ( aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X1),X2) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_270_prod_Oinject,axiom,
    ! [X1: fun_Pr285225112257423215e_term,X2: produc7822875418678951345atural,Y1: fun_Pr285225112257423215e_term,Y2: produc7822875418678951345atural] :
      ( ( aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,X1),X2) = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_271_prod_Oinject,axiom,
    ! [X1: fun_Pr5098669352780829690e_term,X2: produc8923325533196201883nteger,Y1: fun_Pr5098669352780829690e_term,Y2: produc8923325533196201883nteger] :
      ( ( aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X1),X2) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_272_prod_Oinject,axiom,
    ! [X1: fun_Pr7763680713840462443e_term,X2: product_prod_int_int,Y1: fun_Pr7763680713840462443e_term,Y2: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X1),X2) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_273_prod_Oinject,axiom,
    ! [X1: fun_in5730379987311621695e_term,X2: product_prod_int_int,Y1: fun_in5730379987311621695e_term,Y2: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X1),X2) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_274_execute__return,axiom,
    ! [Xa: product_unit] : heap_T875086893843062177t_unit(aa_Pro1108561329362217960t_unit(heap_T7507251653302230130t_unit,Xa)) = comp_P3118722334806803912t_unit(some_P1914260805536162275it_nat,aTP_Lamm_aj(Xa)) ).

% execute_return
tff(fact_275_execute__return,axiom,
    ! [Xa: a] : heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(heap_Time_return_a,Xa)) = comp_P8552900652980440422t_unit(some_P7913643980934408916it_nat,aTP_Lamm_ak(Xa)) ).

% execute_return
tff(fact_276_execute__ureturn,axiom,
    ! [Xa: product_unit] : heap_T875086893843062177t_unit(aa_Pro1108561329362217960t_unit(heap_T4284346855313245393t_unit,Xa)) = comp_P3118722334806803912t_unit(some_P1914260805536162275it_nat,aTP_Lamm_al(Xa)) ).

% execute_ureturn
tff(fact_277_execute__ureturn,axiom,
    ! [Xa: a] : heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(heap_Time_ureturn_a,Xa)) = comp_P8552900652980440422t_unit(some_P7913643980934408916it_nat,aTP_Lamm_am(Xa)) ).

% execute_ureturn
tff(fact_278_prod__induct3,axiom,
    ! [Pa: fun_Pr1347628893830317649eger_o,Xa: produc7509114972854644512nteger] :
      ( ! [A2: fun_Co3460408610320954864e_term,B2: code_integer,C2: code_integer] : aa_Pro4652312591602923194eger_o(Pa,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A2),produc1086072967326762835nteger(B2,C2)))
     => aa_Pro4652312591602923194eger_o(Pa,Xa) ) ).

% prod_induct3
tff(fact_279_prod__induct3,axiom,
    ! [Pa: fun_Pr3733920118687433658ural_o,Xa: produc9140211490168418807atural] :
      ( ! [A2: fun_Pr285225112257423215e_term,B2: code_natural,C2: code_natural] : aa_Pro866224779355485219ural_o(Pa,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A2),produc3574140220909816553atural(B2,C2)))
     => aa_Pro866224779355485219ural_o(Pa,Xa) ) ).

% prod_induct3
tff(fact_280_prod__induct3,axiom,
    ! [Pa: fun_Pr7015759342434529115eger_o,Xa: produc2453742332425399446nteger] :
      ( ! [A2: fun_Pr5098669352780829690e_term,B2: code_integer,C2: code_integer] : aa_Pro4148064003102580676eger_o(Pa,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A2),produc1086072967326762835nteger(B2,C2)))
     => aa_Pro4148064003102580676eger_o(Pa,Xa) ) ).

% prod_induct3
tff(fact_281_prod__induct3,axiom,
    ! [Pa: fun_Pr3540825087398332590_int_o,Xa: produc3043004577601930755nt_int] :
      ( ! [A2: fun_Pr7763680713840462443e_term,B2: int,C2: int] : aa_Pro2601676399929874711_int_o(Pa,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B2),C2)))
     => aa_Pro2601676399929874711_int_o(Pa,Xa) ) ).

% prod_induct3
tff(fact_282_prod__induct3,axiom,
    ! [Pa: fun_Pr1196100989263358594_int_o,Xa: produc4788131091073775919nt_int] :
      ( ! [A2: fun_in5730379987311621695e_term,B2: int,C2: int] : aa_Pro5451197003417746155_int_o(Pa,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B2),C2)))
     => aa_Pro5451197003417746155_int_o(Pa,Xa) ) ).

% prod_induct3
tff(fact_283_timeFrame_Ocases,axiom,
    ! [Xa: produc4453839368661128058it_nat] :
      ( ! [N2: nat,R2: a,H4: heap_e7401611519738050253t_unit,N4: nat] : Xa != produc4111269673004989362it_nat(N2,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(R2,produc584006145561248582it_nat(H4,N4))))
     => ~ ! [N2: nat] : Xa != produc4111269673004989362it_nat(N2,none_P2651198173097904984it_nat) ) ).

% timeFrame.cases
tff(fact_284_timeFrame_Ocases,axiom,
    ! [Xa: produc3911288613690379145it_nat] :
      ( ! [N2: nat,R2: product_unit,H4: heap_e7401611519738050253t_unit,N4: nat] : Xa != produc638857205735767105it_nat(N2,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H4,N4))))
     => ~ ! [N2: nat] : Xa != produc638857205735767105it_nat(N2,none_P9117596204409417319it_nat) ) ).

% timeFrame.cases
tff(fact_285_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc7509114972854644512nteger] :
      ~ ! [A2: fun_Co3460408610320954864e_term,B2: produc8923325533196201883nteger] : Y3 != aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A2),B2) ).

% old.prod.exhaust
tff(fact_286_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc9140211490168418807atural] :
      ~ ! [A2: fun_Pr285225112257423215e_term,B2: produc7822875418678951345atural] : Y3 != aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A2),B2) ).

% old.prod.exhaust
tff(fact_287_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc2453742332425399446nteger] :
      ~ ! [A2: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] : Y3 != aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A2),B2) ).

% old.prod.exhaust
tff(fact_288_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc3043004577601930755nt_int] :
      ~ ! [A2: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] : Y3 != aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A2),B2) ).

% old.prod.exhaust
tff(fact_289_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc4788131091073775919nt_int] :
      ~ ! [A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] : Y3 != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A2),B2) ).

% old.prod.exhaust
tff(fact_290_surj__pair,axiom,
    ! [P: produc7509114972854644512nteger] :
    ? [X4: fun_Co3460408610320954864e_term,Y4: produc8923325533196201883nteger] : P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X4),Y4) ).

% surj_pair
tff(fact_291_surj__pair,axiom,
    ! [P: produc9140211490168418807atural] :
    ? [X4: fun_Pr285225112257423215e_term,Y4: produc7822875418678951345atural] : P = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,X4),Y4) ).

% surj_pair
tff(fact_292_surj__pair,axiom,
    ! [P: produc2453742332425399446nteger] :
    ? [X4: fun_Pr5098669352780829690e_term,Y4: produc8923325533196201883nteger] : P = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X4),Y4) ).

% surj_pair
tff(fact_293_surj__pair,axiom,
    ! [P: produc3043004577601930755nt_int] :
    ? [X4: fun_Pr7763680713840462443e_term,Y4: product_prod_int_int] : P = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Y4) ).

% surj_pair
tff(fact_294_surj__pair,axiom,
    ! [P: produc4788131091073775919nt_int] :
    ? [X4: fun_in5730379987311621695e_term,Y4: product_prod_int_int] : P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y4) ).

% surj_pair
tff(fact_295_prod__cases,axiom,
    ! [Pa: fun_Pr1347628893830317649eger_o,P: produc7509114972854644512nteger] :
      ( ! [A2: fun_Co3460408610320954864e_term,B2: produc8923325533196201883nteger] : aa_Pro4652312591602923194eger_o(Pa,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A2),B2))
     => aa_Pro4652312591602923194eger_o(Pa,P) ) ).

% prod_cases
tff(fact_296_prod__cases,axiom,
    ! [Pa: fun_Pr3733920118687433658ural_o,P: produc9140211490168418807atural] :
      ( ! [A2: fun_Pr285225112257423215e_term,B2: produc7822875418678951345atural] : aa_Pro866224779355485219ural_o(Pa,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A2),B2))
     => aa_Pro866224779355485219ural_o(Pa,P) ) ).

% prod_cases
tff(fact_297_prod__cases,axiom,
    ! [Pa: fun_Pr7015759342434529115eger_o,P: produc2453742332425399446nteger] :
      ( ! [A2: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] : aa_Pro4148064003102580676eger_o(Pa,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A2),B2))
     => aa_Pro4148064003102580676eger_o(Pa,P) ) ).

% prod_cases
tff(fact_298_prod__cases,axiom,
    ! [Pa: fun_Pr3540825087398332590_int_o,P: produc3043004577601930755nt_int] :
      ( ! [A2: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] : aa_Pro2601676399929874711_int_o(Pa,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A2),B2))
     => aa_Pro2601676399929874711_int_o(Pa,P) ) ).

% prod_cases
tff(fact_299_prod__cases,axiom,
    ! [Pa: fun_Pr1196100989263358594_int_o,P: produc4788131091073775919nt_int] :
      ( ! [A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] : aa_Pro5451197003417746155_int_o(Pa,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A2),B2))
     => aa_Pro5451197003417746155_int_o(Pa,P) ) ).

% prod_cases
tff(fact_300_Pair__inject,axiom,
    ! [A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger,A4: fun_Co3460408610320954864e_term,B4: produc8923325533196201883nteger] :
      ( ( aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_301_Pair__inject,axiom,
    ! [A: fun_Pr285225112257423215e_term,B: produc7822875418678951345atural,A4: fun_Pr285225112257423215e_term,B4: produc7822875418678951345atural] :
      ( ( aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A),B) = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_302_Pair__inject,axiom,
    ! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,A4: fun_Pr5098669352780829690e_term,B4: produc8923325533196201883nteger] :
      ( ( aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_303_Pair__inject,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A4: fun_Pr7763680713840462443e_term,B4: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_304_Pair__inject,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_305_comp__def,axiom,
    ! [F: fun_char_char,G: fun_Co8487158583234504619r_char,X: code_integer] : aa_Code_integer_char(comp_c4562247934908617188nteger(F,G),X) = aa_char_char(F,aa_Code_integer_char(G,X)) ).

% comp_def
tff(fact_306_comp__def,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,X: int] : aa_int_num(comp_nat_num_int(F,G),X) = aa_nat_num(F,aa_int_nat(G,X)) ).

% comp_def
tff(fact_307_comp__def,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,X: nat] : aa_nat_nat(comp_nat_nat_nat(F,G),X) = aa_nat_nat(F,aa_nat_nat(G,X)) ).

% comp_def
tff(fact_308_comp__def,axiom,
    ! [F: fun_int_nat,G: fun_int_int,X: int] : aa_int_nat(comp_int_nat_int(F,G),X) = aa_int_nat(F,aa_int_int(G,X)) ).

% comp_def
tff(fact_309_comp__def,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,X: code_integer] : aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(F,G),X) = aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,X)) ).

% comp_def
tff(fact_310_comp__assoc,axiom,
    ! [F: fun_char_char,G: fun_Co8487158583234504619r_char,Ha: fun_Co8499238673658658103nteger] : comp_C3856975138083468656nteger(comp_c4562247934908617188nteger(F,G),Ha) = comp_c4562247934908617188nteger(F,comp_C3856975138083468656nteger(G,Ha)) ).

% comp_assoc
tff(fact_311_comp__assoc,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,Ha: fun_int_int] : comp_int_num_int(comp_nat_num_int(F,G),Ha) = comp_nat_num_int(F,comp_int_nat_int(G,Ha)) ).

% comp_assoc
tff(fact_312_comp__assoc,axiom,
    ! [F: fun_char_char,G: fun_char_char,Ha: fun_Co8487158583234504619r_char] : comp_c4562247934908617188nteger(comp_char_char_char(F,G),Ha) = comp_c4562247934908617188nteger(F,comp_c4562247934908617188nteger(G,Ha)) ).

% comp_assoc
tff(fact_313_comp__assoc,axiom,
    ! [F: fun_num_num,G: fun_nat_num,Ha: fun_int_nat] : comp_nat_num_int(comp_num_num_nat(F,G),Ha) = comp_num_num_int(F,comp_nat_num_int(G,Ha)) ).

% comp_assoc
tff(fact_314_comp__assoc,axiom,
    ! [F: fun_nat_num,G: fun_nat_nat,Ha: fun_int_nat] : comp_nat_num_int(comp_nat_num_nat(F,G),Ha) = comp_nat_num_int(F,comp_nat_nat_int(G,Ha)) ).

% comp_assoc
tff(fact_315_comp__assoc,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Ha: fun_nat_nat] : comp_nat_nat_nat(comp_nat_nat_nat(F,G),Ha) = comp_nat_nat_nat(F,comp_nat_nat_nat(G,Ha)) ).

% comp_assoc
tff(fact_316_comp__assoc,axiom,
    ! [F: fun_nat_nat,G: fun_int_nat,Ha: fun_int_int] : comp_int_nat_int(comp_nat_nat_int(F,G),Ha) = comp_nat_nat_int(F,comp_int_nat_int(G,Ha)) ).

% comp_assoc
tff(fact_317_comp__assoc,axiom,
    ! [F: fun_int_nat,G: fun_int_int,Ha: fun_int_int] : comp_int_nat_int(comp_int_nat_int(F,G),Ha) = comp_int_nat_int(F,comp_int_int_int(G,Ha)) ).

% comp_assoc
tff(fact_318_comp__assoc,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,Ha: fun_Co8499238673658658103nteger] : comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(F,G),Ha) = comp_f6739083541385432444nteger(F,comp_C7233611567195229654nteger(G,Ha)) ).

% comp_assoc
tff(fact_319_comp__assoc,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_fu1763652423599918571nteger,Ha: fun_Co1981900195452921617nteger] : comp_f6739083541385432444nteger(comp_f3848972703483044950nteger(F,G),Ha) = comp_f6739083541385432444nteger(F,comp_f228016600672082096nteger(G,Ha)) ).

% comp_assoc
tff(fact_320_comp__eq__dest,axiom,
    ! [A: fun_char_char,B: fun_Co8487158583234504619r_char,C: fun_char_char,D: fun_Co8487158583234504619r_char,V: code_integer] :
      ( ( comp_c4562247934908617188nteger(A,B) = comp_c4562247934908617188nteger(C,D) )
     => ( aa_char_char(A,aa_Code_integer_char(B,V)) = aa_char_char(C,aa_Code_integer_char(D,V)) ) ) ).

% comp_eq_dest
tff(fact_321_comp__eq__dest,axiom,
    ! [A: fun_nat_num,B: fun_int_nat,C: fun_nat_num,D: fun_int_nat,V: int] :
      ( ( comp_nat_num_int(A,B) = comp_nat_num_int(C,D) )
     => ( aa_nat_num(A,aa_int_nat(B,V)) = aa_nat_num(C,aa_int_nat(D,V)) ) ) ).

% comp_eq_dest
tff(fact_322_comp__eq__dest,axiom,
    ! [A: fun_nat_nat,B: fun_nat_nat,C: fun_nat_nat,D: fun_nat_nat,V: nat] :
      ( ( comp_nat_nat_nat(A,B) = comp_nat_nat_nat(C,D) )
     => ( aa_nat_nat(A,aa_nat_nat(B,V)) = aa_nat_nat(C,aa_nat_nat(D,V)) ) ) ).

% comp_eq_dest
tff(fact_323_comp__eq__dest,axiom,
    ! [A: fun_int_nat,B: fun_int_int,C: fun_int_nat,D: fun_int_int,V: int] :
      ( ( comp_int_nat_int(A,B) = comp_int_nat_int(C,D) )
     => ( aa_int_nat(A,aa_int_int(B,V)) = aa_int_nat(C,aa_int_int(D,V)) ) ) ).

% comp_eq_dest
tff(fact_324_comp__eq__dest,axiom,
    ! [A: fun_fu4800041728187187127nteger,B: fun_Co1981900195452921617nteger,C: fun_fu4800041728187187127nteger,D: fun_Co1981900195452921617nteger,V: code_integer] :
      ( ( comp_f6739083541385432444nteger(A,B) = comp_f6739083541385432444nteger(C,D) )
     => ( aa_fun2509385170507305080nteger(A,aa_Cod4161879620478718930nteger(B,V)) = aa_fun2509385170507305080nteger(C,aa_Cod4161879620478718930nteger(D,V)) ) ) ).

% comp_eq_dest
tff(fact_325_comp__eq__elim,axiom,
    ! [A: fun_char_char,B: fun_Co8487158583234504619r_char,C: fun_char_char,D: fun_Co8487158583234504619r_char] :
      ( ( comp_c4562247934908617188nteger(A,B) = comp_c4562247934908617188nteger(C,D) )
     => ! [V2: code_integer] : aa_char_char(A,aa_Code_integer_char(B,V2)) = aa_char_char(C,aa_Code_integer_char(D,V2)) ) ).

% comp_eq_elim
tff(fact_326_comp__eq__elim,axiom,
    ! [A: fun_nat_num,B: fun_int_nat,C: fun_nat_num,D: fun_int_nat] :
      ( ( comp_nat_num_int(A,B) = comp_nat_num_int(C,D) )
     => ! [V2: int] : aa_nat_num(A,aa_int_nat(B,V2)) = aa_nat_num(C,aa_int_nat(D,V2)) ) ).

% comp_eq_elim
tff(fact_327_comp__eq__elim,axiom,
    ! [A: fun_nat_nat,B: fun_nat_nat,C: fun_nat_nat,D: fun_nat_nat] :
      ( ( comp_nat_nat_nat(A,B) = comp_nat_nat_nat(C,D) )
     => ! [V2: nat] : aa_nat_nat(A,aa_nat_nat(B,V2)) = aa_nat_nat(C,aa_nat_nat(D,V2)) ) ).

% comp_eq_elim
tff(fact_328_comp__eq__elim,axiom,
    ! [A: fun_int_nat,B: fun_int_int,C: fun_int_nat,D: fun_int_int] :
      ( ( comp_int_nat_int(A,B) = comp_int_nat_int(C,D) )
     => ! [V2: int] : aa_int_nat(A,aa_int_int(B,V2)) = aa_int_nat(C,aa_int_int(D,V2)) ) ).

% comp_eq_elim
tff(fact_329_comp__eq__elim,axiom,
    ! [A: fun_fu4800041728187187127nteger,B: fun_Co1981900195452921617nteger,C: fun_fu4800041728187187127nteger,D: fun_Co1981900195452921617nteger] :
      ( ( comp_f6739083541385432444nteger(A,B) = comp_f6739083541385432444nteger(C,D) )
     => ! [V2: code_integer] : aa_fun2509385170507305080nteger(A,aa_Cod4161879620478718930nteger(B,V2)) = aa_fun2509385170507305080nteger(C,aa_Cod4161879620478718930nteger(D,V2)) ) ).

% comp_eq_elim
tff(fact_330_comp__eq__dest__lhs,axiom,
    ! [A: fun_char_char,B: fun_Co8487158583234504619r_char,C: fun_Co8487158583234504619r_char,V: code_integer] :
      ( ( comp_c4562247934908617188nteger(A,B) = C )
     => ( aa_char_char(A,aa_Code_integer_char(B,V)) = aa_Code_integer_char(C,V) ) ) ).

% comp_eq_dest_lhs
tff(fact_331_comp__eq__dest__lhs,axiom,
    ! [A: fun_nat_num,B: fun_int_nat,C: fun_int_num,V: int] :
      ( ( comp_nat_num_int(A,B) = C )
     => ( aa_nat_num(A,aa_int_nat(B,V)) = aa_int_num(C,V) ) ) ).

% comp_eq_dest_lhs
tff(fact_332_comp__eq__dest__lhs,axiom,
    ! [A: fun_nat_nat,B: fun_nat_nat,C: fun_nat_nat,V: nat] :
      ( ( comp_nat_nat_nat(A,B) = C )
     => ( aa_nat_nat(A,aa_nat_nat(B,V)) = aa_nat_nat(C,V) ) ) ).

% comp_eq_dest_lhs
tff(fact_333_comp__eq__dest__lhs,axiom,
    ! [A: fun_int_nat,B: fun_int_int,C: fun_int_nat,V: int] :
      ( ( comp_int_nat_int(A,B) = C )
     => ( aa_int_nat(A,aa_int_int(B,V)) = aa_int_nat(C,V) ) ) ).

% comp_eq_dest_lhs
tff(fact_334_comp__eq__dest__lhs,axiom,
    ! [A: fun_fu4800041728187187127nteger,B: fun_Co1981900195452921617nteger,C: fun_Co1561183069634336733nteger,V: code_integer] :
      ( ( comp_f6739083541385432444nteger(A,B) = C )
     => ( aa_fun2509385170507305080nteger(A,aa_Cod4161879620478718930nteger(B,V)) = aa_Cod1978467581589928094nteger(C,V) ) ) ).

% comp_eq_dest_lhs
tff(fact_335_effectI,axiom,
    ! [C: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,R: product_unit,H: heap_e7401611519738050253t_unit,N: nat] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(C),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(H,N))) )
     => aa_nat_o(aa_Pro847370707247602192_nat_o(heap_T6553295506729943825t_unit(C,Ha,H),R),N) ) ).

% effectI
tff(fact_336_effectI,axiom,
    ! [C: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,R: a,H: heap_e7401611519738050253t_unit,N: nat] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(C),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(R,produc584006145561248582it_nat(H,N))) )
     => aa_nat_o(heap_Time_effect_a(C,Ha,H,R),N) ) ).

% effectI
tff(fact_337_effect__def,axiom,
    ! [C: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: product_unit,N: nat] :
      ( aa_nat_o(aa_Pro847370707247602192_nat_o(heap_T6553295506729943825t_unit(C,Ha,H),R),N)
    <=> ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(C),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(H,N))) ) ) ).

% effect_def
tff(fact_338_effect__def,axiom,
    ! [C: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: a,N: nat] :
      ( aa_nat_o(heap_Time_effect_a(C,Ha,H,R),N)
    <=> ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(C),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(R,produc584006145561248582it_nat(H,N))) ) ) ).

% effect_def
tff(fact_339_ureturn__def,axiom,
    ! [Xa: a] : aa_a_H8829438773920479562Heap_a(heap_Time_ureturn_a,Xa) = heap_Time_heap_a(aTP_Lamm_am(Xa)) ).

% ureturn_def
tff(fact_340_return__def,axiom,
    ! [Xa: a] : aa_a_H8829438773920479562Heap_a(heap_Time_return_a,Xa) = heap_Time_heap_a(aTP_Lamm_ak(Xa)) ).

% return_def
tff(fact_341_execute__bind__eq__SomeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,Xa: product_unit,H: heap_e7401611519738050253t_unit,N: nat,G: fun_Pr292408699144966967t_unit,Y3: product_unit,H3: heap_e7401611519738050253t_unit,N3: nat] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(F),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Xa,produc584006145561248582it_nat(H,N))) )
     => ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_Pro1108561329362217960t_unit(G,Xa)),H) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y3,produc584006145561248582it_nat(H3,N3))) )
       => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(heap_T2633723481742716231t_unit(F,G)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y3,produc584006145561248582it_nat(H3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N3)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_342_execute__bind__eq__SomeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,Xa: product_unit,H: heap_e7401611519738050253t_unit,N: nat,G: fun_Pr5215307506330550056Heap_a,Y3: a,H3: heap_e7401611519738050253t_unit,N3: nat] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(F),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Xa,produc584006145561248582it_nat(H,N))) )
     => ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_Pro7886978635576167513Heap_a(G,Xa)),H) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(Y3,produc584006145561248582it_nat(H3,N3))) )
       => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_T757603679106148408unit_a(F,G)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(Y3,produc584006145561248582it_nat(H3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N3)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_343_execute__bind__eq__SomeI,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,Xa: a,H: heap_e7401611519738050253t_unit,N: nat,G: fun_a_2990225006182978344t_unit,Y3: product_unit,H3: heap_e7401611519738050253t_unit,N3: nat] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(Xa,produc584006145561248582it_nat(H,N))) )
     => ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_a_H4862568901524997081t_unit(G,Xa)),H) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y3,produc584006145561248582it_nat(H3,N3))) )
       => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(heap_T8685611227969916822t_unit(F,G)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y3,produc584006145561248582it_nat(H3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N3)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_344_execute__bind__eq__SomeI,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,Xa: a,H: heap_e7401611519738050253t_unit,N: nat,G: fun_a_6093291656547882009Heap_a,Y3: a,H3: heap_e7401611519738050253t_unit,N3: nat] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(Xa,produc584006145561248582it_nat(H,N))) )
     => ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(G,Xa)),H) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(Y3,produc584006145561248582it_nat(H3,N3))) )
       => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_a_a(F,G)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(Y3,produc584006145561248582it_nat(H3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N3)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_345_success__bind__executeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,Xa: product_unit,H: heap_e7401611519738050253t_unit,N: nat,G: fun_Pr5215307506330550056Heap_a] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(F),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Xa,produc584006145561248582it_nat(H,N))) )
     => ( aa_hea7994211679605476167unit_o(heap_Time_success_a(aa_Pro7886978635576167513Heap_a(G,Xa)),H)
       => aa_hea7994211679605476167unit_o(heap_Time_success_a(heap_T757603679106148408unit_a(F,G)),Ha) ) ) ).

% success_bind_executeI
tff(fact_346_success__bind__executeI,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,Xa: a,H: heap_e7401611519738050253t_unit,N: nat,G: fun_a_6093291656547882009Heap_a] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(Xa,produc584006145561248582it_nat(H,N))) )
     => ( aa_hea7994211679605476167unit_o(heap_Time_success_a(aa_a_H8829438773920479562Heap_a(G,Xa)),H)
       => aa_hea7994211679605476167unit_o(heap_Time_success_a(heap_Time_bind_a_a(F,G)),Ha) ) ) ).

% success_bind_executeI
tff(fact_347_execute__tap,axiom,
    ! [F: fun_he5123621388056232515t_unit,Ha: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(heap_T560649228465745139t_unit(F)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(aa_hea4146273921685161788t_unit(F,Ha),produc584006145561248582it_nat(Ha,one_one_nat))) ).

% execute_tap
tff(fact_348_execute__tap,axiom,
    ! [F: fun_he6342020308413958580unit_a,Ha: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_tap_a(F)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(aa_hea5100639427662539565unit_a(F,Ha),produc584006145561248582it_nat(Ha,one_one_nat))) ).

% execute_tap
tff(fact_349_execute__assert_I1_J,axiom,
    ! [Pa: fun_Product_unit_o,Xa: product_unit,Ha: heap_e7401611519738050253t_unit] :
      ( aa_Product_unit_o(Pa,Xa)
     => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_Pro1108561329362217960t_unit(heap_T4208721593536448476t_unit(Pa),Xa)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Xa,produc584006145561248582it_nat(Ha,one_one_nat))) ) ) ).

% execute_assert(1)
tff(fact_350_execute__assert_I1_J,axiom,
    ! [Pa: fun_a_o,Xa: a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_a_o(Pa,Xa)
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(heap_Time_assert_a(Pa),Xa)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(Xa,produc584006145561248582it_nat(Ha,one_one_nat))) ) ) ).

% execute_assert(1)
tff(fact_351_tap__def,axiom,
    ! [F: fun_he6342020308413958580unit_a] : heap_Time_tap_a(F) = heap_Time_Heap_a2(aTP_Lamm_an(F)) ).

% tap_def
tff(fact_352_tap__def,axiom,
    ! [F: fun_he5123621388056232515t_unit] : heap_T560649228465745139t_unit(F) = heap_T6183433275982383450t_unit(aTP_Lamm_ao(F)) ).

% tap_def
tff(fact_353_prod__cases3,axiom,
    ! [Y3: produc7509114972854644512nteger] :
      ~ ! [A2: fun_Co3460408610320954864e_term,B2: code_integer,C2: code_integer] : Y3 != aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A2),produc1086072967326762835nteger(B2,C2)) ).

% prod_cases3
tff(fact_354_prod__cases3,axiom,
    ! [Y3: produc9140211490168418807atural] :
      ~ ! [A2: fun_Pr285225112257423215e_term,B2: code_natural,C2: code_natural] : Y3 != aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A2),produc3574140220909816553atural(B2,C2)) ).

% prod_cases3
tff(fact_355_prod__cases3,axiom,
    ! [Y3: produc2453742332425399446nteger] :
      ~ ! [A2: fun_Pr5098669352780829690e_term,B2: code_integer,C2: code_integer] : Y3 != aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A2),produc1086072967326762835nteger(B2,C2)) ).

% prod_cases3
tff(fact_356_prod__cases3,axiom,
    ! [Y3: produc3043004577601930755nt_int] :
      ~ ! [A2: fun_Pr7763680713840462443e_term,B2: int,C2: int] : Y3 != aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B2),C2)) ).

% prod_cases3
tff(fact_357_prod__cases3,axiom,
    ! [Y3: produc4788131091073775919nt_int] :
      ~ ! [A2: fun_in5730379987311621695e_term,B2: int,C2: int] : Y3 != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B2),C2)) ).

% prod_cases3
tff(fact_358_execute__bind_I1_J,axiom,
    ! [F: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,Xa: product_unit,H: heap_e7401611519738050253t_unit,N: nat,G: fun_Pr5215307506330550056Heap_a] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(F),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Xa,produc584006145561248582it_nat(H,N))) )
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_T757603679106148408unit_a(F,G)),Ha) = heap_T7616092557645711335rame_a(N,aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_Pro7886978635576167513Heap_a(G,Xa)),H)) ) ) ).

% execute_bind(1)
tff(fact_359_execute__bind_I1_J,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,Xa: a,H: heap_e7401611519738050253t_unit,N: nat,G: fun_a_6093291656547882009Heap_a] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(Xa,produc584006145561248582it_nat(H,N))) )
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_a_a(F,G)),Ha) = heap_T7616092557645711335rame_a(N,aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(G,Xa)),H)) ) ) ).

% execute_bind(1)
tff(fact_360_timeFrame_Oelims,axiom,
    ! [Xa: nat,Xaa: option3562590408128118217it_nat,Y3: option3562590408128118217it_nat] :
      ( ( heap_T7616092557645711335rame_a(Xa,Xaa) = Y3 )
     => ( ! [R2: a,H4: heap_e7401611519738050253t_unit,N4: nat] :
            ( ( Xaa = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(R2,produc584006145561248582it_nat(H4,N4))) )
           => ( Y3 != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(R2,produc584006145561248582it_nat(H4,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),N4)))) ) )
       => ~ ( ( Xaa = none_P2651198173097904984it_nat )
           => ( Y3 != none_P2651198173097904984it_nat ) ) ) ) ).

% timeFrame.elims
tff(fact_361_timeFrame_Oelims,axiom,
    ! [Xa: nat,Xaa: option8956607266484857688it_nat,Y3: option8956607266484857688it_nat] :
      ( ( heap_T3616969660504097270t_unit(Xa,Xaa) = Y3 )
     => ( ! [R2: product_unit,H4: heap_e7401611519738050253t_unit,N4: nat] :
            ( ( Xaa = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H4,N4))) )
           => ( Y3 != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H4,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),N4)))) ) )
       => ~ ( ( Xaa = none_P9117596204409417319it_nat )
           => ( Y3 != none_P9117596204409417319it_nat ) ) ) ) ).

% timeFrame.elims
tff(fact_362_timeFrame_Osimps_I1_J,axiom,
    ! [N: nat,R: product_unit,Ha: heap_e7401611519738050253t_unit,N3: nat] : heap_T3616969660504097270t_unit(N,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(Ha,N3)))) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N3)))) ).

% timeFrame.simps(1)
tff(fact_363_K__record__comp,axiom,
    ! [C: char,F: fun_Co8487158583234504619r_char,X: code_integer] : aa_Code_integer_char(comp_c4562247934908617188nteger(aTP_Lamm_ap(C),F),X) = C ).

% K_record_comp
tff(fact_364_K__record__comp,axiom,
    ! [C: num,F: fun_int_nat,X: int] : aa_int_num(comp_nat_num_int(aTP_Lamm_aq(C),F),X) = C ).

% K_record_comp
tff(fact_365_K__record__comp,axiom,
    ! [C: nat,F: fun_nat_nat,X: nat] : aa_nat_nat(comp_nat_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ar,C),F),X) = C ).

% K_record_comp
tff(fact_366_K__record__comp,axiom,
    ! [C: nat,F: fun_int_int,X: int] : aa_int_nat(comp_int_nat_int(aa_nat_fun_int_nat(aTP_Lamm_as,C),F),X) = C ).

% K_record_comp
tff(fact_367_K__record__comp,axiom,
    ! [C: fun_Pr5876161306168176259nteger,F: fun_Co1981900195452921617nteger,X: code_integer] : aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(aTP_Lamm_at(C),F),X) = C ).

% K_record_comp
tff(fact_368_fun_Omap__ident,axiom,
    ! [T: fun_Co8487158583234504619r_char] : comp_c4562247934908617188nteger(aTP_Lamm_au,T) = T ).

% fun.map_ident
tff(fact_369_fun_Omap__ident,axiom,
    ! [T: fun_nat_nat] : comp_nat_nat_nat(aTP_Lamm_av,T) = T ).

% fun.map_ident
tff(fact_370_pred__equals__eq2,axiom,
    ! [R4: set_Pr1200682188532975830nteger,S2: set_Pr1200682188532975830nteger] :
      ( ! [X3: fun_Co3460408610320954864e_term,Xa2: produc8923325533196201883nteger] :
          ( member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X3),Xa2),R4)
        <=> member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X3),Xa2),S2) )
    <=> ( R4 = S2 ) ) ).

% pred_equals_eq2
tff(fact_371_pred__equals__eq2,axiom,
    ! [R4: set_Pr620054291978045485atural,S2: set_Pr620054291978045485atural] :
      ( ! [X3: fun_Pr285225112257423215e_term,Xa2: produc7822875418678951345atural] :
          ( member6441364043385305230atural(aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,X3),Xa2),R4)
        <=> member6441364043385305230atural(aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,X3),Xa2),S2) )
    <=> ( R4 = S2 ) ) ).

% pred_equals_eq2
tff(fact_372_pred__equals__eq2,axiom,
    ! [R4: set_Pr3486312109254157132nteger,S2: set_Pr3486312109254157132nteger] :
      ( ! [X3: fun_Pr5098669352780829690e_term,Xa2: produc8923325533196201883nteger] :
          ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X3),Xa2),R4)
        <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X3),Xa2),S2) )
    <=> ( R4 = S2 ) ) ).

% pred_equals_eq2
tff(fact_373_pred__equals__eq2,axiom,
    ! [R4: set_Pr6700895561065273913nt_int,S2: set_Pr6700895561065273913nt_int] :
      ( ! [X3: fun_Pr7763680713840462443e_term,Xa2: product_prod_int_int] :
          ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Xa2),R4)
        <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Xa2),S2) )
    <=> ( R4 = S2 ) ) ).

% pred_equals_eq2
tff(fact_374_pred__equals__eq2,axiom,
    ! [R4: set_Pr4166680992137330533nt_int,S2: set_Pr4166680992137330533nt_int] :
      ( ! [X3: fun_in5730379987311621695e_term,Xa2: product_prod_int_int] :
          ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Xa2),R4)
        <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Xa2),S2) )
    <=> ( R4 = S2 ) ) ).

% pred_equals_eq2
tff(fact_375_divides__aux__eq,axiom,
    ! [Q2: nat,R: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R))
    <=> ( R = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_376_divides__aux__eq,axiom,
    ! [Q2: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
    <=> ( R = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_377_divides__aux__eq,axiom,
    ! [Q2: code_integer,R: code_integer] :
      ( aa_Pro4926521453222654329eger_o(unique5706413561485394159nteger,produc1086072967326762835nteger(Q2,R))
    <=> ( R = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_378_wait__def,axiom,
    ! [N: nat] : heap_Time_wait(N) = heap_T6183433275982383450t_unit(aTP_Lamm_aw(N)) ).

% wait_def
tff(fact_379_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_380_fun_Omap__comp,axiom,
    ! [G: fun_num_num,F: fun_nat_num,V: fun_int_nat] : comp_num_num_int(G,comp_nat_num_int(F,V)) = comp_nat_num_int(comp_num_num_nat(G,F),V) ).

% fun.map_comp
tff(fact_381_fun_Omap__comp,axiom,
    ! [G: fun_nat_nat,F: fun_int_nat,V: fun_int_int] : comp_nat_nat_int(G,comp_int_nat_int(F,V)) = comp_int_nat_int(comp_nat_nat_int(G,F),V) ).

% fun.map_comp
tff(fact_382_fun_Omap__comp,axiom,
    ! [G: fun_char_char,F: fun_Co8487158583234504619r_char,V: fun_Co8499238673658658103nteger] : comp_c4562247934908617188nteger(G,comp_C3856975138083468656nteger(F,V)) = comp_C3856975138083468656nteger(comp_c4562247934908617188nteger(G,F),V) ).

% fun.map_comp
tff(fact_383_fun_Omap__comp,axiom,
    ! [G: fun_char_char,F: fun_char_char,V: fun_Co8487158583234504619r_char] : comp_c4562247934908617188nteger(G,comp_c4562247934908617188nteger(F,V)) = comp_c4562247934908617188nteger(comp_char_char_char(G,F),V) ).

% fun.map_comp
tff(fact_384_fun_Omap__comp,axiom,
    ! [G: fun_nat_num,F: fun_nat_nat,V: fun_int_nat] : comp_nat_num_int(G,comp_nat_nat_int(F,V)) = comp_nat_num_int(comp_nat_num_nat(G,F),V) ).

% fun.map_comp
tff(fact_385_fun_Omap__comp,axiom,
    ! [G: fun_nat_num,F: fun_int_nat,V: fun_int_int] : comp_nat_num_int(G,comp_int_nat_int(F,V)) = comp_int_num_int(comp_nat_num_int(G,F),V) ).

% fun.map_comp
tff(fact_386_fun_Omap__comp,axiom,
    ! [G: fun_nat_nat,F: fun_nat_nat,V: fun_nat_nat] : comp_nat_nat_nat(G,comp_nat_nat_nat(F,V)) = comp_nat_nat_nat(comp_nat_nat_nat(G,F),V) ).

% fun.map_comp
tff(fact_387_fun_Omap__comp,axiom,
    ! [G: fun_int_nat,F: fun_int_int,V: fun_int_int] : comp_int_nat_int(G,comp_int_int_int(F,V)) = comp_int_nat_int(comp_int_nat_int(G,F),V) ).

% fun.map_comp
tff(fact_388_fun_Omap__comp,axiom,
    ! [G: fun_fu4800041728187187127nteger,F: fun_Co1981900195452921617nteger,V: fun_Co8499238673658658103nteger] : comp_f6739083541385432444nteger(G,comp_C7233611567195229654nteger(F,V)) = comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(G,F),V) ).

% fun.map_comp
tff(fact_389_fun_Omap__comp,axiom,
    ! [G: fun_fu4800041728187187127nteger,F: fun_fu1763652423599918571nteger,V: fun_Co1981900195452921617nteger] : comp_f6739083541385432444nteger(G,comp_f228016600672082096nteger(F,V)) = comp_f6739083541385432444nteger(comp_f3848972703483044950nteger(G,F),V) ).

% fun.map_comp
tff(fact_390_old_Ounit_Oexhaust,axiom,
    ! [Y3: product_unit] : Y3 = product_Unity ).

% old.unit.exhaust
tff(fact_391_unit__all__impI,axiom,
    ! [Pa: fun_Product_unit_o,Q: fun_Product_unit_o] :
      ( ( aa_Product_unit_o(Pa,product_Unity)
       => aa_Product_unit_o(Q,product_Unity) )
     => ! [X: product_unit] :
          ( aa_Product_unit_o(Pa,X)
         => aa_Product_unit_o(Q,X) ) ) ).

% unit_all_impI
tff(fact_392_eq__sym__Unity__conv,axiom,
    ! [Xa: $o] :
      ( ( (Xa)
      <=> ( product_Unity = product_Unity ) )
    <=> (Xa) ) ).

% eq_sym_Unity_conv
tff(fact_393_timeFrame_Osimps_I2_J,axiom,
    ! [N: nat] : heap_T7616092557645711335rame_a(N,none_P2651198173097904984it_nat) = none_P2651198173097904984it_nat ).

% timeFrame.simps(2)
tff(fact_394_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_395_default__unit__def,axiom,
    defaul566961228789861419t_unit = product_Unity ).

% default_unit_def
tff(fact_396_timeFrame_Opelims,axiom,
    ! [Xa: nat,Xaa: option3562590408128118217it_nat,Y3: option3562590408128118217it_nat] :
      ( ( heap_T7616092557645711335rame_a(Xa,Xaa) = Y3 )
     => ( accp_P6263092265436569219it_nat(heap_T8132184524487034138_rel_a,produc4111269673004989362it_nat(Xa,Xaa))
       => ( ! [R2: a,H4: heap_e7401611519738050253t_unit,N4: nat] :
              ( ( Xaa = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(R2,produc584006145561248582it_nat(H4,N4))) )
             => ( ( Y3 = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(R2,produc584006145561248582it_nat(H4,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),N4)))) )
               => ~ accp_P6263092265436569219it_nat(heap_T8132184524487034138_rel_a,produc4111269673004989362it_nat(Xa,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(R2,produc584006145561248582it_nat(H4,N4))))) ) )
         => ~ ( ( Xaa = none_P2651198173097904984it_nat )
             => ( ( Y3 = none_P2651198173097904984it_nat )
               => ~ accp_P6263092265436569219it_nat(heap_T8132184524487034138_rel_a,produc4111269673004989362it_nat(Xa,none_P2651198173097904984it_nat)) ) ) ) ) ) ).

% timeFrame.pelims
tff(fact_397_timeFrame_Opelims,axiom,
    ! [Xa: nat,Xaa: option8956607266484857688it_nat,Y3: option8956607266484857688it_nat] :
      ( ( heap_T3616969660504097270t_unit(Xa,Xaa) = Y3 )
     => ( accp_P414730952086964626it_nat(heap_T996182799752388649t_unit,produc638857205735767105it_nat(Xa,Xaa))
       => ( ! [R2: product_unit,H4: heap_e7401611519738050253t_unit,N4: nat] :
              ( ( Xaa = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H4,N4))) )
             => ( ( Y3 = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H4,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),N4)))) )
               => ~ accp_P414730952086964626it_nat(heap_T996182799752388649t_unit,produc638857205735767105it_nat(Xa,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H4,N4))))) ) )
         => ~ ( ( Xaa = none_P9117596204409417319it_nat )
             => ( ( Y3 = none_P9117596204409417319it_nat )
               => ~ accp_P414730952086964626it_nat(heap_T996182799752388649t_unit,produc638857205735767105it_nat(Xa,none_P9117596204409417319it_nat)) ) ) ) ) ) ).

% timeFrame.pelims
tff(fact_398_type__copy__map__cong0,axiom,
    ! [M2: fun_Co8487158583234504619r_char,G: fun_Co8499238673658658103nteger,Xa: code_integer,N5: fun_char_char,Ha: fun_Co8487158583234504619r_char,F: fun_char_char] :
      ( ( aa_Code_integer_char(M2,aa_Cod1783974652533396728nteger(G,Xa)) = aa_char_char(N5,aa_Code_integer_char(Ha,Xa)) )
     => ( aa_Code_integer_char(comp_C3856975138083468656nteger(comp_c4562247934908617188nteger(F,M2),G),Xa) = aa_Code_integer_char(comp_c4562247934908617188nteger(comp_char_char_char(F,N5),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_399_type__copy__map__cong0,axiom,
    ! [M2: fun_int_nat,G: fun_int_int,Xa: int,N5: fun_nat_nat,Ha: fun_int_nat,F: fun_nat_num] :
      ( ( aa_int_nat(M2,aa_int_int(G,Xa)) = aa_nat_nat(N5,aa_int_nat(Ha,Xa)) )
     => ( aa_int_num(comp_int_num_int(comp_nat_num_int(F,M2),G),Xa) = aa_int_num(comp_nat_num_int(comp_nat_num_nat(F,N5),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_400_type__copy__map__cong0,axiom,
    ! [M2: fun_nat_nat,G: fun_int_nat,Xa: int,N5: fun_int_nat,Ha: fun_int_int,F: fun_nat_nat] :
      ( ( aa_nat_nat(M2,aa_int_nat(G,Xa)) = aa_int_nat(N5,aa_int_int(Ha,Xa)) )
     => ( aa_int_nat(comp_nat_nat_int(comp_nat_nat_nat(F,M2),G),Xa) = aa_int_nat(comp_int_nat_int(comp_nat_nat_int(F,N5),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_401_type__copy__map__cong0,axiom,
    ! [M2: fun_int_int,G: fun_nat_int,Xa: nat,N5: fun_nat_int,Ha: fun_nat_nat,F: fun_int_nat] :
      ( ( aa_int_int(M2,aa_nat_int(G,Xa)) = aa_nat_int(N5,aa_nat_nat(Ha,Xa)) )
     => ( aa_nat_nat(comp_int_nat_nat(comp_int_nat_int(F,M2),G),Xa) = aa_nat_nat(comp_nat_nat_nat(comp_int_nat_nat(F,N5),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_402_type__copy__map__cong0,axiom,
    ! [M2: fun_char_char,G: fun_Co8487158583234504619r_char,Xa: code_integer,N5: fun_Co8487158583234504619r_char,Ha: fun_Co8499238673658658103nteger,F: fun_char_char] :
      ( ( aa_char_char(M2,aa_Code_integer_char(G,Xa)) = aa_Code_integer_char(N5,aa_Cod1783974652533396728nteger(Ha,Xa)) )
     => ( aa_Code_integer_char(comp_c4562247934908617188nteger(comp_char_char_char(F,M2),G),Xa) = aa_Code_integer_char(comp_C3856975138083468656nteger(comp_c4562247934908617188nteger(F,N5),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_403_type__copy__map__cong0,axiom,
    ! [M2: fun_nat_nat,G: fun_int_nat,Xa: int,N5: fun_int_nat,Ha: fun_int_int,F: fun_nat_num] :
      ( ( aa_nat_nat(M2,aa_int_nat(G,Xa)) = aa_int_nat(N5,aa_int_int(Ha,Xa)) )
     => ( aa_int_num(comp_nat_num_int(comp_nat_num_nat(F,M2),G),Xa) = aa_int_num(comp_int_num_int(comp_nat_num_int(F,N5),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_404_type__copy__map__cong0,axiom,
    ! [M2: fun_nat_int,G: fun_nat_nat,Xa: nat,N5: fun_int_int,Ha: fun_nat_int,F: fun_int_nat] :
      ( ( aa_nat_int(M2,aa_nat_nat(G,Xa)) = aa_int_int(N5,aa_nat_int(Ha,Xa)) )
     => ( aa_nat_nat(comp_nat_nat_nat(comp_int_nat_nat(F,M2),G),Xa) = aa_nat_nat(comp_int_nat_nat(comp_int_nat_int(F,N5),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_405_type__copy__map__cong0,axiom,
    ! [M2: fun_nat_nat,G: fun_nat_nat,Xa: nat,N5: fun_nat_nat,Ha: fun_nat_nat,F: fun_nat_nat] :
      ( ( aa_nat_nat(M2,aa_nat_nat(G,Xa)) = aa_nat_nat(N5,aa_nat_nat(Ha,Xa)) )
     => ( aa_nat_nat(comp_nat_nat_nat(comp_nat_nat_nat(F,M2),G),Xa) = aa_nat_nat(comp_nat_nat_nat(comp_nat_nat_nat(F,N5),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_406_type__copy__map__cong0,axiom,
    ! [M2: fun_int_nat,G: fun_int_int,Xa: int,N5: fun_nat_nat,Ha: fun_int_nat,F: fun_nat_nat] :
      ( ( aa_int_nat(M2,aa_int_int(G,Xa)) = aa_nat_nat(N5,aa_int_nat(Ha,Xa)) )
     => ( aa_int_nat(comp_int_nat_int(comp_nat_nat_int(F,M2),G),Xa) = aa_int_nat(comp_nat_nat_int(comp_nat_nat_nat(F,N5),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_407_type__copy__map__cong0,axiom,
    ! [M2: fun_int_int,G: fun_int_int,Xa: int,N5: fun_int_int,Ha: fun_int_int,F: fun_int_nat] :
      ( ( aa_int_int(M2,aa_int_int(G,Xa)) = aa_int_int(N5,aa_int_int(Ha,Xa)) )
     => ( aa_int_nat(comp_int_nat_int(comp_int_nat_int(F,M2),G),Xa) = aa_int_nat(comp_int_nat_int(comp_int_nat_int(F,N5),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_408_comp__apply__eq,axiom,
    ! [F: fun_char_char,G: fun_Co8487158583234504619r_char,Xa: code_integer,Ha: fun_char_char,K: fun_Co8487158583234504619r_char] :
      ( ( aa_char_char(F,aa_Code_integer_char(G,Xa)) = aa_char_char(Ha,aa_Code_integer_char(K,Xa)) )
     => ( aa_Code_integer_char(comp_c4562247934908617188nteger(F,G),Xa) = aa_Code_integer_char(comp_c4562247934908617188nteger(Ha,K),Xa) ) ) ).

% comp_apply_eq
tff(fact_409_comp__apply__eq,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,Xa: int,Ha: fun_nat_num,K: fun_int_nat] :
      ( ( aa_nat_num(F,aa_int_nat(G,Xa)) = aa_nat_num(Ha,aa_int_nat(K,Xa)) )
     => ( aa_int_num(comp_nat_num_int(F,G),Xa) = aa_int_num(comp_nat_num_int(Ha,K),Xa) ) ) ).

% comp_apply_eq
tff(fact_410_comp__apply__eq,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Xa: nat,Ha: fun_nat_nat,K: fun_nat_nat] :
      ( ( aa_nat_nat(F,aa_nat_nat(G,Xa)) = aa_nat_nat(Ha,aa_nat_nat(K,Xa)) )
     => ( aa_nat_nat(comp_nat_nat_nat(F,G),Xa) = aa_nat_nat(comp_nat_nat_nat(Ha,K),Xa) ) ) ).

% comp_apply_eq
tff(fact_411_comp__apply__eq,axiom,
    ! [F: fun_int_nat,G: fun_int_int,Xa: int,Ha: fun_int_nat,K: fun_int_int] :
      ( ( aa_int_nat(F,aa_int_int(G,Xa)) = aa_int_nat(Ha,aa_int_int(K,Xa)) )
     => ( aa_int_nat(comp_int_nat_int(F,G),Xa) = aa_int_nat(comp_int_nat_int(Ha,K),Xa) ) ) ).

% comp_apply_eq
tff(fact_412_comp__apply__eq,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,Xa: code_integer,Ha: fun_fu4800041728187187127nteger,K: fun_Co1981900195452921617nteger] :
      ( ( aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,Xa)) = aa_fun2509385170507305080nteger(Ha,aa_Cod4161879620478718930nteger(K,Xa)) )
     => ( aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(F,G),Xa) = aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(Ha,K),Xa) ) ) ).

% comp_apply_eq
tff(fact_413_comp__cong,axiom,
    ! [F: fun_char_char,G: fun_Co8487158583234504619r_char,Xa: code_integer,F2: fun_char_char,G2: fun_Co8487158583234504619r_char,X5: code_integer] :
      ( ( aa_char_char(F,aa_Code_integer_char(G,Xa)) = aa_char_char(F2,aa_Code_integer_char(G2,X5)) )
     => ( aa_Code_integer_char(comp_c4562247934908617188nteger(F,G),Xa) = aa_Code_integer_char(comp_c4562247934908617188nteger(F2,G2),X5) ) ) ).

% comp_cong
tff(fact_414_comp__cong,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,Xa: int,F2: fun_nat_num,G2: fun_int_nat,X5: int] :
      ( ( aa_nat_num(F,aa_int_nat(G,Xa)) = aa_nat_num(F2,aa_int_nat(G2,X5)) )
     => ( aa_int_num(comp_nat_num_int(F,G),Xa) = aa_int_num(comp_nat_num_int(F2,G2),X5) ) ) ).

% comp_cong
tff(fact_415_comp__cong,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Xa: nat,F2: fun_nat_nat,G2: fun_nat_nat,X5: nat] :
      ( ( aa_nat_nat(F,aa_nat_nat(G,Xa)) = aa_nat_nat(F2,aa_nat_nat(G2,X5)) )
     => ( aa_nat_nat(comp_nat_nat_nat(F,G),Xa) = aa_nat_nat(comp_nat_nat_nat(F2,G2),X5) ) ) ).

% comp_cong
tff(fact_416_comp__cong,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Xa: nat,F2: fun_int_nat,G2: fun_int_int,X5: int] :
      ( ( aa_nat_nat(F,aa_nat_nat(G,Xa)) = aa_int_nat(F2,aa_int_int(G2,X5)) )
     => ( aa_nat_nat(comp_nat_nat_nat(F,G),Xa) = aa_int_nat(comp_int_nat_int(F2,G2),X5) ) ) ).

% comp_cong
tff(fact_417_comp__cong,axiom,
    ! [F: fun_int_nat,G: fun_int_int,Xa: int,F2: fun_nat_nat,G2: fun_nat_nat,X5: nat] :
      ( ( aa_int_nat(F,aa_int_int(G,Xa)) = aa_nat_nat(F2,aa_nat_nat(G2,X5)) )
     => ( aa_int_nat(comp_int_nat_int(F,G),Xa) = aa_nat_nat(comp_nat_nat_nat(F2,G2),X5) ) ) ).

% comp_cong
tff(fact_418_comp__cong,axiom,
    ! [F: fun_int_nat,G: fun_int_int,Xa: int,F2: fun_int_nat,G2: fun_int_int,X5: int] :
      ( ( aa_int_nat(F,aa_int_int(G,Xa)) = aa_int_nat(F2,aa_int_int(G2,X5)) )
     => ( aa_int_nat(comp_int_nat_int(F,G),Xa) = aa_int_nat(comp_int_nat_int(F2,G2),X5) ) ) ).

% comp_cong
tff(fact_419_comp__cong,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,Xa: code_integer,F2: fun_fu4800041728187187127nteger,G2: fun_Co1981900195452921617nteger,X5: code_integer] :
      ( ( aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,Xa)) = aa_fun2509385170507305080nteger(F2,aa_Cod4161879620478718930nteger(G2,X5)) )
     => ( aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(F,G),Xa) = aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(F2,G2),X5) ) ) ).

% comp_cong
tff(fact_420_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_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),A)),A) != zero_zero_int ) ) ).

% Ints_odd_nonzero
tff(fact_421_Ints__odd__nonzero,axiom,
    ! [A: code_integer] :
      ( member_Code_integer(A,ring_11222124179247155820nteger)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),A) != zero_z3403309356797280102nteger ) ) ).

% Ints_odd_nonzero
tff(fact_422_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A) != zero_zero_rat ) ) ).

% Ints_odd_nonzero
tff(fact_423_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% is_num_normalize(1)
tff(fact_424_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% is_num_normalize(1)
tff(fact_425_ssubst__Pair__rhs,axiom,
    ! [R: fun_Co3460408610320954864e_term,S: produc8923325533196201883nteger,R4: set_Pr1200682188532975830nteger,S3: produc8923325533196201883nteger] :
      ( member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,R),S),R4)
     => ( ( S3 = S )
       => member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,R),S3),R4) ) ) ).

% ssubst_Pair_rhs
tff(fact_426_ssubst__Pair__rhs,axiom,
    ! [R: fun_Pr285225112257423215e_term,S: produc7822875418678951345atural,R4: set_Pr620054291978045485atural,S3: produc7822875418678951345atural] :
      ( member6441364043385305230atural(aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,R),S),R4)
     => ( ( S3 = S )
       => member6441364043385305230atural(aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,R),S3),R4) ) ) ).

% ssubst_Pair_rhs
tff(fact_427_ssubst__Pair__rhs,axiom,
    ! [R: fun_Pr5098669352780829690e_term,S: produc8923325533196201883nteger,R4: set_Pr3486312109254157132nteger,S3: produc8923325533196201883nteger] :
      ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,R),S),R4)
     => ( ( S3 = S )
       => member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,R),S3),R4) ) ) ).

% ssubst_Pair_rhs
tff(fact_428_ssubst__Pair__rhs,axiom,
    ! [R: fun_Pr7763680713840462443e_term,S: product_prod_int_int,R4: set_Pr6700895561065273913nt_int,S3: product_prod_int_int] :
      ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,R),S),R4)
     => ( ( S3 = S )
       => member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,R),S3),R4) ) ) ).

% ssubst_Pair_rhs
tff(fact_429_ssubst__Pair__rhs,axiom,
    ! [R: fun_in5730379987311621695e_term,S: product_prod_int_int,R4: set_Pr4166680992137330533nt_int,S3: product_prod_int_int] :
      ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,R),S),R4)
     => ( ( S3 = S )
       => member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,R),S3),R4) ) ) ).

% ssubst_Pair_rhs
tff(fact_430_Ints__0,axiom,
    member_int(zero_zero_int,ring_1_Ints_int) ).

% Ints_0
tff(fact_431_Ints__0,axiom,
    member_Code_integer(zero_z3403309356797280102nteger,ring_11222124179247155820nteger) ).

% Ints_0
tff(fact_432_Ints__0,axiom,
    member_rat(zero_zero_rat,ring_1_Ints_rat) ).

% Ints_0
tff(fact_433_Ints__add,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),ring_1_Ints_rat) ) ) ).

% Ints_add
tff(fact_434_Ints__add,axiom,
    ! [A: int,B: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B),ring_1_Ints_int) ) ) ).

% Ints_add
tff(fact_435_Ints__1,axiom,
    member_int(one_one_int,ring_1_Ints_int) ).

% Ints_1
tff(fact_436_Ints__1,axiom,
    member_Code_integer(one_one_Code_integer,ring_11222124179247155820nteger) ).

% Ints_1
tff(fact_437_Ints__1,axiom,
    member_rat(one_one_rat,ring_1_Ints_rat) ).

% Ints_1
tff(fact_438_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_439_Ints__double__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( member_Code_integer(A,ring_11222124179247155820nteger)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_440_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_441_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_442_dbl__inc__simps_I2_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,zero_z3403309356797280102nteger) = one_one_Code_integer ).

% dbl_inc_simps(2)
tff(fact_443_dbl__inc__simps_I2_J,axiom,
    neg_nu5219082963157363817nc_rat(zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_444_Ints__odd__less__0,axiom,
    ! [A: code_integer] :
      ( member_Code_integer(A,ring_11222124179247155820nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),A)),zero_z3403309356797280102nteger)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ) ).

% Ints_odd_less_0
tff(fact_445_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A)),zero_zero_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% Ints_odd_less_0
tff(fact_446_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Ints_odd_less_0
tff(fact_447_dbl__inc__def,axiom,
    ! [Xa: code_integer] : aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,Xa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Xa),Xa)),one_one_Code_integer) ).

% dbl_inc_def
tff(fact_448_dbl__inc__def,axiom,
    ! [Xa: rat] : neg_nu5219082963157363817nc_rat(Xa) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Xa)),one_one_rat) ).

% dbl_inc_def
tff(fact_449_dbl__inc__def,axiom,
    ! [Xa: int] : aa_int_int(neg_nu5851722552734809277nc_int,Xa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Xa)),one_one_int) ).

% dbl_inc_def
tff(fact_450_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_451_add__neg__numeral__special_I8_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_452_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_453_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_454_add__neg__numeral__special_I7_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_455_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_456_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_457_dbl__dec__simps_I3_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,one_one_Code_integer) = one_one_Code_integer ).

% dbl_dec_simps(3)
tff(fact_458_dbl__dec__simps_I3_J,axiom,
    neg_nu3179335615603231917ec_rat(one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_459_case__optionE,axiom,
    ! [Pa: $o,Q: fun_Pr5746006151868627512_nat_o,Xa: option3562590408128118217it_nat] :
      ( aa_opt9064029888611036241_nat_o(case_o3267218206291580230it_nat((Pa),Q),Xa)
     => ( ( ( Xa = none_P2651198173097904984it_nat )
         => ~ (Pa) )
       => ~ ! [Y4: produc3260487557148687353it_nat] :
              ( ( Xa = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Y4) )
             => ~ aa_Pro6426884464393220769_nat_o(Q,Y4) ) ) ) ).

% case_optionE
tff(fact_460_case__optionE,axiom,
    ! [Pa: $o,Q: fun_Pr8032424337255632745_nat_o,Xa: option8956607266484857688it_nat] :
      ( aa_opt8360315729766332930_nat_o(case_o2686588417244861013it_nat((Pa),Q),Xa)
     => ( ( ( Xa = none_P9117596204409417319it_nat )
         => ~ (Pa) )
       => ~ ! [Y4: produc8664842809031399944it_nat] :
              ( ( Xa = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Y4) )
             => ~ aa_Pro4744234731172327378_nat_o(Q,Y4) ) ) ) ).

% case_optionE
tff(fact_461_case__optionE,axiom,
    ! [Pa: $o,Q: fun_num_o,Xa: option_num] :
      ( aa_option_num_o(case_option_o_num((Pa),Q),Xa)
     => ( ( ( Xa = none_num )
         => ~ (Pa) )
       => ~ ! [Y4: num] :
              ( ( Xa = aa_num_option_num(some_num,Y4) )
             => ~ aa_num_o(Q,Y4) ) ) ) ).

% case_optionE
tff(fact_462_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_463_dbl__simps_I2_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu8804712462038260780nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% dbl_simps(2)
tff(fact_464_dbl__simps_I2_J,axiom,
    neg_numeral_dbl_rat(zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_465_option_Ocollapse,axiom,
    ! [Option: option3562590408128118217it_nat] :
      ( ( Option != none_P2651198173097904984it_nat )
     => ( aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,the_Pr3616839707675578981it_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_466_option_Ocollapse,axiom,
    ! [Option: option8956607266484857688it_nat] :
      ( ( Option != none_P9117596204409417319it_nat )
     => ( aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,the_Pr7776125533777466484it_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_467_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) = Option ) ) ).

% option.collapse
tff(fact_468_curryI,axiom,
    ! [F: fun_Pr1347628893830317649eger_o,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4652312591602923194eger_o(F,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B))
     => aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(produc4448170916228617336eger_o(F),A),B) ) ).

% curryI
tff(fact_469_curryI,axiom,
    ! [F: fun_Pr3733920118687433658ural_o,A: fun_Pr285225112257423215e_term,B: produc7822875418678951345atural] :
      ( aa_Pro866224779355485219ural_o(F,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A),B))
     => aa_Pro6073791466322389475ural_o(aa_fun9089857049899810327ural_o(produc5480829236514930401ural_o(F),A),B) ) ).

% curryI
tff(fact_470_curryI,axiom,
    ! [F: fun_Pr7015759342434529115eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4148064003102580676eger_o(F,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B))
     => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(produc177509963858859394eger_o(F),A),B) ) ).

% curryI
tff(fact_471_curryI,axiom,
    ! [F: fun_Pr3540825087398332590_int_o,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_Pro2601676399929874711_int_o(F,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(produc443495763101479893_int_o(F),A),B) ) ).

% curryI
tff(fact_472_curryI,axiom,
    ! [F: fun_Pr1196100989263358594_int_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_Pro5451197003417746155_int_o(F,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(produc2168430993391648169_int_o(F),A),B) ) ).

% curryI
tff(fact_473_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_474_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_475_add_Oinverse__inverse,axiom,
    ! [A: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A)) = A ).

% add.inverse_inverse
tff(fact_476_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_477_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_478_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_479_verit__minus__simplify_I4_J,axiom,
    ! [B: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,B)) = B ).

% verit_minus_simplify(4)
tff(fact_480_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = B ).

% verit_minus_simplify(4)
tff(fact_481_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,B)) = B ).

% verit_minus_simplify(4)
tff(fact_482_less__nat__zero__code,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_483_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% neq0_conv
tff(fact_484_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_485_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% nat_add_left_cancel_less
tff(fact_486_not__gr__zero,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
    <=> ( N = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_487_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_488_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_489_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_490_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_491_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_492_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_493_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_494_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_495_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_496_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_497_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_498_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_499_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_500_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_501_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_502_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_503_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_504_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_505_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_506_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_507_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_508_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% neg_less_iff_less
tff(fact_509_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% neg_less_iff_less
tff(fact_510_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% neg_less_iff_less
tff(fact_511_minus__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_512_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_add_distrib
tff(fact_513_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_514_minus__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = B ).

% minus_add_cancel
tff(fact_515_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = B ).

% minus_add_cancel
tff(fact_516_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = B ).

% minus_add_cancel
tff(fact_517_add__minus__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B)) = B ).

% add_minus_cancel
tff(fact_518_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) = B ).

% add_minus_cancel
tff(fact_519_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_520_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% add_gr_0
tff(fact_521_less__one,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),one_one_nat)
    <=> ( N = zero_zero_nat ) ) ).

% less_one
tff(fact_522_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_523_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_524_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_525_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_526_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_527_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_528_less__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ).

% less_add_same_cancel2
tff(fact_529_less__add__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A))
    <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B) ) ).

% less_add_same_cancel2
tff(fact_530_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_531_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_532_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_533_less__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ).

% less_add_same_cancel1
tff(fact_534_less__add__same__cancel1,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B))
    <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B) ) ).

% less_add_same_cancel1
tff(fact_535_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_536_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_537_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_538_add__less__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% add_less_same_cancel2
tff(fact_539_add__less__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),B)
    <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural) ) ).

% add_less_same_cancel2
tff(fact_540_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_541_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_542_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_543_add__less__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% add_less_same_cancel1
tff(fact_544_add__less__same__cancel1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A)),B)
    <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural) ) ).

% add_less_same_cancel1
tff(fact_545_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_546_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_547_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_548_less__neg__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% less_neg_neg
tff(fact_549_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_550_less__neg__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_551_neg__less__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_pos
tff(fact_552_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_pos
tff(fact_553_neg__less__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_pos
tff(fact_554_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_555_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_556_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_557_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_0_iff_less
tff(fact_558_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_0_iff_less
tff(fact_559_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_0_iff_less
tff(fact_560_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_561_ab__left__minus,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_562_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_563_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_564_add_Oright__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_565_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_566_dbl__inc__simps_I4_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_inc_simps(4)
tff(fact_567_dbl__inc__simps_I4_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_568_dbl__inc__simps_I4_J,axiom,
    neg_nu5219082963157363817nc_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_569_dbl__dec__simps_I2_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_dec_simps(2)
tff(fact_570_dbl__dec__simps_I2_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_571_dbl__dec__simps_I2_J,axiom,
    neg_nu3179335615603231917ec_rat(zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_572_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% equation_minus_iff
tff(fact_573_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% equation_minus_iff
tff(fact_574_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% equation_minus_iff
tff(fact_575_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(uminus_uminus_int,B) = A ) ) ).

% minus_equation_iff
tff(fact_576_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A ) ) ).

% minus_equation_iff
tff(fact_577_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(uminus_uminus_rat,B) = A ) ) ).

% minus_equation_iff
tff(fact_578_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% less_minus_iff
tff(fact_579_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% less_minus_iff
tff(fact_580_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% less_minus_iff
tff(fact_581_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_less_iff
tff(fact_582_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_less_iff
tff(fact_583_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_less_iff
tff(fact_584_verit__comp__simplify1_I1_J,axiom,
    ! [A: literal] : ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A),A) ).

% verit_comp_simplify1(1)
tff(fact_585_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_586_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(ord_less_num(A),A) ).

% verit_comp_simplify1(1)
tff(fact_587_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_588_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% verit_comp_simplify1(1)
tff(fact_589_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_590_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_591_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_592_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_593_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_594_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_595_linorder__neqE__linordered__idom,axiom,
    ! [Xa: rat,Y3: rat] :
      ( ( Xa != Y3 )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_596_linorder__neqE__linordered__idom,axiom,
    ! [Xa: int,Y3: int] :
      ( ( Xa != Y3 )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_597_less__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% less_minus_one_simps(2)
tff(fact_598_less__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% less_minus_one_simps(2)
tff(fact_599_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_600_less__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(4)
tff(fact_601_less__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(4)
tff(fact_602_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_603_option_Ocase__eq__if,axiom,
    ! [F1: int,F22: fun_num_int,Option: option_num] :
      aa_option_num_int(case_option_int_num(F1,F22),Option) = $ite(Option = none_num,F1,aa_num_int(F22,aa_option_num_num(the_num,Option))) ).

% option.case_eq_if
tff(fact_604_option_Ocase__eq__if,axiom,
    ! [F1: option_num,F22: fun_num_option_num,Option: option_num] :
      aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option) = $ite(Option = none_num,F1,aa_num_option_num(F22,aa_option_num_num(the_num,Option))) ).

% option.case_eq_if
tff(fact_605_option_Ocase__eq__if,axiom,
    ! [F1: num,F22: fun_num_num,Option: option_num] :
      aa_option_num_num(case_option_num_num(F1,F22),Option) = $ite(Option = none_num,F1,aa_num_num(F22,aa_option_num_num(the_num,Option))) ).

% option.case_eq_if
tff(fact_606_option_Ocase__distrib,axiom,
    ! [Ha: fun_int_int,F1: int,F22: fun_num_int,Option: option_num] : aa_int_int(Ha,aa_option_num_int(case_option_int_num(F1,F22),Option)) = aa_option_num_int(case_option_int_num(aa_int_int(Ha,F1),aa_fun329774391934724804um_int(aTP_Lamm_ax(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_607_option_Ocase__distrib,axiom,
    ! [Ha: fun_int_option_num,F1: int,F22: fun_num_int,Option: option_num] : aa_int_option_num(Ha,aa_option_num_int(case_option_int_num(F1,F22),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_int_option_num(Ha,F1),aa_fun3858197967482829954on_num(aTP_Lamm_ay(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_608_option_Ocase__distrib,axiom,
    ! [Ha: fun_int_num,F1: int,F22: fun_num_int,Option: option_num] : aa_int_num(Ha,aa_option_num_int(case_option_int_num(F1,F22),Option)) = aa_option_num_num(case_option_num_num(aa_int_num(Ha,F1),aa_fun5215869348392759218um_num(aTP_Lamm_az(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_609_option_Ocase__distrib,axiom,
    ! [Ha: fun_option_num_int,F1: option_num,F22: fun_num_option_num,Option: option_num] : aa_option_num_int(Ha,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option)) = aa_option_num_int(case_option_int_num(aa_option_num_int(Ha,F1),aa_fun2256179333612926594um_int(aTP_Lamm_ba(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_610_option_Ocase__distrib,axiom,
    ! [Ha: fun_op8989173174992673105on_num,F1: option_num,F22: fun_num_option_num,Option: option_num] : aa_opt5369159329878751634on_num(Ha,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_opt5369159329878751634on_num(Ha,F1),aa_fun3636977833914307392on_num(aTP_Lamm_bb(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_611_option_Ocase__distrib,axiom,
    ! [Ha: fun_option_num_num,F1: option_num,F22: fun_num_option_num,Option: option_num] : aa_option_num_num(Ha,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option)) = aa_option_num_num(case_option_num_num(aa_option_num_num(Ha,F1),aa_fun7142274290070961008um_num(aTP_Lamm_bc(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_612_option_Ocase__distrib,axiom,
    ! [Ha: fun_num_int,F1: num,F22: fun_num_num,Option: option_num] : aa_num_int(Ha,aa_option_num_num(case_option_num_num(F1,F22),Option)) = aa_option_num_int(case_option_int_num(aa_num_int(Ha,F1),aa_fun3162900327506486194um_int(aTP_Lamm_bd(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_613_option_Ocase__distrib,axiom,
    ! [Ha: fun_num_option_num,F1: num,F22: fun_num_num,Option: option_num] : aa_num_option_num(Ha,aa_option_num_num(case_option_num_num(F1,F22),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(Ha,F1),aa_fun612943910911948912on_num(aTP_Lamm_be(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_614_option_Ocase__distrib,axiom,
    ! [Ha: fun_num_num,F1: num,F22: fun_num_num,Option: option_num] : aa_num_num(Ha,aa_option_num_num(case_option_num_num(F1,F22),Option)) = aa_option_num_num(case_option_num_num(aa_num_num(Ha,F1),aa_fun8048995283964520608um_num(aTP_Lamm_bf(Ha),F22)),Option) ).

% option.case_distrib
tff(fact_615_less__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% less_minus_one_simps(1)
tff(fact_616_less__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% less_minus_one_simps(1)
tff(fact_617_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_618_less__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(3)
tff(fact_619_less__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(3)
tff(fact_620_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_621_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% add.inverse_distrib_swap
tff(fact_622_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% add.inverse_distrib_swap
tff(fact_623_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_624_group__cancel_Oneg1,axiom,
    ! [A3: int,K: int,A: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A3) = 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_625_group__cancel_Oneg1,axiom,
    ! [A3: code_integer,K: code_integer,A: code_integer] :
      ( ( A3 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_626_group__cancel_Oneg1,axiom,
    ! [A3: rat,K: rat,A: rat] :
      ( ( A3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_627_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% is_num_normalize(8)
tff(fact_628_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% is_num_normalize(8)
tff(fact_629_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% is_num_normalize(8)
tff(fact_630_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_631_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_632_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_633_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
    <=> ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_634_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_635_not__less__zero,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),zero_zero_nat) ).

% not_less_zero
tff(fact_636_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% gr_zeroI
tff(fact_637_less__numeral__extra_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_numeral_extra(3)
tff(fact_638_less__numeral__extra_I3_J,axiom,
    ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),zero_z2226904508553997617atural) ).

% less_numeral_extra(3)
tff(fact_639_less__numeral__extra_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_zero_rat) ).

% less_numeral_extra(3)
tff(fact_640_less__numeral__extra_I3_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_zero_nat) ).

% less_numeral_extra(3)
tff(fact_641_less__numeral__extra_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_numeral_extra(3)
tff(fact_642_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_643_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_644_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_645_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_646_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_647_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_648_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_strict_right_mono
tff(fact_649_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_strict_right_mono
tff(fact_650_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_strict_right_mono
tff(fact_651_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_strict_left_mono
tff(fact_652_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_strict_left_mono
tff(fact_653_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_strict_left_mono
tff(fact_654_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_strict_mono
tff(fact_655_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_strict_mono
tff(fact_656_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_strict_mono
tff(fact_657_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_658_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_659_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_660_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_661_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_662_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_663_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_664_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_665_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_666_less__numeral__extra_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),one_one_Code_integer) ).

% less_numeral_extra(4)
tff(fact_667_less__numeral__extra_I4_J,axiom,
    ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),one_one_Code_natural) ).

% less_numeral_extra(4)
tff(fact_668_less__numeral__extra_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),one_one_rat) ).

% less_numeral_extra(4)
tff(fact_669_less__numeral__extra_I4_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),one_one_nat) ).

% less_numeral_extra(4)
tff(fact_670_less__numeral__extra_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),one_one_int) ).

% less_numeral_extra(4)
tff(fact_671_infinite__descent0,axiom,
    ! [Pa: fun_nat_o,N: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( ~ aa_nat_o(Pa,N2)
             => ? [M3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
                  & ~ aa_nat_o(Pa,M3) ) ) )
       => aa_nat_o(Pa,N) ) ) ).

% infinite_descent0
tff(fact_672_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_673_less__zeroE,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),zero_zero_nat) ).

% less_zeroE
tff(fact_674_not__less0,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),zero_zero_nat) ).

% not_less0
tff(fact_675_not__gr0,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
    <=> ( N = zero_zero_nat ) ) ).

% not_gr0
tff(fact_676_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% gr0I
tff(fact_677_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_678_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% less_add_eq_less
tff(fact_679_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_less_add2
tff(fact_680_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_less_add1
tff(fact_681_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(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_682_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I) ).

% not_add_less2
tff(fact_683_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I) ).

% not_add_less1
tff(fact_684_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_685_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% add_lessD1
tff(fact_686_uminus__unit__def,axiom,
    ! [Uu: product_unit] : uminus2952777764628376836t_unit(Uu) = product_Unity ).

% uminus_unit_def
tff(fact_687_option_Osplit__sel,axiom,
    ! [Pa: fun_int_o,F1: int,F22: fun_num_int,Option: option_num] :
      ( aa_int_o(Pa,aa_option_num_int(case_option_int_num(F1,F22),Option))
    <=> ( ( ( Option = none_num )
         => aa_int_o(Pa,F1) )
        & ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
         => aa_int_o(Pa,aa_num_int(F22,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel
tff(fact_688_option_Osplit__sel,axiom,
    ! [Pa: fun_option_num_o,F1: option_num,F22: fun_num_option_num,Option: option_num] :
      ( aa_option_num_o(Pa,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option))
    <=> ( ( ( Option = none_num )
         => aa_option_num_o(Pa,F1) )
        & ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
         => aa_option_num_o(Pa,aa_num_option_num(F22,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel
tff(fact_689_option_Osplit__sel,axiom,
    ! [Pa: fun_num_o,F1: num,F22: fun_num_num,Option: option_num] :
      ( aa_num_o(Pa,aa_option_num_num(case_option_num_num(F1,F22),Option))
    <=> ( ( ( Option = none_num )
         => aa_num_o(Pa,F1) )
        & ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
         => aa_num_o(Pa,aa_num_num(F22,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel
tff(fact_690_option_Osplit__sel__asm,axiom,
    ! [Pa: fun_int_o,F1: int,F22: fun_num_int,Option: option_num] :
      ( aa_int_o(Pa,aa_option_num_int(case_option_int_num(F1,F22),Option))
    <=> ~ ( ( ( Option = none_num )
            & ~ aa_int_o(Pa,F1) )
          | ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
            & ~ aa_int_o(Pa,aa_num_int(F22,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_691_option_Osplit__sel__asm,axiom,
    ! [Pa: fun_option_num_o,F1: option_num,F22: fun_num_option_num,Option: option_num] :
      ( aa_option_num_o(Pa,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),Option))
    <=> ~ ( ( ( Option = none_num )
            & ~ aa_option_num_o(Pa,F1) )
          | ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
            & ~ aa_option_num_o(Pa,aa_num_option_num(F22,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_692_option_Osplit__sel__asm,axiom,
    ! [Pa: fun_num_o,F1: num,F22: fun_num_num,Option: option_num] :
      ( aa_num_o(Pa,aa_option_num_num(case_option_num_num(F1,F22),Option))
    <=> ~ ( ( ( Option = none_num )
            & ~ aa_num_o(Pa,F1) )
          | ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
            & ~ aa_num_o(Pa,aa_num_num(F22,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_693_Ints__minus,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => member_int(aa_int_int(uminus_uminus_int,A),ring_1_Ints_int) ) ).

% Ints_minus
tff(fact_694_Ints__minus,axiom,
    ! [A: code_integer] :
      ( member_Code_integer(A,ring_11222124179247155820nteger)
     => member_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),ring_11222124179247155820nteger) ) ).

% Ints_minus
tff(fact_695_Ints__minus,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => member_rat(aa_rat_rat(uminus_uminus_rat,A),ring_1_Ints_rat) ) ).

% Ints_minus
tff(fact_696_minus__in__Ints__iff,axiom,
    ! [Xa: int] :
      ( member_int(aa_int_int(uminus_uminus_int,Xa),ring_1_Ints_int)
    <=> member_int(Xa,ring_1_Ints_int) ) ).

% minus_in_Ints_iff
tff(fact_697_minus__in__Ints__iff,axiom,
    ! [Xa: code_integer] :
      ( member_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xa),ring_11222124179247155820nteger)
    <=> member_Code_integer(Xa,ring_11222124179247155820nteger) ) ).

% minus_in_Ints_iff
tff(fact_698_minus__in__Ints__iff,axiom,
    ! [Xa: rat] :
      ( member_rat(aa_rat_rat(uminus_uminus_rat,Xa),ring_1_Ints_rat)
    <=> member_rat(Xa,ring_1_Ints_rat) ) ).

% minus_in_Ints_iff
tff(fact_699_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
      <=> ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( aa_option_num_num(the_num,Option) = aa_option_num_num(the_num,Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_700_option_Oexpand,axiom,
    ! [Option: option3562590408128118217it_nat,Option2: option3562590408128118217it_nat] :
      ( ( ( Option = none_P2651198173097904984it_nat )
      <=> ( Option2 = none_P2651198173097904984it_nat ) )
     => ( ( ( Option != none_P2651198173097904984it_nat )
         => ( ( Option2 != none_P2651198173097904984it_nat )
           => ( the_Pr3616839707675578981it_nat(Option) = the_Pr3616839707675578981it_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_701_option_Osel,axiom,
    ! [X2: produc8664842809031399944it_nat] : the_Pr7776125533777466484it_nat(aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2)) = X2 ).

% option.sel
tff(fact_702_option_Osel,axiom,
    ! [X2: num] : aa_option_num_num(the_num,aa_num_option_num(some_num,X2)) = X2 ).

% option.sel
tff(fact_703_option_Osimps_I4_J,axiom,
    ! [F1: int,F22: fun_num_int] : aa_option_num_int(case_option_int_num(F1,F22),none_num) = F1 ).

% option.simps(4)
tff(fact_704_option_Osimps_I4_J,axiom,
    ! [F1: option_num,F22: fun_num_option_num] : aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),none_num) = F1 ).

% option.simps(4)
tff(fact_705_option_Osimps_I4_J,axiom,
    ! [F1: num,F22: fun_num_num] : aa_option_num_num(case_option_num_num(F1,F22),none_num) = F1 ).

% option.simps(4)
tff(fact_706_option_Osimps_I5_J,axiom,
    ! [F1: int,F22: fun_num_int,X2: num] : aa_option_num_int(case_option_int_num(F1,F22),aa_num_option_num(some_num,X2)) = aa_num_int(F22,X2) ).

% option.simps(5)
tff(fact_707_option_Osimps_I5_J,axiom,
    ! [F1: option_num,F22: fun_num_option_num,X2: num] : aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F22),aa_num_option_num(some_num,X2)) = aa_num_option_num(F22,X2) ).

% option.simps(5)
tff(fact_708_option_Osimps_I5_J,axiom,
    ! [F1: num,F22: fun_num_num,X2: num] : aa_option_num_num(case_option_num_num(F1,F22),aa_num_option_num(some_num,X2)) = aa_num_num(F22,X2) ).

% option.simps(5)
tff(fact_709_curryE,axiom,
    ! [F: fun_Pr1347628893830317649eger_o,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(produc4448170916228617336eger_o(F),A),B)
     => aa_Pro4652312591602923194eger_o(F,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B)) ) ).

% curryE
tff(fact_710_curryE,axiom,
    ! [F: fun_Pr3733920118687433658ural_o,A: fun_Pr285225112257423215e_term,B: produc7822875418678951345atural] :
      ( aa_Pro6073791466322389475ural_o(aa_fun9089857049899810327ural_o(produc5480829236514930401ural_o(F),A),B)
     => aa_Pro866224779355485219ural_o(F,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A),B)) ) ).

% curryE
tff(fact_711_curryE,axiom,
    ! [F: fun_Pr7015759342434529115eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(produc177509963858859394eger_o(F),A),B)
     => aa_Pro4148064003102580676eger_o(F,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B)) ) ).

% curryE
tff(fact_712_curryE,axiom,
    ! [F: fun_Pr3540825087398332590_int_o,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(produc443495763101479893_int_o(F),A),B)
     => aa_Pro2601676399929874711_int_o(F,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B)) ) ).

% curryE
tff(fact_713_curryE,axiom,
    ! [F: fun_Pr1196100989263358594_int_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(produc2168430993391648169_int_o(F),A),B)
     => aa_Pro5451197003417746155_int_o(F,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B)) ) ).

% curryE
tff(fact_714_curryD,axiom,
    ! [F: fun_Pr1347628893830317649eger_o,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(produc4448170916228617336eger_o(F),A),B)
     => aa_Pro4652312591602923194eger_o(F,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B)) ) ).

% curryD
tff(fact_715_curryD,axiom,
    ! [F: fun_Pr3733920118687433658ural_o,A: fun_Pr285225112257423215e_term,B: produc7822875418678951345atural] :
      ( aa_Pro6073791466322389475ural_o(aa_fun9089857049899810327ural_o(produc5480829236514930401ural_o(F),A),B)
     => aa_Pro866224779355485219ural_o(F,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A),B)) ) ).

% curryD
tff(fact_716_curryD,axiom,
    ! [F: fun_Pr7015759342434529115eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(produc177509963858859394eger_o(F),A),B)
     => aa_Pro4148064003102580676eger_o(F,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B)) ) ).

% curryD
tff(fact_717_curryD,axiom,
    ! [F: fun_Pr3540825087398332590_int_o,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(produc443495763101479893_int_o(F),A),B)
     => aa_Pro2601676399929874711_int_o(F,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B)) ) ).

% curryD
tff(fact_718_curryD,axiom,
    ! [F: fun_Pr1196100989263358594_int_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(produc2168430993391648169_int_o(F),A),B)
     => aa_Pro5451197003417746155_int_o(F,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B)) ) ).

% curryD
tff(fact_719_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_num] :
      ( ( Option = none_num )
    <=> aa_option_num_o(case_option_o_num($true,aTP_Lamm_bg),Option) ) ).

% option.disc_eq_case(1)
tff(fact_720_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option3562590408128118217it_nat] :
      ( ( Option = none_P2651198173097904984it_nat )
    <=> aa_opt9064029888611036241_nat_o(case_o3267218206291580230it_nat($true,aTP_Lamm_bh),Option) ) ).

% option.disc_eq_case(1)
tff(fact_721_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
    <=> aa_option_num_o(case_option_o_num($false,aTP_Lamm_bi),Option) ) ).

% option.disc_eq_case(2)
tff(fact_722_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option3562590408128118217it_nat] :
      ( ( Option != none_P2651198173097904984it_nat )
    <=> aa_opt9064029888611036241_nat_o(case_o3267218206291580230it_nat($false,aTP_Lamm_bj),Option) ) ).

% option.disc_eq_case(2)
tff(fact_723_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_724_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_725_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_726_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_727_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_728_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_729_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B ) ) ).

% add.inverse_unique
tff(fact_730_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).

% add.inverse_unique
tff(fact_731_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_732_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_733_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_734_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_735_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_736_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_737_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_738_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_739_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_740_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_741_add__less__zeroD,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Xa),Y3)),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y3),zero_z3403309356797280102nteger) ) ) ).

% add_less_zeroD
tff(fact_742_add__less__zeroD,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y3)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_743_add__less__zeroD,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y3)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_744_pos__add__strict,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% pos_add_strict
tff(fact_745_pos__add__strict,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)) ) ) ).

% pos_add_strict
tff(fact_746_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% pos_add_strict
tff(fact_747_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% pos_add_strict
tff(fact_748_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% pos_add_strict
tff(fact_749_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C2: nat] :
            ( ( B = 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_750_add__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% add_pos_pos
tff(fact_751_add__pos__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) ) ) ).

% add_pos_pos
tff(fact_752_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_pos
tff(fact_753_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_pos
tff(fact_754_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_pos
tff(fact_755_add__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_neg_neg
tff(fact_756_add__neg__neg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),zero_z2226904508553997617atural) ) ) ).

% add_neg_neg
tff(fact_757_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_758_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_759_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_760_not__one__less__zero,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),zero_z3403309356797280102nteger) ).

% not_one_less_zero
tff(fact_761_not__one__less__zero,axiom,
    ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),zero_z2226904508553997617atural) ).

% not_one_less_zero
tff(fact_762_not__one__less__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),zero_zero_rat) ).

% not_one_less_zero
tff(fact_763_not__one__less__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),zero_zero_nat) ).

% not_one_less_zero
tff(fact_764_not__one__less__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),zero_zero_int) ).

% not_one_less_zero
tff(fact_765_zero__less__one,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% zero_less_one
tff(fact_766_zero__less__one,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),one_one_Code_natural) ).

% zero_less_one
tff(fact_767_zero__less__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one
tff(fact_768_zero__less__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one
tff(fact_769_zero__less__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% zero_less_one
tff(fact_770_less__numeral__extra_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% less_numeral_extra(1)
tff(fact_771_less__numeral__extra_I1_J,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),one_one_Code_natural) ).

% less_numeral_extra(1)
tff(fact_772_less__numeral__extra_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% less_numeral_extra(1)
tff(fact_773_less__numeral__extra_I1_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% less_numeral_extra(1)
tff(fact_774_less__numeral__extra_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% less_numeral_extra(1)
tff(fact_775_add__mono1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),one_one_Code_integer)) ) ).

% add_mono1
tff(fact_776_add__mono1,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),one_one_Code_natural)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),one_one_Code_natural)) ) ).

% add_mono1
tff(fact_777_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),one_one_rat)) ) ).

% add_mono1
tff(fact_778_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).

% add_mono1
tff(fact_779_add__mono1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)) ) ).

% add_mono1
tff(fact_780_less__add__one,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)) ).

% less_add_one
tff(fact_781_less__add__one,axiom,
    ! [A: code_natural] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),one_one_Code_natural)) ).

% less_add_one
tff(fact_782_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)) ).

% less_add_one
tff(fact_783_less__add__one,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).

% less_add_one
tff(fact_784_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).

% less_add_one
tff(fact_785_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_786_dbl__def,axiom,
    ! [Xa: rat] : neg_numeral_dbl_rat(Xa) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Xa) ).

% dbl_def
tff(fact_787_dbl__def,axiom,
    ! [Xa: int] : aa_int_int(neg_numeral_dbl_int,Xa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Xa) ).

% dbl_def
tff(fact_788_option_Oexhaust__sel,axiom,
    ! [Option: option3562590408128118217it_nat] :
      ( ( Option != none_P2651198173097904984it_nat )
     => ( Option = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,the_Pr3616839707675578981it_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_789_option_Oexhaust__sel,axiom,
    ! [Option: option8956607266484857688it_nat] :
      ( ( Option != none_P9117596204409417319it_nat )
     => ( Option = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,the_Pr7776125533777466484it_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_790_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) ) ) ).

% option.exhaust_sel
tff(fact_791_zero__less__two,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),one_one_Code_integer)) ).

% zero_less_two
tff(fact_792_zero__less__two,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),one_one_Code_natural)) ).

% zero_less_two
tff(fact_793_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_794_zero__less__two,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)) ).

% zero_less_two
tff(fact_795_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)) ).

% zero_less_two
tff(fact_796_disjE__realizer2,axiom,
    ! [Pa: $o,Q: fun_num_o,Xa: option_num,R4: fun_int_o,F: int,G: fun_num_int] :
      ( aa_option_num_o(case_option_o_num((Pa),Q),Xa)
     => ( ( (Pa)
         => aa_int_o(R4,F) )
       => ( ! [Q3: num] :
              ( aa_num_o(Q,Q3)
             => aa_int_o(R4,aa_num_int(G,Q3)) )
         => aa_int_o(R4,aa_option_num_int(case_option_int_num(F,G),Xa)) ) ) ) ).

% disjE_realizer2
tff(fact_797_disjE__realizer2,axiom,
    ! [Pa: $o,Q: fun_num_o,Xa: option_num,R4: fun_option_num_o,F: option_num,G: fun_num_option_num] :
      ( aa_option_num_o(case_option_o_num((Pa),Q),Xa)
     => ( ( (Pa)
         => aa_option_num_o(R4,F) )
       => ( ! [Q3: num] :
              ( aa_num_o(Q,Q3)
             => aa_option_num_o(R4,aa_num_option_num(G,Q3)) )
         => aa_option_num_o(R4,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F,G),Xa)) ) ) ) ).

% disjE_realizer2
tff(fact_798_disjE__realizer2,axiom,
    ! [Pa: $o,Q: fun_num_o,Xa: option_num,R4: fun_num_o,F: num,G: fun_num_num] :
      ( aa_option_num_o(case_option_o_num((Pa),Q),Xa)
     => ( ( (Pa)
         => aa_num_o(R4,F) )
       => ( ! [Q3: num] :
              ( aa_num_o(Q,Q3)
             => aa_num_o(R4,aa_num_num(G,Q3)) )
         => aa_num_o(R4,aa_option_num_num(case_option_num_num(F,G),Xa)) ) ) ) ).

% disjE_realizer2
tff(fact_799_sgn__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% sgn_neg
tff(fact_800_sgn__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% sgn_neg
tff(fact_801_sgn__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( sgn_sgn_rat(A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% sgn_neg
tff(fact_802_of__nat__code,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,N) = aa_Cod1783974652533396728nteger(semiri4055485073559036834nteger(aTP_Lamm_bk,N),zero_z3403309356797280102nteger) ).

% of_nat_code
tff(fact_803_of__nat__code,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,N) = semiri7787848453975740701ux_rat(aTP_Lamm_bl,N,zero_zero_rat) ).

% of_nat_code
tff(fact_804_of__nat__code,axiom,
    ! [N: nat] : aa_nat_Code_natural(semiri3763490453095760265atural,N) = semiri2879080225315754349atural(aTP_Lamm_bm,N,zero_z2226904508553997617atural) ).

% of_nat_code
tff(fact_805_of__nat__code,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_int_int(aa_nat_fun_int_int(semiri8420488043553186161ux_int(aTP_Lamm_bn),N),zero_zero_int) ).

% of_nat_code
tff(fact_806_of__nat__code,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_bo),N),zero_zero_nat) ).

% of_nat_code
tff(fact_807_add_Ogroup__axioms,axiom,
    group_int(plus_plus_int,zero_zero_int,uminus_uminus_int) ).

% add.group_axioms
tff(fact_808_add_Ogroup__axioms,axiom,
    group_Code_integer(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger,uminus1351360451143612070nteger) ).

% add.group_axioms
tff(fact_809_add_Ogroup__axioms,axiom,
    group_rat(plus_plus_rat,zero_zero_rat,uminus_uminus_rat) ).

% add.group_axioms
tff(fact_810_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_1_neg
tff(fact_811_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_1_neg
tff(fact_812_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat(A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% sgn_1_neg
tff(fact_813_sgn__if,axiom,
    ! [Xa: int] :
      aa_int_int(sgn_sgn_int,Xa) = $ite(
        Xa = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xa),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% sgn_if
tff(fact_814_sgn__if,axiom,
    ! [Xa: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,Xa) = $ite(
        Xa = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Xa),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ) ).

% sgn_if
tff(fact_815_sgn__if,axiom,
    ! [Xa: rat] :
      sgn_sgn_rat(Xa) = $ite(
        Xa = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_if
tff(fact_816_is__num_Osimps,axiom,
    ! [A: int] :
      ( aa_int_o(neg_nu7441568979088483369um_int,A)
    <=> ( ( A = one_one_int )
        | ? [X3: int] :
            ( ( A = aa_int_int(uminus_uminus_int,X3) )
            & aa_int_o(neg_nu7441568979088483369um_int,X3) )
        | ? [X3: int,Y: int] :
            ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y) )
            & aa_int_o(neg_nu7441568979088483369um_int,X3)
            & aa_int_o(neg_nu7441568979088483369um_int,Y) ) ) ) ).

% is_num.simps
tff(fact_817_is__num_Osimps,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(neg_nu2435721316971906650nteger,A)
    <=> ( ( A = one_one_Code_integer )
        | ? [X3: code_integer] :
            ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X3) )
            & aa_Code_integer_o(neg_nu2435721316971906650nteger,X3) )
        | ? [X3: code_integer,Y: code_integer] :
            ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),Y) )
            & aa_Code_integer_o(neg_nu2435721316971906650nteger,X3)
            & aa_Code_integer_o(neg_nu2435721316971906650nteger,Y) ) ) ) ).

% is_num.simps
tff(fact_818_is__num_Osimps,axiom,
    ! [A: rat] :
      ( aa_rat_o(neg_nu6808929389511037909um_rat,A)
    <=> ( ( A = one_one_rat )
        | ? [X3: rat] :
            ( ( A = aa_rat_rat(uminus_uminus_rat,X3) )
            & aa_rat_o(neg_nu6808929389511037909um_rat,X3) )
        | ? [X3: rat,Y: rat] :
            ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y) )
            & aa_rat_o(neg_nu6808929389511037909um_rat,X3)
            & aa_rat_o(neg_nu6808929389511037909um_rat,Y) ) ) ) ).

% is_num.simps
tff(fact_819_is__num_Ocases,axiom,
    ! [A: int] :
      ( aa_int_o(neg_nu7441568979088483369um_int,A)
     => ( ( A != one_one_int )
       => ( ! [X4: int] :
              ( ( A = aa_int_int(uminus_uminus_int,X4) )
             => ~ aa_int_o(neg_nu7441568979088483369um_int,X4) )
         => ~ ! [X4: int,Y4: int] :
                ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),Y4) )
               => ( aa_int_o(neg_nu7441568979088483369um_int,X4)
                 => ~ aa_int_o(neg_nu7441568979088483369um_int,Y4) ) ) ) ) ) ).

% is_num.cases
tff(fact_820_is__num_Ocases,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(neg_nu2435721316971906650nteger,A)
     => ( ( A != one_one_Code_integer )
       => ( ! [X4: code_integer] :
              ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X4) )
             => ~ aa_Code_integer_o(neg_nu2435721316971906650nteger,X4) )
         => ~ ! [X4: code_integer,Y4: code_integer] :
                ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X4),Y4) )
               => ( aa_Code_integer_o(neg_nu2435721316971906650nteger,X4)
                 => ~ aa_Code_integer_o(neg_nu2435721316971906650nteger,Y4) ) ) ) ) ) ).

% is_num.cases
tff(fact_821_is__num_Ocases,axiom,
    ! [A: rat] :
      ( aa_rat_o(neg_nu6808929389511037909um_rat,A)
     => ( ( A != one_one_rat )
       => ( ! [X4: rat] :
              ( ( A = aa_rat_rat(uminus_uminus_rat,X4) )
             => ~ aa_rat_o(neg_nu6808929389511037909um_rat,X4) )
         => ~ ! [X4: rat,Y4: rat] :
                ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),Y4) )
               => ( aa_rat_o(neg_nu6808929389511037909um_rat,X4)
                 => ~ aa_rat_o(neg_nu6808929389511037909um_rat,Y4) ) ) ) ) ) ).

% is_num.cases
tff(fact_822_Ints__nonzero__abs__less1,axiom,
    ! [Xa: code_integer] :
      ( member_Code_integer(Xa,ring_11222124179247155820nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xa)),one_one_Code_integer)
       => ( Xa = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_823_Ints__nonzero__abs__less1,axiom,
    ! [Xa: rat] :
      ( member_rat(Xa,ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(Xa)),one_one_rat)
       => ( Xa = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_824_Ints__nonzero__abs__less1,axiom,
    ! [Xa: int] :
      ( member_int(Xa,ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xa)),one_one_int)
       => ( Xa = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_825_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_826_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_827_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_828_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_829_abs__abs,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ).

% abs_abs
tff(fact_830_abs__abs,axiom,
    ! [A: rat] : abs_abs_rat(abs_abs_rat(A)) = abs_abs_rat(A) ).

% abs_abs
tff(fact_831_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_832_abs__idempotent,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ).

% abs_idempotent
tff(fact_833_abs__idempotent,axiom,
    ! [A: rat] : abs_abs_rat(abs_abs_rat(A)) = abs_abs_rat(A) ).

% abs_idempotent
tff(fact_834_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_835_sgn__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) ).

% sgn_sgn
tff(fact_836_sgn__sgn,axiom,
    ! [A: rat] : sgn_sgn_rat(sgn_sgn_rat(A)) = sgn_sgn_rat(A) ).

% sgn_sgn
tff(fact_837_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_838_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_839_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = abs_abs_rat(A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_840_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_841_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_842_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_843_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_844_abs__zero,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_845_abs__zero,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_846_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_847_abs__0,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_848_abs__0,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_849_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ).

% abs_add_abs
tff(fact_850_abs__add__abs,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B)) ).

% abs_add_abs
tff(fact_851_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_852_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_853_abs__1,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_854_abs__1,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_1
tff(fact_855_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_856_abs__minus,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ).

% abs_minus
tff(fact_857_abs__minus,axiom,
    ! [A: rat] : abs_abs_rat(aa_rat_rat(uminus_uminus_rat,A)) = abs_abs_rat(A) ).

% abs_minus
tff(fact_858_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_859_abs__minus__cancel,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ).

% abs_minus_cancel
tff(fact_860_abs__minus__cancel,axiom,
    ! [A: rat] : abs_abs_rat(aa_rat_rat(uminus_uminus_rat,A)) = abs_abs_rat(A) ).

% abs_minus_cancel
tff(fact_861_abs__of__nat,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).

% abs_of_nat
tff(fact_862_abs__of__nat,axiom,
    ! [N: nat] : abs_abs_rat(aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% abs_of_nat
tff(fact_863_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_864_sgn__0,axiom,
    aa_int_int(sgn_sgn_int,zero_zero_int) = zero_zero_int ).

% sgn_0
tff(fact_865_sgn__0,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% sgn_0
tff(fact_866_sgn__0,axiom,
    sgn_sgn_rat(zero_zero_rat) = zero_zero_rat ).

% sgn_0
tff(fact_867_sgn__1,axiom,
    aa_int_int(sgn_sgn_int,one_one_int) = one_one_int ).

% sgn_1
tff(fact_868_sgn__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% sgn_1
tff(fact_869_sgn__1,axiom,
    sgn_sgn_rat(one_one_rat) = one_one_rat ).

% sgn_1
tff(fact_870_sgn__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)) ).

% sgn_minus
tff(fact_871_sgn__minus,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) ).

% sgn_minus
tff(fact_872_sgn__minus,axiom,
    ! [A: rat] : sgn_sgn_rat(aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,sgn_sgn_rat(A)) ).

% sgn_minus
tff(fact_873_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_874_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),abs_abs_rat(A))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_875_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_876_of__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% of_nat_0
tff(fact_877_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_878_of__nat__0,axiom,
    aa_nat_Code_natural(semiri3763490453095760265atural,zero_zero_nat) = zero_z2226904508553997617atural ).

% of_nat_0
tff(fact_879_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_880_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_881_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_882_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_883_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_z2226904508553997617atural = aa_nat_Code_natural(semiri3763490453095760265atural,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_884_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_885_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_886_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = zero_z3403309356797280102nteger )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_887_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = zero_zero_rat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_888_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_Code_natural(semiri3763490453095760265atural,M) = zero_z2226904508553997617atural )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_889_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_890_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_891_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_iff
tff(fact_892_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_iff
tff(fact_893_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_iff
tff(fact_894_abs__neg__one,axiom,
    aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,one_one_int)) = one_one_int ).

% abs_neg_one
tff(fact_895_abs__neg__one,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_896_abs__neg__one,axiom,
    abs_abs_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_897_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_add
tff(fact_898_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_899_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_900_sgn__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_less
tff(fact_901_sgn__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,sgn_sgn_rat(A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% sgn_less
tff(fact_902_sgn__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(sgn_sgn_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_less
tff(fact_903_sgn__greater,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% sgn_greater
tff(fact_904_sgn__greater,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),sgn_sgn_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_greater
tff(fact_905_sgn__greater,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(sgn_sgn_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_greater
tff(fact_906_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_907_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_908_of__nat__1,axiom,
    aa_nat_Code_natural(semiri3763490453095760265atural,one_one_nat) = one_one_Code_natural ).

% of_nat_1
tff(fact_909_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_910_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_911_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_912_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_913_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_Code_natural = aa_nat_Code_natural(semiri3763490453095760265atural,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_914_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_915_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_916_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,N) = one_one_Code_integer )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_917_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,N) = one_one_rat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_918_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_Code_natural(semiri3763490453095760265atural,N) = one_one_Code_natural )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_919_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_920_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_921_sgn__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer ) ) ).

% sgn_pos
tff(fact_922_sgn__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( sgn_sgn_rat(A) = one_one_rat ) ) ).

% sgn_pos
tff(fact_923_sgn__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(sgn_sgn_int,A) = one_one_int ) ) ).

% sgn_pos
tff(fact_924_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_925_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
tff(fact_926_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( abs_abs_rat(sgn_sgn_rat(A)) = one_one_rat ) ) ).

% abs_sgn_eq_1
tff(fact_927_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% of_nat_0_less_iff
tff(fact_928_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_nat_Code_natural(semiri3763490453095760265atural,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% of_nat_0_less_iff
tff(fact_929_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% of_nat_0_less_iff
tff(fact_930_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% of_nat_0_less_iff
tff(fact_931_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% of_nat_0_less_iff
tff(fact_932_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ).

% nat_neq_iff
tff(fact_933_less__not__refl,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N) ).

% less_not_refl
tff(fact_934_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_935_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),T)
     => ( S != T ) ) ).

% less_not_refl3
tff(fact_936_less__irrefl__nat,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N) ).

% less_irrefl_nat
tff(fact_937_nat__less__induct,axiom,
    ! [Pa: fun_nat_o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
             => aa_nat_o(Pa,M3) )
         => aa_nat_o(Pa,N2) )
     => aa_nat_o(Pa,N) ) ).

% nat_less_induct
tff(fact_938_infinite__descent,axiom,
    ! [Pa: fun_nat_o,N: nat] :
      ( ! [N2: nat] :
          ( ~ aa_nat_o(Pa,N2)
         => ? [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
              & ~ aa_nat_o(Pa,M3) ) )
     => aa_nat_o(Pa,N) ) ).

% infinite_descent
tff(fact_939_linorder__neqE__nat,axiom,
    ! [Xa: nat,Y3: nat] :
      ( ( Xa != Y3 )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xa) ) ) ).

% linorder_neqE_nat
tff(fact_940_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(2)
tff(fact_941_same__sgn__abs__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ) ) ).

% same_sgn_abs_add
tff(fact_942_same__sgn__abs__add,axiom,
    ! [B: rat,A: rat] :
      ( ( sgn_sgn_rat(B) = sgn_sgn_rat(A) )
     => ( abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B)) ) ) ).

% same_sgn_abs_add
tff(fact_943_same__sgn__abs__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% same_sgn_abs_add
tff(fact_944_nat__less__as__int,axiom,
    ! [X: nat,Xa3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Xa3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa3)) ) ).

% nat_less_as_int
tff(fact_945_abs__sgn__eq,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = $ite(A = zero_zero_int,zero_zero_int,one_one_int) ).

% abs_sgn_eq
tff(fact_946_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = $ite(A = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,one_one_Code_integer) ).

% abs_sgn_eq
tff(fact_947_abs__sgn__eq,axiom,
    ! [A: rat] :
      abs_abs_rat(sgn_sgn_rat(A)) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% abs_sgn_eq
tff(fact_948_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_949_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_950_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_951_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_952_abs__one,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_953_abs__one,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_one
tff(fact_954_abs__eq__iff,axiom,
    ! [Xa: int,Y3: int] :
      ( ( aa_int_int(abs_abs_int,Xa) = aa_int_int(abs_abs_int,Y3) )
    <=> ( ( Xa = Y3 )
        | ( Xa = aa_int_int(uminus_uminus_int,Y3) ) ) ) ).

% abs_eq_iff
tff(fact_955_abs__eq__iff,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xa) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y3) )
    <=> ( ( Xa = Y3 )
        | ( Xa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y3) ) ) ) ).

% abs_eq_iff
tff(fact_956_abs__eq__iff,axiom,
    ! [Xa: rat,Y3: rat] :
      ( ( abs_abs_rat(Xa) = abs_abs_rat(Y3) )
    <=> ( ( Xa = Y3 )
        | ( Xa = aa_rat_rat(uminus_uminus_rat,Y3) ) ) ) ).

% abs_eq_iff
tff(fact_957_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_imp_less
tff(fact_958_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_imp_less
tff(fact_959_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_imp_less
tff(fact_960_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ).

% less_imp_of_nat_less
tff(fact_961_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% less_imp_of_nat_less
tff(fact_962_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ).

% less_imp_of_nat_less
tff(fact_963_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_964_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) )
       => ~ ! [N2: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ) ).

% int_cases3
tff(fact_965_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).

% int_cases4
tff(fact_966_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N2: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% neg_int_cases
tff(fact_967_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% pos_int_cases
tff(fact_968_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_969_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))),Z) ).

% zadd_int_left
tff(fact_970_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_971_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_972_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_973_sgn__0__0,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
tff(fact_974_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_0_0
tff(fact_975_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_976_sgn__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
tff(fact_977_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
tff(fact_978_same__sgn__sgn__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) ) ) ).

% same_sgn_sgn_add
tff(fact_979_same__sgn__sgn__add,axiom,
    ! [B: rat,A: rat] :
      ( ( sgn_sgn_rat(B) = sgn_sgn_rat(A) )
     => ( sgn_sgn_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = sgn_sgn_rat(A) ) ) ).

% same_sgn_sgn_add
tff(fact_980_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(sgn_sgn_int,A) ) ) ).

% same_sgn_sgn_add
tff(fact_981_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_982_Ints__abs,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => member_int(aa_int_int(abs_abs_int,A),ring_1_Ints_int) ) ).

% Ints_abs
tff(fact_983_Ints__abs,axiom,
    ! [A: code_integer] :
      ( member_Code_integer(A,ring_11222124179247155820nteger)
     => member_Code_integer(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A),ring_11222124179247155820nteger) ) ).

% Ints_abs
tff(fact_984_Ints__abs,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => member_rat(abs_abs_rat(A),ring_1_Ints_rat) ) ).

% Ints_abs
tff(fact_985_Ints__of__nat,axiom,
    ! [N: nat] : member_int(aa_nat_int(semiri1314217659103216013at_int,N),ring_1_Ints_int) ).

% Ints_of_nat
tff(fact_986_is__num__normalize_I6_J,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(neg_nu6808929389511037909um_rat,Xa)
     => ( aa_rat_o(neg_nu6808929389511037909um_rat,Y3)
       => aa_rat_o(neg_nu6808929389511037909um_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y3)) ) ) ).

% is_num_normalize(6)
tff(fact_987_is__num__normalize_I6_J,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(neg_nu7441568979088483369um_int,Xa)
     => ( aa_int_o(neg_nu7441568979088483369um_int,Y3)
       => aa_int_o(neg_nu7441568979088483369um_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y3)) ) ) ).

% is_num_normalize(6)
tff(fact_988_is__num__add__commute,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(neg_nu6808929389511037909um_rat,Xa)
     => ( aa_rat_o(neg_nu6808929389511037909um_rat,Y3)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y3),Xa) ) ) ) ).

% is_num_add_commute
tff(fact_989_is__num__add__commute,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(neg_nu7441568979088483369um_int,Xa)
     => ( aa_int_o(neg_nu7441568979088483369um_int,Y3)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),Xa) ) ) ) ).

% is_num_add_commute
tff(fact_990_is__num__add__left__commute,axiom,
    ! [Xa: rat,Y3: rat,Z: rat] :
      ( aa_rat_o(neg_nu6808929389511037909um_rat,Xa)
     => ( aa_rat_o(neg_nu6808929389511037909um_rat,Y3)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y3),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Z)) ) ) ) ).

% is_num_add_left_commute
tff(fact_991_is__num__add__left__commute,axiom,
    ! [Xa: int,Y3: int,Z: int] :
      ( aa_int_o(neg_nu7441568979088483369um_int,Xa)
     => ( aa_int_o(neg_nu7441568979088483369um_int,Y3)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Z)) ) ) ) ).

% is_num_add_left_commute
tff(fact_992_is__num__normalize_I4_J,axiom,
    aa_int_o(neg_nu7441568979088483369um_int,one_one_int) ).

% is_num_normalize(4)
tff(fact_993_is__num__normalize_I4_J,axiom,
    aa_Code_integer_o(neg_nu2435721316971906650nteger,one_one_Code_integer) ).

% is_num_normalize(4)
tff(fact_994_is__num__normalize_I4_J,axiom,
    aa_rat_o(neg_nu6808929389511037909um_rat,one_one_rat) ).

% is_num_normalize(4)
tff(fact_995_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_996_abs__not__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_997_abs__not__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_998_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = A ) ) ).

% abs_of_pos
tff(fact_999_abs__of__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_pos
tff(fact_1000_abs__of__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_1001_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_less_iff
tff(fact_1002_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_less_iff
tff(fact_1003_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_less_iff
tff(fact_1004_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger) ).

% of_nat_less_0_iff
tff(fact_1005_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),zero_z2226904508553997617atural) ).

% of_nat_less_0_iff
tff(fact_1006_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_1007_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_1008_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_1009_sgn__not__eq__imp,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) != aa_int_int(sgn_sgn_int,A) )
     => ( ( aa_int_int(sgn_sgn_int,A) != zero_zero_int )
       => ( ( aa_int_int(sgn_sgn_int,B) != zero_zero_int )
         => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_1010_sgn__not__eq__imp,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) != aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) != zero_z3403309356797280102nteger )
       => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) != zero_z3403309356797280102nteger )
         => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_1011_sgn__not__eq__imp,axiom,
    ! [B: rat,A: rat] :
      ( ( sgn_sgn_rat(B) != sgn_sgn_rat(A) )
     => ( ( sgn_sgn_rat(A) != zero_zero_rat )
       => ( ( sgn_sgn_rat(B) != zero_zero_rat )
         => ( sgn_sgn_rat(A) = aa_rat_rat(uminus_uminus_rat,sgn_sgn_rat(B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_1012_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_1013_sgn__minus__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% sgn_minus_1
tff(fact_1014_sgn__minus__1,axiom,
    sgn_sgn_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% sgn_minus_1
tff(fact_1015_abs__if__raw,axiom,
    ! [X: int] :
      aa_int_int(abs_abs_int,X) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int),aa_int_int(uminus_uminus_int,X),X) ).

% abs_if_raw
tff(fact_1016_abs__if__raw,axiom,
    ! [X: code_integer] :
      aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X),X) ).

% abs_if_raw
tff(fact_1017_abs__if__raw,axiom,
    ! [X: rat] :
      abs_abs_rat(X) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X),X) ).

% abs_if_raw
tff(fact_1018_abs__if,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,A) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int),aa_int_int(uminus_uminus_int,A),A) ).

% abs_if
tff(fact_1019_abs__if,axiom,
    ! [A: code_integer] :
      aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) ).

% abs_if
tff(fact_1020_abs__if,axiom,
    ! [A: rat] :
      abs_abs_rat(A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_if
tff(fact_1021_abs__of__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_1022_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_1023_abs__of__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( abs_abs_rat(A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_neg
tff(fact_1024_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% sgn_1_pos
tff(fact_1025_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat(A) = one_one_rat )
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_1_pos
tff(fact_1026_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = one_one_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_1_pos
tff(fact_1027_abs__add__one__gt__zero,axiom,
    ! [Xa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xa))) ).

% abs_add_one_gt_zero
tff(fact_1028_abs__add__one__gt__zero,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),abs_abs_rat(Xa))) ).

% abs_add_one_gt_zero
tff(fact_1029_abs__add__one__gt__zero,axiom,
    ! [Xa: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,Xa))) ).

% abs_add_one_gt_zero
tff(fact_1030_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc: nat] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F,produc487386426758144856at_nat(A,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Acc))))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_1031_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [Xa: fun_nat_fun_nat_nat,Xaa: nat,Xb: nat,Xc: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xa,Xaa),Xb),Xc) = Y3 )
     => ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(Xa,produc487386426758144856at_nat(Xaa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xc))))
       => ~ ( ( Y3 = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xaa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(Xa,Xaa),Xc))) )
           => ~ accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(Xa,produc487386426758144856at_nat(Xaa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_1032_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [Xa: fun_nat_fun_nat_nat,Xaa: nat,Xb: nat,Xc: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xa,Xaa),Xb),Xc) = Y3 )
     => ( Y3 = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xaa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(Xa,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_1033_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_1034_sgn__of__nat,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_1035_sgn__of__nat,axiom,
    ! [N: nat] : sgn_sgn_rat(aa_nat_rat(semiri681578069525770553at_rat,N)) = zero_n2052037380579107095ol_rat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_1036_sgn__of__nat,axiom,
    ! [N: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_1037_Ints__nonzero__abs__ge1,axiom,
    ! [Xa: code_integer] :
      ( member_Code_integer(Xa,ring_11222124179247155820nteger)
     => ( ( Xa != zero_z3403309356797280102nteger )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xa)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_1038_Ints__nonzero__abs__ge1,axiom,
    ! [Xa: rat] :
      ( member_rat(Xa,ring_1_Ints_rat)
     => ( ( Xa != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),abs_abs_rat(Xa)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_1039_Ints__nonzero__abs__ge1,axiom,
    ! [Xa: int] :
      ( member_int(Xa,ring_1_Ints_int)
     => ( ( Xa != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(abs_abs_int,Xa)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_1040_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_num_num(plus_plus_num(aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_1041_Ints__eq__abs__less1,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( member_Code_integer(Xa,ring_11222124179247155820nteger)
     => ( member_Code_integer(Y3,ring_11222124179247155820nteger)
       => ( ( Xa = Y3 )
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xa),Y3))),one_one_Code_integer) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_1042_Ints__eq__abs__less1,axiom,
    ! [Xa: rat,Y3: rat] :
      ( member_rat(Xa,ring_1_Ints_rat)
     => ( member_rat(Y3,ring_1_Ints_rat)
       => ( ( Xa = Y3 )
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(minus_minus_rat(Xa,Y3))),one_one_rat) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_1043_Ints__eq__abs__less1,axiom,
    ! [Xa: int,Y3: int] :
      ( member_int(Xa,ring_1_Ints_int)
     => ( member_int(Y3,ring_1_Ints_int)
       => ( ( Xa = Y3 )
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y3))),one_one_int) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_1044_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1045_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_nat_Code_natural(power_7079662738309270450atural(aa_nat_Code_natural(semiri3763490453095760265atural,Xa)),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1046_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Xa)),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1047_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1048_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1049_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_1050_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( abs_abs_rat(A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_1051_abs__of__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_1052_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 )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% diff_is_0_eq
tff(fact_1053_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_1054_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_1055_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_1056_le0,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),N) ).

% le0
tff(fact_1057_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A) ).

% bot_nat_0.extremum
tff(fact_1058_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,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_1059_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),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_1060_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,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_1061_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_1062_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% nat_add_left_cancel_le
tff(fact_1063_le__zero__eq,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),zero_zero_nat)
    <=> ( N = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_1064_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_1065_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_1066_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_1067_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_1068_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_1069_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_1070_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: code_natural] : minus_7197305767214868737atural(A,A) = zero_z2226904508553997617atural ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1071_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_1072_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_1073_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),A) = zero_z3403309356797280102nteger ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1074_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1075_diff__zero,axiom,
    ! [A: code_natural] : minus_7197305767214868737atural(A,zero_z2226904508553997617atural) = A ).

% diff_zero
tff(fact_1076_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_1077_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_1078_diff__zero,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),zero_z3403309356797280102nteger) = A ).

% diff_zero
tff(fact_1079_diff__zero,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_zero
tff(fact_1080_zero__diff,axiom,
    ! [A: code_natural] : minus_7197305767214868737atural(zero_z2226904508553997617atural,A) = zero_z2226904508553997617atural ).

% zero_diff
tff(fact_1081_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_1082_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_1083_diff__0__right,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),zero_z3403309356797280102nteger) = A ).

% diff_0_right
tff(fact_1084_diff__0__right,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_0_right
tff(fact_1085_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_1086_diff__self,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),A) = zero_z3403309356797280102nteger ).

% diff_self
tff(fact_1087_diff__self,axiom,
    ! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).

% diff_self
tff(fact_1088_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% neg_le_iff_le
tff(fact_1089_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% neg_le_iff_le
tff(fact_1090_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% neg_le_iff_le
tff(fact_1091_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_1092_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_1093_add__diff__cancel__right_H,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_1094_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),B) = A ).

% add_diff_cancel_right'
tff(fact_1095_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_1096_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_1097_add__diff__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) ).

% add_diff_cancel_right
tff(fact_1098_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = minus_minus_rat(A,B) ).

% add_diff_cancel_right
tff(fact_1099_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_1100_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_1101_add__diff__cancel__left_H,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_1102_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),A) = B ).

% add_diff_cancel_left'
tff(fact_1103_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_1104_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_1105_add__diff__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) ).

% add_diff_cancel_left
tff(fact_1106_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) = minus_minus_rat(A,B) ).

% add_diff_cancel_left
tff(fact_1107_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ).

% diff_add_cancel
tff(fact_1108_diff__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),B) = A ).

% diff_add_cancel
tff(fact_1109_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,B)),B) = A ).

% diff_add_cancel
tff(fact_1110_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel
tff(fact_1111_add__diff__cancel,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B) = A ).

% add_diff_cancel
tff(fact_1112_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),B) = A ).

% add_diff_cancel
tff(fact_1113_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% of_nat_le_iff
tff(fact_1114_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% of_nat_le_iff
tff(fact_1115_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% of_nat_le_iff
tff(fact_1116_minus__diff__eq,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% minus_diff_eq
tff(fact_1117_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A) ).

% minus_diff_eq
tff(fact_1118_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,minus_minus_rat(A,B)) = minus_minus_rat(B,A) ).

% minus_diff_eq
tff(fact_1119_power__one,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),N) = one_one_Code_integer ).

% power_one
tff(fact_1120_power__one,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(one_one_rat),N) = one_one_rat ).

% power_one
tff(fact_1121_power__one,axiom,
    ! [N: nat] : aa_nat_Code_natural(power_7079662738309270450atural(one_one_Code_natural),N) = one_one_Code_natural ).

% power_one
tff(fact_1122_power__one,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),N) = one_one_nat ).

% power_one
tff(fact_1123_power__one,axiom,
    ! [N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,one_one_int),N) = one_one_int ).

% power_one
tff(fact_1124_nat__zero__less__power__iff,axiom,
    ! [Xa: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1125_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% zero_less_diff
tff(fact_1126_of__bool__less__eq__iff,axiom,
    ! [Pa: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((Pa))),zero_n2052037380579107095ol_rat((Q)))
    <=> ( (Pa)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_1127_of__bool__less__eq__iff,axiom,
    ! [Pa: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(Pa))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( (Pa)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_1128_of__bool__less__eq__iff,axiom,
    ! [Pa: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( (Pa)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_1129_of__bool__eq__0__iff,axiom,
    ! [Pa: $o] :
      ( ( zero_n356916108424825756nteger((Pa)) = zero_z3403309356797280102nteger )
    <=> ~ (Pa) ) ).

% of_bool_eq_0_iff
tff(fact_1130_of__bool__eq__0__iff,axiom,
    ! [Pa: $o] :
      ( ( zero_n2052037380579107095ol_rat((Pa)) = zero_zero_rat )
    <=> ~ (Pa) ) ).

% of_bool_eq_0_iff
tff(fact_1131_of__bool__eq__0__iff,axiom,
    ! [Pa: $o] :
      ( ( zero_n8403883297036319079atural((Pa)) = zero_z2226904508553997617atural )
    <=> ~ (Pa) ) ).

% of_bool_eq_0_iff
tff(fact_1132_of__bool__eq__0__iff,axiom,
    ! [Pa: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(Pa)) = zero_zero_int )
    <=> ~ (Pa) ) ).

% of_bool_eq_0_iff
tff(fact_1133_of__bool__eq__0__iff,axiom,
    ! [Pa: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(Pa)) = zero_zero_nat )
    <=> ~ (Pa) ) ).

% of_bool_eq_0_iff
tff(fact_1134_of__bool__eq_I1_J,axiom,
    zero_n356916108424825756nteger($false) = zero_z3403309356797280102nteger ).

% of_bool_eq(1)
tff(fact_1135_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat($false) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_1136_of__bool__eq_I1_J,axiom,
    zero_n8403883297036319079atural($false) = zero_z2226904508553997617atural ).

% of_bool_eq(1)
tff(fact_1137_of__bool__eq_I1_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$false) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_1138_of__bool__eq_I1_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$false) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_1139_of__bool__less__iff,axiom,
    ! [Pa: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((Pa))),zero_n2052037380579107095ol_rat((Q)))
    <=> ( ~ (Pa)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_1140_of__bool__less__iff,axiom,
    ! [Pa: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(Pa))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( ~ (Pa)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_1141_of__bool__less__iff,axiom,
    ! [Pa: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( ~ (Pa)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_1142_power__one__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),one_one_nat) = A ).

% power_one_right
tff(fact_1143_power__one__right,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),one_one_nat) = A ).

% power_one_right
tff(fact_1144_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% zle_diff1_eq
tff(fact_1145_of__bool__eq__1__iff,axiom,
    ! [Pa: $o] :
      ( ( zero_n356916108424825756nteger((Pa)) = one_one_Code_integer )
    <=> (Pa) ) ).

% of_bool_eq_1_iff
tff(fact_1146_of__bool__eq__1__iff,axiom,
    ! [Pa: $o] :
      ( ( zero_n2052037380579107095ol_rat((Pa)) = one_one_rat )
    <=> (Pa) ) ).

% of_bool_eq_1_iff
tff(fact_1147_of__bool__eq__1__iff,axiom,
    ! [Pa: $o] :
      ( ( zero_n8403883297036319079atural((Pa)) = one_one_Code_natural )
    <=> (Pa) ) ).

% of_bool_eq_1_iff
tff(fact_1148_of__bool__eq__1__iff,axiom,
    ! [Pa: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(Pa)) = one_one_int )
    <=> (Pa) ) ).

% of_bool_eq_1_iff
tff(fact_1149_of__bool__eq__1__iff,axiom,
    ! [Pa: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(Pa)) = one_one_nat )
    <=> (Pa) ) ).

% of_bool_eq_1_iff
tff(fact_1150_of__bool__eq_I2_J,axiom,
    zero_n356916108424825756nteger($true) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_1151_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat($true) = one_one_rat ).

% of_bool_eq(2)
tff(fact_1152_of__bool__eq_I2_J,axiom,
    zero_n8403883297036319079atural($true) = one_one_Code_natural ).

% of_bool_eq(2)
tff(fact_1153_of__bool__eq_I2_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).

% of_bool_eq(2)
tff(fact_1154_of__bool__eq_I2_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_1155_negative__zle,axiom,
    ! [N: nat,M: nat] : aa_int_o(aa_int_fun_int_o(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_1156_of__nat__of__bool,axiom,
    ! [Pa: $o] : aa_nat_int(semiri1314217659103216013at_int,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))) = aa_o_int(zero_n2684676970156552555ol_int,(Pa)) ).

% of_nat_of_bool
tff(fact_1157_of__nat__of__bool,axiom,
    ! [Pa: $o] : aa_nat_nat(semiri1316708129612266289at_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))) = aa_o_nat(zero_n2687167440665602831ol_nat,(Pa)) ).

% of_nat_of_bool
tff(fact_1158_abs__bool__eq,axiom,
    ! [Pa: $o] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_n356916108424825756nteger((Pa))) = zero_n356916108424825756nteger((Pa)) ).

% abs_bool_eq
tff(fact_1159_abs__bool__eq,axiom,
    ! [Pa: $o] : abs_abs_rat(zero_n2052037380579107095ol_rat((Pa))) = zero_n2052037380579107095ol_rat((Pa)) ).

% abs_bool_eq
tff(fact_1160_abs__bool__eq,axiom,
    ! [Pa: $o] : aa_int_int(abs_abs_int,aa_o_int(zero_n2684676970156552555ol_int,(Pa))) = aa_o_int(zero_n2684676970156552555ol_int,(Pa)) ).

% abs_bool_eq
tff(fact_1161_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1162_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1163_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1164_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1165_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1166_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1167_le__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ).

% le_add_same_cancel2
tff(fact_1168_le__add__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A))
    <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B) ) ).

% le_add_same_cancel2
tff(fact_1169_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_1170_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_1171_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_1172_le__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ).

% le_add_same_cancel1
tff(fact_1173_le__add__same__cancel1,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B))
    <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B) ) ).

% le_add_same_cancel1
tff(fact_1174_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_1175_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_1176_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_1177_add__le__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% add_le_same_cancel2
tff(fact_1178_add__le__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),B)
    <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural) ) ).

% add_le_same_cancel2
tff(fact_1179_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_1180_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_1181_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_1182_add__le__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% add_le_same_cancel1
tff(fact_1183_add__le__same__cancel1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A)),B)
    <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural) ) ).

% add_le_same_cancel1
tff(fact_1184_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_1185_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_1186_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_1187_diff__ge__0__iff__ge,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_1188_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),minus_minus_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_1189_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_1190_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_1191_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_1192_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_1193_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_le_0_iff_le
tff(fact_1194_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_le_0_iff_le
tff(fact_1195_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_le_0_iff_le
tff(fact_1196_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_1197_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_1198_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_1199_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_eq_nonneg
tff(fact_1200_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_less_eq_nonneg
tff(fact_1201_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_less_eq_nonneg
tff(fact_1202_diff__gt__0__iff__gt,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_1203_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),minus_minus_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_1204_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_1205_le__add__diff__inverse2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_1206_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_1207_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_1208_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_1209_le__add__diff__inverse,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_1210_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),minus_minus_rat(A,B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_1211_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_1212_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_1213_diff__add__zero,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) = zero_z2226904508553997617atural ).

% diff_add_zero
tff(fact_1214_diff__add__zero,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ).

% diff_add_zero
tff(fact_1215_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_1216_diff__numeral__special_I9_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% diff_numeral_special(9)
tff(fact_1217_diff__numeral__special_I9_J,axiom,
    minus_minus_rat(one_one_rat,one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_1218_verit__minus__simplify_I3_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B) = aa_int_int(uminus_uminus_int,B) ).

% verit_minus_simplify(3)
tff(fact_1219_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,zero_z3403309356797280102nteger),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_1220_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : minus_minus_rat(zero_zero_rat,B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_1221_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_1222_diff__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,zero_z3403309356797280102nteger),A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_1223_diff__0,axiom,
    ! [A: rat] : minus_minus_rat(zero_zero_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_1224_diff__minus__eq__add,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) ).

% diff_minus_eq_add
tff(fact_1225_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) ).

% diff_minus_eq_add
tff(fact_1226_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) ).

% diff_minus_eq_add
tff(fact_1227_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% uminus_add_conv_diff
tff(fact_1228_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A) ).

% uminus_add_conv_diff
tff(fact_1229_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = minus_minus_rat(B,A) ).

% uminus_add_conv_diff
tff(fact_1230_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = A ) ) ).

% abs_of_nonneg
tff(fact_1231_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_nonneg
tff(fact_1232_abs__of__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_1233_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% abs_le_self_iff
tff(fact_1234_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% abs_le_self_iff
tff(fact_1235_abs__le__self__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% abs_le_self_iff
tff(fact_1236_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_1237_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_1238_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_1239_power__inject__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),M) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_1240_power__inject__exp,axiom,
    ! [A: code_natural,M: nat,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
     => ( ( aa_nat_Code_natural(power_7079662738309270450atural(A),M) = aa_nat_Code_natural(power_7079662738309270450atural(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_1241_power__inject__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),M) = aa_nat_rat(power_power_rat(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_1242_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_1243_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_1244_zero__less__of__bool__iff,axiom,
    ! [Pa: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((Pa)))
    <=> (Pa) ) ).

% zero_less_of_bool_iff
tff(fact_1245_zero__less__of__bool__iff,axiom,
    ! [Pa: $o] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),zero_n8403883297036319079atural((Pa)))
    <=> (Pa) ) ).

% zero_less_of_bool_iff
tff(fact_1246_zero__less__of__bool__iff,axiom,
    ! [Pa: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((Pa)))
    <=> (Pa) ) ).

% zero_less_of_bool_iff
tff(fact_1247_zero__less__of__bool__iff,axiom,
    ! [Pa: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(Pa)))
    <=> (Pa) ) ).

% zero_less_of_bool_iff
tff(fact_1248_zero__less__of__bool__iff,axiom,
    ! [Pa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(Pa)))
    <=> (Pa) ) ).

% zero_less_of_bool_iff
tff(fact_1249_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((Pa))),one_one_Code_integer)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_1250_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_n8403883297036319079atural((Pa))),one_one_Code_natural)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_1251_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((Pa))),one_one_rat)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_1252_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(Pa))),one_one_int)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_1253_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))),one_one_nat)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_1254_of__bool__not__iff,axiom,
    ! [Pa: $o] : zero_n356916108424825756nteger(~ (Pa)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),zero_n356916108424825756nteger((Pa))) ).

% of_bool_not_iff
tff(fact_1255_of__bool__not__iff,axiom,
    ! [Pa: $o] : zero_n2052037380579107095ol_rat(~ (Pa)) = minus_minus_rat(one_one_rat,zero_n2052037380579107095ol_rat((Pa))) ).

% of_bool_not_iff
tff(fact_1256_of__bool__not__iff,axiom,
    ! [Pa: $o] : aa_o_int(zero_n2684676970156552555ol_int,~ (Pa)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_o_int(zero_n2684676970156552555ol_int,(Pa))) ).

% of_bool_not_iff
tff(fact_1257_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int)
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_1258_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% zle_add1_eq_le
tff(fact_1259_power__increasing__iff,axiom,
    ! [B: code_integer,Xa: nat,Y3: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y3) ) ) ).

% power_increasing_iff
tff(fact_1260_power__increasing__iff,axiom,
    ! [B: code_natural,Xa: nat,Y3: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(B),Xa)),aa_nat_Code_natural(power_7079662738309270450atural(B),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y3) ) ) ).

% power_increasing_iff
tff(fact_1261_power__increasing__iff,axiom,
    ! [B: rat,Xa: nat,Y3: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Xa)),aa_nat_rat(power_power_rat(B),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y3) ) ) ).

% power_increasing_iff
tff(fact_1262_power__increasing__iff,axiom,
    ! [B: nat,Xa: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y3) ) ) ).

% power_increasing_iff
tff(fact_1263_power__increasing__iff,axiom,
    ! [B: int,Xa: nat,Y3: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Xa)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y3) ) ) ).

% power_increasing_iff
tff(fact_1264_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_1265_diff__numeral__special_I12_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_1266_diff__numeral__special_I12_J,axiom,
    minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_1267_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1268_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),zero_z2226904508553997617atural)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1269_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1270_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1271_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1272_power__strict__increasing__iff,axiom,
    ! [B: code_integer,Xa: nat,Y3: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y3) ) ) ).

% power_strict_increasing_iff
tff(fact_1273_power__strict__increasing__iff,axiom,
    ! [B: code_natural,Xa: nat,Y3: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(B),Xa)),aa_nat_Code_natural(power_7079662738309270450atural(B),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y3) ) ) ).

% power_strict_increasing_iff
tff(fact_1274_power__strict__increasing__iff,axiom,
    ! [B: rat,Xa: nat,Y3: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Xa)),aa_nat_rat(power_power_rat(B),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y3) ) ) ).

% power_strict_increasing_iff
tff(fact_1275_power__strict__increasing__iff,axiom,
    ! [B: nat,Xa: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y3) ) ) ).

% power_strict_increasing_iff
tff(fact_1276_power__strict__increasing__iff,axiom,
    ! [B: int,Xa: nat,Y3: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Xa)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y3) ) ) ).

% power_strict_increasing_iff
tff(fact_1277_power__eq__0__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N) = zero_z3403309356797280102nteger )
    <=> ( ( A = zero_z3403309356797280102nteger )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% power_eq_0_iff
tff(fact_1278_power__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% power_eq_0_iff
tff(fact_1279_power__eq__0__iff,axiom,
    ! [A: code_natural,N: nat] :
      ( ( aa_nat_Code_natural(power_7079662738309270450atural(A),N) = zero_z2226904508553997617atural )
    <=> ( ( A = zero_z2226904508553997617atural )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% power_eq_0_iff
tff(fact_1280_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 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% power_eq_0_iff
tff(fact_1281_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% power_eq_0_iff
tff(fact_1282_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = zero_n356916108424825756nteger(A != zero_z3403309356797280102nteger) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_1283_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: rat] : sgn_sgn_rat(abs_abs_rat(A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_1284_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(abs_abs_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_1285_sgn__abs,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = zero_n356916108424825756nteger(A != zero_z3403309356797280102nteger) ).

% sgn_abs
tff(fact_1286_sgn__abs,axiom,
    ! [A: rat] : abs_abs_rat(sgn_sgn_rat(A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% sgn_abs
tff(fact_1287_sgn__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% sgn_abs
tff(fact_1288_power__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_1289_power__decreasing__iff,axiom,
    ! [B: code_natural,M: nat,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),one_one_Code_natural)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(B),M)),aa_nat_Code_natural(power_7079662738309270450atural(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_1290_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_1291_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_1292_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_1293_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1294_power__strict__decreasing__iff,axiom,
    ! [B: code_natural,M: nat,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),one_one_Code_natural)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(B),M)),aa_nat_Code_natural(power_7079662738309270450atural(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1295_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1296_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1297_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1298_power__mono__iff,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))
          <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1299_power__mono__iff,axiom,
    ! [A: code_natural,B: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A),N)),aa_nat_Code_natural(power_7079662738309270450atural(B),N))
          <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1300_power__mono__iff,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1301_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1302_power__mono__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1303_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),N))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_1304_zero__less__power__abs__iff,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A)),N))
    <=> ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_1305_zero__less__power__abs__iff,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),N))
    <=> ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_1306_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(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_1307_negative__zle__0,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),zero_zero_int) ).

% negative_zle_0
tff(fact_1308_zabs__def,axiom,
    ! [I: int] :
      aa_int_int(abs_abs_int,I) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int),aa_int_int(uminus_uminus_int,I),I) ).

% zabs_def
tff(fact_1309_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_1310_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_1311_less__int__code_I1_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_int_code(1)
tff(fact_1312_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% int_cases2
tff(fact_1313_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ aa_int_o(aa_int_fun_int_o(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_1314_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_1315_power__le__imp__le__exp,axiom,
    ! [A: code_natural,M: nat,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A),M)),aa_nat_Code_natural(power_7079662738309270450atural(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_1316_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_1317_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_1318_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_1319_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ).

% int_ops(6)
tff(fact_1320_zsgn__def,axiom,
    ! [I: int] :
      aa_int_int(sgn_sgn_int,I) = $ite(
        I = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_1321_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% add1_zle_eq
tff(fact_1322_int__gr__induct,axiom,
    ! [K: int,I: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2)
             => ( aa_int_o(Pa,I2)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(Pa,I) ) ) ) ).

% int_gr_induct
tff(fact_1323_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ).

% le_imp_0_less
tff(fact_1324_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_1325_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_1326_int__less__induct,axiom,
    ! [I: int,K: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K)
             => ( aa_int_o(Pa,I2)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(Pa,I) ) ) ) ).

% int_less_induct
tff(fact_1327_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_1328_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_1329_nat__leq__as__int,axiom,
    ! [X: nat,Xa3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa3)) ) ).

% nat_leq_as_int
tff(fact_1330_int__if,axiom,
    ! [Pa: $o,A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,
        $ite((Pa),A,B)) = $ite((Pa),aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_if
tff(fact_1331_nat__int__comparison_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_int(semiri1314217659103216013at_int,A) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ).

% nat_int_comparison(1)
tff(fact_1332_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(3)
tff(fact_1333_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N2: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M4)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_diff_cases
tff(fact_1334_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_1335_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% zle_int
tff(fact_1336_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ) ) ).

% of_nat_diff
tff(fact_1337_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).

% of_nat_diff
tff(fact_1338_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_1339_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_1340_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z)
    <=> ? [N6: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,N6)) ) ).

% zle_iff_zadd
tff(fact_1341_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ~ ! [N2: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% nonneg_int_cases
tff(fact_1342_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ? [N2: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% zero_le_imp_eq_int
tff(fact_1343_zero__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ).

% zero_le_power
tff(fact_1344_zero__le__power,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) ) ).

% zero_le_power
tff(fact_1345_zero__le__power,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)) ) ).

% zero_le_power
tff(fact_1346_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ).

% zero_le_power
tff(fact_1347_zero__le__power,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ).

% zero_le_power
tff(fact_1348_power__mono,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)) ) ) ).

% power_mono
tff(fact_1349_power__mono,axiom,
    ! [A: code_natural,B: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A),N)),aa_nat_Code_natural(power_7079662738309270450atural(B),N)) ) ) ).

% power_mono
tff(fact_1350_power__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)) ) ) ).

% power_mono
tff(fact_1351_power__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)) ) ) ).

% power_mono
tff(fact_1352_power__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N)) ) ) ).

% power_mono
tff(fact_1353_less__eq__int__code_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% less_eq_int_code(1)
tff(fact_1354_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_1355_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1356_diff__eq__diff__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1357_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1358_diff__mono,axiom,
    ! [A: code_integer,B: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,D),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),D)) ) ) ).

% diff_mono
tff(fact_1359_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D)) ) ) ).

% diff_mono
tff(fact_1360_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_mono
tff(fact_1361_diff__left__mono,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B)) ) ).

% diff_left_mono
tff(fact_1362_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B)) ) ).

% diff_left_mono
tff(fact_1363_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_left_mono
tff(fact_1364_diff__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) ) ).

% diff_right_mono
tff(fact_1365_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C)) ) ).

% diff_right_mono
tff(fact_1366_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_right_mono
tff(fact_1367_diff__eq__diff__less__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),D) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1368_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1369_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1370_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).

% diff_right_commute
tff(fact_1371_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% diff_right_commute
tff(fact_1372_diff__right__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C) ).

% diff_right_commute
tff(fact_1373_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : minus_minus_rat(minus_minus_rat(A,C),B) = minus_minus_rat(minus_minus_rat(A,B),C) ).

% diff_right_commute
tff(fact_1374_power__increasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N5)) ) ) ).

% power_increasing
tff(fact_1375_power__increasing,axiom,
    ! [N: nat,N5: nat,A: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),A)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A),N)),aa_nat_Code_natural(power_7079662738309270450atural(A),N5)) ) ) ).

% power_increasing
tff(fact_1376_power__increasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N5)) ) ) ).

% power_increasing
tff(fact_1377_power__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5)) ) ) ).

% power_increasing
tff(fact_1378_power__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N5)) ) ) ).

% power_increasing
tff(fact_1379_one__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ).

% one_le_power
tff(fact_1380_one__le__power,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),A)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) ) ).

% one_le_power
tff(fact_1381_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N)) ) ).

% one_le_power
tff(fact_1382_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ).

% one_le_power
tff(fact_1383_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ).

% one_le_power
tff(fact_1384_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),A) ).

% verit_comp_simplify1(2)
tff(fact_1385_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1386_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(ord_less_eq_num(A),A) ).

% verit_comp_simplify1(2)
tff(fact_1387_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1388_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1389_of__bool__eq__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = aa_o_int(zero_n2684676970156552555ol_int,(Q2)) )
    <=> ( (P)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_1390_of__bool__eq__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = aa_o_nat(zero_n2687167440665602831ol_nat,(Q2)) )
    <=> ( (P)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_1391_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% verit_la_disequality
tff(fact_1392_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(ord_less_eq_num(A),B)
      | ~ aa_num_o(ord_less_eq_num(B),A) ) ).

% verit_la_disequality
tff(fact_1393_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% verit_la_disequality
tff(fact_1394_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% verit_la_disequality
tff(fact_1395_int__induct,axiom,
    ! [Pa: fun_int_o,K: int,I: int] :
      ( aa_int_o(Pa,K)
     => ( ! [I2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
           => ( aa_int_o(Pa,I2)
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(Pa,I2)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(Pa,I) ) ) ) ).

% int_induct
tff(fact_1396_int__ge__induct,axiom,
    ! [K: int,I: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I)
     => ( aa_int_o(Pa,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
             => ( aa_int_o(Pa,I2)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(Pa,I) ) ) ) ).

% int_ge_induct
tff(fact_1397_int__le__induct,axiom,
    ! [I: int,K: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(Pa,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(Pa,I2)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(Pa,I) ) ) ) ).

% int_le_induct
tff(fact_1398_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N5)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).

% power_decreasing
tff(fact_1399_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),one_one_Code_natural)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A),N5)),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) ) ) ) ).

% power_decreasing
tff(fact_1400_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N5)),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% power_decreasing
tff(fact_1401_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_decreasing
tff(fact_1402_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N5)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ) ).

% power_decreasing
tff(fact_1403_zero__less__eq__of__bool,axiom,
    ! [Pa: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((Pa))) ).

% zero_less_eq_of_bool
tff(fact_1404_zero__less__eq__of__bool,axiom,
    ! [Pa: $o] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),zero_n8403883297036319079atural((Pa))) ).

% zero_less_eq_of_bool
tff(fact_1405_zero__less__eq__of__bool,axiom,
    ! [Pa: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((Pa))) ).

% zero_less_eq_of_bool
tff(fact_1406_zero__less__eq__of__bool,axiom,
    ! [Pa: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(Pa))) ).

% zero_less_eq_of_bool
tff(fact_1407_zero__less__eq__of__bool,axiom,
    ! [Pa: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))) ).

% zero_less_eq_of_bool
tff(fact_1408_le__iff__diff__le__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),zero_z3403309356797280102nteger) ) ).

% le_iff_diff_le_0
tff(fact_1409_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_1410_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_1411_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((Pa))),one_one_Code_integer) ).

% of_bool_less_eq_one
tff(fact_1412_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_n8403883297036319079atural((Pa))),one_one_Code_natural) ).

% of_bool_less_eq_one
tff(fact_1413_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((Pa))),one_one_rat) ).

% of_bool_less_eq_one
tff(fact_1414_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(Pa))),one_one_int) ).

% of_bool_less_eq_one
tff(fact_1415_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))),one_one_nat) ).

% of_bool_less_eq_one
tff(fact_1416_add__le__add__imp__diff__le,axiom,
    ! [I: code_integer,K: code_integer,N: code_integer,J: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),N)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),K))
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),N)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),K))
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,N),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1417_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(N,K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1418_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1419_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1420_add__le__imp__le__diff,axiom,
    ! [I: code_integer,K: code_integer,N: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),N)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,N),K)) ) ).

% add_le_imp_le_diff
tff(fact_1421_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),minus_minus_rat(N,K)) ) ).

% add_le_imp_le_diff
tff(fact_1422_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N)
     => aa_nat_o(aa_nat_fun_nat_o(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_1423_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N)
     => aa_int_o(aa_int_fun_int_o(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_1424_diff__le__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).

% diff_le_eq
tff(fact_1425_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_le_eq
tff(fact_1426_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_le_eq
tff(fact_1427_le__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ) ).

% le_diff_eq
tff(fact_1428_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),minus_minus_rat(C,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% le_diff_eq
tff(fact_1429_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% le_diff_eq
tff(fact_1430_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% diff_add
tff(fact_1431_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A)) ) ).

% le_add_diff
tff(fact_1432_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1433_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1434_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1435_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1436_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1437_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1438_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1439_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1440_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) ) ) ).

% diff_less_mono
tff(fact_1441_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
       => ( aa_nat_o(aa_nat_fun_nat_o(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))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ) ).

% less_diff_iff
tff(fact_1442_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B))) ).

% abs_triangle_ineq2
tff(fact_1443_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B))),abs_abs_rat(minus_minus_rat(A,B))) ).

% abs_triangle_ineq2
tff(fact_1444_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq2
tff(fact_1445_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B))) ).

% abs_triangle_ineq3
tff(fact_1446_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B)))),abs_abs_rat(minus_minus_rat(A,B))) ).

% abs_triangle_ineq3
tff(fact_1447_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq3
tff(fact_1448_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_1449_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B))),abs_abs_rat(minus_minus_rat(B,A))) ).

% abs_triangle_ineq2_sym
tff(fact_1450_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_1451_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ).

% le_diff_conv
tff(fact_1452_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1453_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,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_1454_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),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_1455_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_1456_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).

% of_nat_mono
tff(fact_1457_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_1458_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_1459_power__not__zero,axiom,
    ! [A: code_integer,N: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N) != zero_z3403309356797280102nteger ) ) ).

% power_not_zero
tff(fact_1460_power__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),N) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_1461_power__not__zero,axiom,
    ! [A: code_natural,N: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_nat_Code_natural(power_7079662738309270450atural(A),N) != zero_z2226904508553997617atural ) ) ).

% power_not_zero
tff(fact_1462_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_1463_power__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_1464_power__less__imp__less__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_1465_power__less__imp__less__base,axiom,
    ! [A: code_natural,N: nat,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A),N)),aa_nat_Code_natural(power_7079662738309270450atural(B),N))
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_1466_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_1467_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_1468_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_1469_power__le__one,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),one_one_Code_integer) ) ) ).

% power_le_one
tff(fact_1470_power__le__one,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),one_one_Code_natural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A),N)),one_one_Code_natural) ) ) ).

% power_le_one
tff(fact_1471_power__le__one,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),one_one_rat) ) ) ).

% power_le_one
tff(fact_1472_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat) ) ) ).

% power_le_one
tff(fact_1473_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),one_one_int) ) ) ).

% power_le_one
tff(fact_1474_pred__subset__eq2,axiom,
    ! [R4: set_Pr1200682188532975830nteger,S2: set_Pr1200682188532975830nteger] :
      ( ord_le7899862361585801933eger_o(aTP_Lamm_bp(R4),aTP_Lamm_bp(S2))
    <=> ord_le2645283422117218614nteger(R4,S2) ) ).

% pred_subset_eq2
tff(fact_1475_pred__subset__eq2,axiom,
    ! [R4: set_Pr620054291978045485atural,S2: set_Pr620054291978045485atural] :
      ( ord_le825093278501044790ural_o(aTP_Lamm_bq(R4),aTP_Lamm_bq(S2))
    <=> ord_le66118011446560909atural(R4,S2) ) ).

% pred_subset_eq2
tff(fact_1476_pred__subset__eq2,axiom,
    ! [R4: set_Pr3486312109254157132nteger,S2: set_Pr3486312109254157132nteger] :
      ( ord_le4825817444062779351eger_o(aTP_Lamm_br(R4),aTP_Lamm_br(S2))
    <=> ord_le2932375828722672556nteger(R4,S2) ) ).

% pred_subset_eq2
tff(fact_1477_pred__subset__eq2,axiom,
    ! [R4: set_Pr6700895561065273913nt_int,S2: set_Pr6700895561065273913nt_int] :
      ( ord_le4806730579060245290_int_o(aTP_Lamm_bs(R4),aTP_Lamm_bs(S2))
    <=> ord_le3990897393281686169nt_int(R4,S2) ) ).

% pred_subset_eq2
tff(fact_1478_pred__subset__eq2,axiom,
    ! [R4: set_Pr4166680992137330533nt_int,S2: set_Pr4166680992137330533nt_int] :
      ( ord_le2876595303158420734_int_o(aTP_Lamm_bt(R4),aTP_Lamm_bt(S2))
    <=> ord_le3197784583289274309nt_int(R4,S2) ) ).

% pred_subset_eq2
tff(fact_1479_zero__le__power__abs,axiom,
    ! [A: code_integer,N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),N)) ).

% zero_le_power_abs
tff(fact_1480_zero__le__power__abs,axiom,
    ! [A: rat,N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A)),N)) ).

% zero_le_power_abs
tff(fact_1481_zero__le__power__abs,axiom,
    ! [A: int,N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),N)) ).

% zero_le_power_abs
tff(fact_1482_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),D)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_1483_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),D)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(minus_minus_rat(A,C))),abs_abs_rat(minus_minus_rat(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_1484_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_1485_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))) ).

% abs_triangle_ineq4
tff(fact_1486_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(A,B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) ).

% abs_triangle_ineq4
tff(fact_1487_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq4
tff(fact_1488_abs__diff__le__iff,axiom,
    ! [Xa: code_integer,A: code_integer,R: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xa),A))),R)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),R)),Xa)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),R)) ) ) ).

% abs_diff_le_iff
tff(fact_1489_abs__diff__le__iff,axiom,
    ! [Xa: rat,A: rat,R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(Xa,A))),R)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,R)),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R)) ) ) ).

% abs_diff_le_iff
tff(fact_1490_abs__diff__le__iff,axiom,
    ! [Xa: int,A: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),A))),R)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R)),Xa)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R)) ) ) ).

% abs_diff_le_iff
tff(fact_1491_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ) ).

% less_diff_conv2
tff(fact_1492_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N) = aa_nat_Code_integer(power_8256067586552552935nteger(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1493_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: code_natural,B: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
         => ( ( aa_nat_Code_natural(power_7079662738309270450atural(A),N) = aa_nat_Code_natural(power_7079662738309270450atural(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1494_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1495_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1496_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1497_power__eq__imp__eq__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N) = aa_nat_Code_integer(power_8256067586552552935nteger(B),N) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1498_power__eq__imp__eq__base,axiom,
    ! [A: code_natural,N: nat,B: code_natural] :
      ( ( aa_nat_Code_natural(power_7079662738309270450atural(A),N) = aa_nat_Code_natural(power_7079662738309270450atural(B),N) )
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1499_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1500_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1501_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1502_self__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).

% self_le_power
tff(fact_1503_self__le__power,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) ) ) ).

% self_le_power
tff(fact_1504_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),N)) ) ) ).

% self_le_power
tff(fact_1505_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).

% self_le_power
tff(fact_1506_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ).

% self_le_power
tff(fact_1507_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_1508_eq__iff__diff__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = zero_z3403309356797280102nteger ) ) ).

% eq_iff_diff_eq_0
tff(fact_1509_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( minus_minus_rat(A,B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_1510_diff__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,D),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),D)) ) ) ).

% diff_strict_mono
tff(fact_1511_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D)) ) ) ).

% diff_strict_mono
tff(fact_1512_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_strict_mono
tff(fact_1513_diff__eq__diff__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),D) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_1514_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_1515_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_1516_diff__strict__left__mono,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B)) ) ).

% diff_strict_left_mono
tff(fact_1517_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B)) ) ).

% diff_strict_left_mono
tff(fact_1518_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_strict_left_mono
tff(fact_1519_diff__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) ) ).

% diff_strict_right_mono
tff(fact_1520_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C)) ) ).

% diff_strict_right_mono
tff(fact_1521_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_strict_right_mono
tff(fact_1522_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_1523_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% diff_diff_eq
tff(fact_1524_diff__diff__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).

% diff_diff_eq
tff(fact_1525_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(minus_minus_rat(A,B),C) = minus_minus_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% diff_diff_eq
tff(fact_1526_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_1527_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_1528_add__implies__diff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B) = A )
     => ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) ) ) ).

% add_implies_diff
tff(fact_1529_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) = A )
     => ( C = minus_minus_rat(A,B) ) ) ).

% add_implies_diff
tff(fact_1530_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_1531_diff__add__eq__diff__diff__swap,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_1532_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = minus_minus_rat(minus_minus_rat(A,C),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_1533_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_add_eq
tff(fact_1534_diff__add__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ).

% diff_add_eq
tff(fact_1535_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),B) ).

% diff_add_eq
tff(fact_1536_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_diff_eq2
tff(fact_1537_diff__diff__eq2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ).

% diff_diff_eq2
tff(fact_1538_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(A,minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),B) ).

% diff_diff_eq2
tff(fact_1539_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% add_diff_eq
tff(fact_1540_add__diff__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ).

% add_diff_eq
tff(fact_1541_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),C) ).

% add_diff_eq
tff(fact_1542_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_1543_eq__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_1544_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = minus_minus_rat(C,B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_1545_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).

% diff_eq_eq
tff(fact_1546_diff__eq__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = C )
    <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B) ) ) ).

% diff_eq_eq
tff(fact_1547_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat(A,B) = C )
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) ) ) ).

% diff_eq_eq
tff(fact_1548_group__cancel_Osub1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A3),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_1549_group__cancel_Osub1,axiom,
    ! [A3: code_integer,K: code_integer,A: code_integer,B: code_integer] :
      ( ( A3 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),A) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A3),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_1550_group__cancel_Osub1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( minus_minus_rat(A3,B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),minus_minus_rat(A,B)) ) ) ).

% group_cancel.sub1
tff(fact_1551_minus__diff__commute,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),B) ).

% minus_diff_commute
tff(fact_1552_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ).

% minus_diff_commute
tff(fact_1553_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B),A) = minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A),B) ).

% minus_diff_commute
tff(fact_1554_zero__le,axiom,
    ! [Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa) ).

% zero_le
tff(fact_1555_le__numeral__extra_I3_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% le_numeral_extra(3)
tff(fact_1556_le__numeral__extra_I3_J,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),zero_z2226904508553997617atural) ).

% le_numeral_extra(3)
tff(fact_1557_le__numeral__extra_I3_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_zero_rat) ).

% le_numeral_extra(3)
tff(fact_1558_le__numeral__extra_I3_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_zero_nat) ).

% le_numeral_extra(3)
tff(fact_1559_le__numeral__extra_I3_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% le_numeral_extra(3)
tff(fact_1560_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_1561_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_1562_verit__comp__simplify1_I3_J,axiom,
    ! [B4: literal,A4: literal] :
      ( ~ aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,B4),A4)
    <=> aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_1563_verit__comp__simplify1_I3_J,axiom,
    ! [B4: rat,A4: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B4),A4)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_1564_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A4: num] :
      ( ~ aa_num_o(ord_less_eq_num(B4),A4)
    <=> aa_num_o(ord_less_num(A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_1565_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A4: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B4),A4)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_1566_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A4: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),A4)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_1567_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).

% abs_minus_commute
tff(fact_1568_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A)) ).

% abs_minus_commute
tff(fact_1569_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(minus_minus_rat(A,B)) = abs_abs_rat(minus_minus_rat(B,A)) ).

% abs_minus_commute
tff(fact_1570_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N)),K) ) ).

% less_imp_diff_less
tff(fact_1571_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),L)
       => aa_nat_o(aa_nat_fun_nat_o(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_1572_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_1573_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_1574_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_1575_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_1576_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_1577_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_1578_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C3: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% le_iff_add
tff(fact_1579_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_right_mono
tff(fact_1580_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_right_mono
tff(fact_1581_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_right_mono
tff(fact_1582_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% less_eqE
tff(fact_1583_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_left_mono
tff(fact_1584_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_left_mono
tff(fact_1585_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_left_mono
tff(fact_1586_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_mono
tff(fact_1587_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_mono
tff(fact_1588_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_mono
tff(fact_1589_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1590_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1591_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1592_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1593_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1594_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1595_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1596_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1597_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1598_le__numeral__extra_I4_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),one_one_Code_integer) ).

% le_numeral_extra(4)
tff(fact_1599_le__numeral__extra_I4_J,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),one_one_Code_natural) ).

% le_numeral_extra(4)
tff(fact_1600_le__numeral__extra_I4_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),one_one_rat) ).

% le_numeral_extra(4)
tff(fact_1601_le__numeral__extra_I4_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),one_one_nat) ).

% le_numeral_extra(4)
tff(fact_1602_le__numeral__extra_I4_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),one_one_int) ).

% le_numeral_extra(4)
tff(fact_1603_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_1604_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_1605_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_1606_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_1607_subrelI,axiom,
    ! [R: set_Pr1200682188532975830nteger,S: set_Pr1200682188532975830nteger] :
      ( ! [X4: fun_Co3460408610320954864e_term,Y4: produc8923325533196201883nteger] :
          ( member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X4),Y4),R)
         => member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X4),Y4),S) )
     => ord_le2645283422117218614nteger(R,S) ) ).

% subrelI
tff(fact_1608_subrelI,axiom,
    ! [R: set_Pr620054291978045485atural,S: set_Pr620054291978045485atural] :
      ( ! [X4: fun_Pr285225112257423215e_term,Y4: produc7822875418678951345atural] :
          ( member6441364043385305230atural(aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,X4),Y4),R)
         => member6441364043385305230atural(aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,X4),Y4),S) )
     => ord_le66118011446560909atural(R,S) ) ).

% subrelI
tff(fact_1609_subrelI,axiom,
    ! [R: set_Pr3486312109254157132nteger,S: set_Pr3486312109254157132nteger] :
      ( ! [X4: fun_Pr5098669352780829690e_term,Y4: produc8923325533196201883nteger] :
          ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X4),Y4),R)
         => member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X4),Y4),S) )
     => ord_le2932375828722672556nteger(R,S) ) ).

% subrelI
tff(fact_1610_subrelI,axiom,
    ! [R: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int] :
      ( ! [X4: fun_Pr7763680713840462443e_term,Y4: product_prod_int_int] :
          ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Y4),R)
         => member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Y4),S) )
     => ord_le3990897393281686169nt_int(R,S) ) ).

% subrelI
tff(fact_1611_subrelI,axiom,
    ! [R: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int] :
      ( ! [X4: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
          ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y4),R)
         => member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y4),S) )
     => ord_le3197784583289274309nt_int(R,S) ) ).

% subrelI
tff(fact_1612_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_minus_iff
tff(fact_1613_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_minus_iff
tff(fact_1614_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% le_minus_iff
tff(fact_1615_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_le_iff
tff(fact_1616_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_le_iff
tff(fact_1617_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_le_iff
tff(fact_1618_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_imp_neg_le
tff(fact_1619_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_imp_neg_le
tff(fact_1620_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% le_imp_neg_le
tff(fact_1621_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(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_1622_le__0__eq,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),zero_zero_nat)
    <=> ( N = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_1623_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_1624_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_1625_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),N) ).

% less_eq_nat.simps(1)
tff(fact_1626_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_1627_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( ( M != N )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% le_neq_implies_less
tff(fact_1628_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | ( M = N ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% less_or_eq_imp_le
tff(fact_1629_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_1630_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% less_imp_le_nat
tff(fact_1631_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_1632_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
    <=> ? [K3: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_1633_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_le_add2
tff(fact_1634_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_le_add1
tff(fact_1635_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(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_1636_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_1637_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_1638_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) ) ).

% add_leD2
tff(fact_1639_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% add_leD1
tff(fact_1640_le__add2,axiom,
    ! [N: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% le_add2
tff(fact_1641_le__add1,axiom,
    ! [N: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) ).

% le_add1
tff(fact_1642_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) ) ) ).

% add_leE
tff(fact_1643_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% abs_le_D1
tff(fact_1644_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% abs_le_D1
tff(fact_1645_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% abs_le_D1
tff(fact_1646_abs__ge__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) ).

% abs_ge_self
tff(fact_1647_abs__ge__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),abs_abs_rat(A)) ).

% abs_ge_self
tff(fact_1648_abs__ge__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(abs_abs_int,A)) ).

% abs_ge_self
tff(fact_1649_zero__less__power,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ).

% zero_less_power
tff(fact_1650_zero__less__power,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) ) ).

% zero_less_power
tff(fact_1651_zero__less__power,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)) ) ).

% zero_less_power
tff(fact_1652_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ).

% zero_less_power
tff(fact_1653_zero__less__power,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ).

% zero_less_power
tff(fact_1654_Ints__diff,axiom,
    ! [A: int,B: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B),ring_1_Ints_int) ) ) ).

% Ints_diff
tff(fact_1655_Ints__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( member_Code_integer(A,ring_11222124179247155820nteger)
     => ( member_Code_integer(B,ring_11222124179247155820nteger)
       => member_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B),ring_11222124179247155820nteger) ) ) ).

% Ints_diff
tff(fact_1656_Ints__diff,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(minus_minus_rat(A,B),ring_1_Ints_rat) ) ) ).

% Ints_diff
tff(fact_1657_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,N))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_1658_Ints__power,axiom,
    ! [A: int,N: nat] :
      ( member_int(A,ring_1_Ints_int)
     => member_int(aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N),ring_1_Ints_int) ) ).

% Ints_power
tff(fact_1659_power__0,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),zero_zero_nat) = one_one_Code_integer ).

% power_0
tff(fact_1660_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_1661_power__0,axiom,
    ! [A: code_natural] : aa_nat_Code_natural(power_7079662738309270450atural(A),zero_zero_nat) = one_one_Code_natural ).

% power_0
tff(fact_1662_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_1663_power__0,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_1664_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(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))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% nat_power_less_imp_less
tff(fact_1665_power__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)) ) ) ) ).

% power_strict_mono
tff(fact_1666_power__strict__mono,axiom,
    ! [A: code_natural,B: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A),N)),aa_nat_Code_natural(power_7079662738309270450atural(B),N)) ) ) ) ).

% power_strict_mono
tff(fact_1667_power__strict__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)) ) ) ) ).

% power_strict_mono
tff(fact_1668_power__strict__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)) ) ) ) ).

% power_strict_mono
tff(fact_1669_power__strict__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N)) ) ) ) ).

% power_strict_mono
tff(fact_1670_split__of__bool__asm,axiom,
    ! [Pa: fun_Code_integer_o,P: $o] :
      ( aa_Code_integer_o(Pa,zero_n356916108424825756nteger((P)))
    <=> ~ ( ( (P)
            & ~ aa_Code_integer_o(Pa,one_one_Code_integer) )
          | ( ~ (P)
            & ~ aa_Code_integer_o(Pa,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool_asm
tff(fact_1671_split__of__bool__asm,axiom,
    ! [Pa: fun_rat_o,P: $o] :
      ( aa_rat_o(Pa,zero_n2052037380579107095ol_rat((P)))
    <=> ~ ( ( (P)
            & ~ aa_rat_o(Pa,one_one_rat) )
          | ( ~ (P)
            & ~ aa_rat_o(Pa,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_1672_split__of__bool__asm,axiom,
    ! [Pa: fun_Code_natural_o,P: $o] :
      ( aa_Code_natural_o(Pa,zero_n8403883297036319079atural((P)))
    <=> ~ ( ( (P)
            & ~ aa_Code_natural_o(Pa,one_one_Code_natural) )
          | ( ~ (P)
            & ~ aa_Code_natural_o(Pa,zero_z2226904508553997617atural) ) ) ) ).

% split_of_bool_asm
tff(fact_1673_split__of__bool__asm,axiom,
    ! [Pa: fun_int_o,P: $o] :
      ( aa_int_o(Pa,aa_o_int(zero_n2684676970156552555ol_int,(P)))
    <=> ~ ( ( (P)
            & ~ aa_int_o(Pa,one_one_int) )
          | ( ~ (P)
            & ~ aa_int_o(Pa,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_1674_split__of__bool__asm,axiom,
    ! [Pa: fun_nat_o,P: $o] :
      ( aa_nat_o(Pa,aa_o_nat(zero_n2687167440665602831ol_nat,(P)))
    <=> ~ ( ( (P)
            & ~ aa_nat_o(Pa,one_one_nat) )
          | ( ~ (P)
            & ~ aa_nat_o(Pa,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_1675_split__of__bool,axiom,
    ! [Pa: fun_Code_integer_o,P: $o] :
      ( aa_Code_integer_o(Pa,zero_n356916108424825756nteger((P)))
    <=> ( ( (P)
         => aa_Code_integer_o(Pa,one_one_Code_integer) )
        & ( ~ (P)
         => aa_Code_integer_o(Pa,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool
tff(fact_1676_split__of__bool,axiom,
    ! [Pa: fun_rat_o,P: $o] :
      ( aa_rat_o(Pa,zero_n2052037380579107095ol_rat((P)))
    <=> ( ( (P)
         => aa_rat_o(Pa,one_one_rat) )
        & ( ~ (P)
         => aa_rat_o(Pa,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_1677_split__of__bool,axiom,
    ! [Pa: fun_Code_natural_o,P: $o] :
      ( aa_Code_natural_o(Pa,zero_n8403883297036319079atural((P)))
    <=> ( ( (P)
         => aa_Code_natural_o(Pa,one_one_Code_natural) )
        & ( ~ (P)
         => aa_Code_natural_o(Pa,zero_z2226904508553997617atural) ) ) ) ).

% split_of_bool
tff(fact_1678_split__of__bool,axiom,
    ! [Pa: fun_int_o,P: $o] :
      ( aa_int_o(Pa,aa_o_int(zero_n2684676970156552555ol_int,(P)))
    <=> ( ( (P)
         => aa_int_o(Pa,one_one_int) )
        & ( ~ (P)
         => aa_int_o(Pa,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_1679_split__of__bool,axiom,
    ! [Pa: fun_nat_o,P: $o] :
      ( aa_nat_o(Pa,aa_o_nat(zero_n2687167440665602831ol_nat,(P)))
    <=> ( ( (P)
         => aa_nat_o(Pa,one_one_nat) )
        & ( ~ (P)
         => aa_nat_o(Pa,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_1680_of__bool__def,axiom,
    ! [P: $o] :
      zero_n356916108424825756nteger((P)) = $ite((P),one_one_Code_integer,zero_z3403309356797280102nteger) ).

% of_bool_def
tff(fact_1681_of__bool__def,axiom,
    ! [P: $o] :
      zero_n2052037380579107095ol_rat((P)) = $ite((P),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_1682_of__bool__def,axiom,
    ! [P: $o] :
      zero_n8403883297036319079atural((P)) = $ite((P),one_one_Code_natural,zero_z2226904508553997617atural) ).

% of_bool_def
tff(fact_1683_of__bool__def,axiom,
    ! [P: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,(P)) = $ite((P),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_1684_of__bool__def,axiom,
    ! [P: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = $ite((P),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_1685_less__iff__diff__less__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),zero_z3403309356797280102nteger) ) ).

% less_iff_diff_less_0
tff(fact_1686_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_1687_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_1688_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = $ite(N = zero_zero_nat,one_one_Code_integer,zero_z3403309356797280102nteger) ).

% power_0_left
tff(fact_1689_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),N) = $ite(N = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_1690_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_Code_natural(power_7079662738309270450atural(zero_z2226904508553997617atural),N) = $ite(N = zero_zero_nat,one_one_Code_natural,zero_z2226904508553997617atural) ).

% power_0_left
tff(fact_1691_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = $ite(N = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_1692_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),N) = $ite(N = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_1693_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1694_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),minus_minus_rat(A,B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1695_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1696_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1697_diff__less__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).

% diff_less_eq
tff(fact_1698_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_less_eq
tff(fact_1699_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_less_eq
tff(fact_1700_less__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ) ).

% less_diff_eq
tff(fact_1701_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),minus_minus_rat(C,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% less_diff_eq
tff(fact_1702_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% less_diff_eq
tff(fact_1703_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N5)) ) ) ).

% power_strict_increasing
tff(fact_1704_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A),N)),aa_nat_Code_natural(power_7079662738309270450atural(A),N5)) ) ) ).

% power_strict_increasing
tff(fact_1705_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N5)) ) ) ).

% power_strict_increasing
tff(fact_1706_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5)) ) ) ).

% power_strict_increasing
tff(fact_1707_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N5)) ) ) ).

% power_strict_increasing
tff(fact_1708_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_1709_power__less__imp__less__exp,axiom,
    ! [A: code_natural,M: nat,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A),M)),aa_nat_Code_natural(power_7079662738309270450atural(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_1710_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_1711_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_1712_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_1713_zero__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = zero_z3403309356797280102nteger ) ) ).

% zero_power
tff(fact_1714_zero__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ).

% zero_power
tff(fact_1715_zero__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_Code_natural(power_7079662738309270450atural(zero_z2226904508553997617atural),N) = zero_z2226904508553997617atural ) ) ).

% zero_power
tff(fact_1716_zero__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_1717_zero__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),N) = zero_zero_int ) ) ).

% zero_power
tff(fact_1718_group__cancel_Osub2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_1719_group__cancel_Osub2,axiom,
    ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B3 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),B) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_1720_group__cancel_Osub2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( minus_minus_rat(A,B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),minus_minus_rat(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_1721_diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_1722_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% diff_conv_add_uminus
tff(fact_1723_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_1724_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_1725_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_1726_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_1727_add__nonpos__eq__0__iff,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y3),zero_z3403309356797280102nteger)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Xa),Y3) = zero_z3403309356797280102nteger )
        <=> ( ( Xa = zero_z3403309356797280102nteger )
            & ( Y3 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1728_add__nonpos__eq__0__iff,axiom,
    ! [Xa: code_natural,Y3: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,Xa),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,Y3),zero_z2226904508553997617atural)
       => ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Xa),Y3) = zero_z2226904508553997617atural )
        <=> ( ( Xa = zero_z2226904508553997617atural )
            & ( Y3 = zero_z2226904508553997617atural ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1729_add__nonpos__eq__0__iff,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),zero_zero_rat)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y3) = zero_zero_rat )
        <=> ( ( Xa = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1730_add__nonpos__eq__0__iff,axiom,
    ! [Xa: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y3) = zero_zero_nat )
        <=> ( ( Xa = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1731_add__nonpos__eq__0__iff,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y3) = zero_zero_int )
        <=> ( ( Xa = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1732_add__nonneg__eq__0__iff,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y3)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Xa),Y3) = zero_z3403309356797280102nteger )
        <=> ( ( Xa = zero_z3403309356797280102nteger )
            & ( Y3 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1733_add__nonneg__eq__0__iff,axiom,
    ! [Xa: code_natural,Y3: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),Xa)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),Y3)
       => ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Xa),Y3) = zero_z2226904508553997617atural )
        <=> ( ( Xa = zero_z2226904508553997617atural )
            & ( Y3 = zero_z2226904508553997617atural ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1734_add__nonneg__eq__0__iff,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y3)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y3) = zero_zero_rat )
        <=> ( ( Xa = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1735_add__nonneg__eq__0__iff,axiom,
    ! [Xa: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y3)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y3) = zero_zero_nat )
        <=> ( ( Xa = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1736_add__nonneg__eq__0__iff,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y3) = zero_zero_int )
        <=> ( ( Xa = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1737_add__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_nonpos_nonpos
tff(fact_1738_add__nonpos__nonpos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),zero_z2226904508553997617atural) ) ) ).

% add_nonpos_nonpos
tff(fact_1739_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_1740_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_1741_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_1742_add__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_1743_add__nonneg__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_1744_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_1745_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_1746_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_1747_add__increasing2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% add_increasing2
tff(fact_1748_add__increasing2,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),A)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)) ) ) ).

% add_increasing2
tff(fact_1749_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing2
tff(fact_1750_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing2
tff(fact_1751_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing2
tff(fact_1752_add__decreasing2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_1753_add__decreasing2,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,C),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_1754_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_1755_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_1756_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_1757_add__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% add_increasing
tff(fact_1758_add__increasing,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)) ) ) ).

% add_increasing
tff(fact_1759_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing
tff(fact_1760_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing
tff(fact_1761_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing
tff(fact_1762_add__decreasing,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ) ) ).

% add_decreasing
tff(fact_1763_add__decreasing,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,C),B)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)),B) ) ) ).

% add_decreasing
tff(fact_1764_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_1765_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_1766_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing
tff(fact_1767_not__one__le__zero,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),zero_z3403309356797280102nteger) ).

% not_one_le_zero
tff(fact_1768_not__one__le__zero,axiom,
    ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),zero_z2226904508553997617atural) ).

% not_one_le_zero
tff(fact_1769_not__one__le__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),zero_zero_rat) ).

% not_one_le_zero
tff(fact_1770_not__one__le__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),zero_zero_nat) ).

% not_one_le_zero
tff(fact_1771_not__one__le__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),zero_zero_int) ).

% not_one_le_zero
tff(fact_1772_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1773_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),one_one_Code_natural) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1774_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1775_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1776_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1777_zero__less__one__class_Ozero__le__one,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% zero_less_one_class.zero_le_one
tff(fact_1778_zero__less__one__class_Ozero__le__one,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),one_one_Code_natural) ).

% zero_less_one_class.zero_le_one
tff(fact_1779_zero__less__one__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one_class.zero_le_one
tff(fact_1780_zero__less__one__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one_class.zero_le_one
tff(fact_1781_zero__less__one__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% zero_less_one_class.zero_le_one
tff(fact_1782_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M) ) ) ).

% diff_less
tff(fact_1783_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1784_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1785_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1786_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1787_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1788_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1789_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_1790_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_1791_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_le_less_mono
tff(fact_1792_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_1793_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_1794_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_less_le_mono
tff(fact_1795_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_1796_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ).

% less_diff_conv
tff(fact_1797_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(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_1798_le__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% le_minus_one_simps(2)
tff(fact_1799_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_1800_le__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% le_minus_one_simps(2)
tff(fact_1801_le__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(4)
tff(fact_1802_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_1803_le__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(4)
tff(fact_1804_of__nat__0__le__iff,axiom,
    ! [N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_0_le_iff
tff(fact_1805_of__nat__0__le__iff,axiom,
    ! [N: nat] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(semiri3763490453095760265atural,N)) ).

% of_nat_0_le_iff
tff(fact_1806_of__nat__0__le__iff,axiom,
    ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_0_le_iff
tff(fact_1807_of__nat__0__le__iff,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_0_le_iff
tff(fact_1808_of__nat__0__le__iff,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_0_le_iff
tff(fact_1809_abs__ge__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) ).

% abs_ge_zero
tff(fact_1810_abs__ge__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),abs_abs_rat(A)) ).

% abs_ge_zero
tff(fact_1811_abs__ge__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A)) ).

% abs_ge_zero
tff(fact_1812_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))) ).

% abs_triangle_ineq
tff(fact_1813_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) ).

% abs_triangle_ineq
tff(fact_1814_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq
tff(fact_1815_ex__least__nat__le,axiom,
    ! [Pa: fun_nat_o,N: nat] :
      ( aa_nat_o(Pa,N)
     => ( ~ aa_nat_o(Pa,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N)
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
               => ~ aa_nat_o(Pa,I3) )
            & aa_nat_o(Pa,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_1816_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B) ) ) ).

% abs_leI
tff(fact_1817_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B) ) ) ).

% abs_leI
tff(fact_1818_abs__leI,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B) ) ) ).

% abs_leI
tff(fact_1819_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).

% abs_le_D2
tff(fact_1820_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ).

% abs_le_D2
tff(fact_1821_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% abs_le_D2
tff(fact_1822_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_le_iff
tff(fact_1823_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_le_iff
tff(fact_1824_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_le_iff
tff(fact_1825_abs__ge__minus__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) ).

% abs_ge_minus_self
tff(fact_1826_abs__ge__minus__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),abs_abs_rat(A)) ).

% abs_ge_minus_self
tff(fact_1827_abs__ge__minus__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A)) ).

% abs_ge_minus_self
tff(fact_1828_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M4: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M4)),aa_nat_nat(F,N2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,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_1829_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N5)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_1830_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),one_one_Code_natural)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A),N5)),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_1831_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N5)),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_1832_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_1833_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N5)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_1834_one__less__power,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).

% one_less_power
tff(fact_1835_one__less__power,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) ) ) ).

% one_less_power
tff(fact_1836_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N)) ) ) ).

% one_less_power
tff(fact_1837_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).

% one_less_power
tff(fact_1838_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ).

% one_less_power
tff(fact_1839_abs__diff__less__iff,axiom,
    ! [Xa: code_integer,A: code_integer,R: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xa),A))),R)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),R)),Xa)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),R)) ) ) ).

% abs_diff_less_iff
tff(fact_1840_abs__diff__less__iff,axiom,
    ! [Xa: rat,A: rat,R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(minus_minus_rat(Xa,A))),R)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,R)),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R)) ) ) ).

% abs_diff_less_iff
tff(fact_1841_abs__diff__less__iff,axiom,
    ! [Xa: int,A: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),A))),R)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R)),Xa)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R)) ) ) ).

% abs_diff_less_iff
tff(fact_1842_add__strict__increasing2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_1843_add__strict__increasing2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_1844_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_1845_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_1846_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_1847_add__strict__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% add_strict_increasing
tff(fact_1848_add__strict__increasing,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)) ) ) ).

% add_strict_increasing
tff(fact_1849_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_1850_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_1851_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing
tff(fact_1852_add__pos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_1853_add__pos__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_1854_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_1855_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_1856_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_1857_add__nonpos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_nonpos_neg
tff(fact_1858_add__nonpos__neg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),zero_z2226904508553997617atural) ) ) ).

% add_nonpos_neg
tff(fact_1859_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_1860_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_1861_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_1862_add__nonneg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_1863_add__nonneg__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_1864_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_1865_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_1866_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_1867_add__neg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_neg_nonpos
tff(fact_1868_add__neg__nonpos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),zero_z2226904508553997617atural) ) ) ).

% add_neg_nonpos
tff(fact_1869_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_1870_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_1871_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_1872_discrete,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),B) ) ).

% discrete
tff(fact_1873_discrete,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).

% discrete
tff(fact_1874_discrete,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B) ) ).

% discrete
tff(fact_1875_le__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% le_minus_one_simps(1)
tff(fact_1876_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_1877_le__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% le_minus_one_simps(1)
tff(fact_1878_le__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(3)
tff(fact_1879_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_1880_le__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(3)
tff(fact_1881_nat__diff__split__asm,axiom,
    ! [Pa: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
            & ~ aa_nat_o(Pa,zero_zero_nat) )
          | ? [D2: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2) )
              & ~ aa_nat_o(Pa,D2) ) ) ) ).

% nat_diff_split_asm
tff(fact_1882_nat__diff__split,axiom,
    ! [Pa: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => aa_nat_o(Pa,zero_zero_nat) )
        & ! [D2: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2) )
           => aa_nat_o(Pa,D2) ) ) ) ).

% nat_diff_split
tff(fact_1883_dense__eq0__I,axiom,
    ! [Xa: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Xa)),E2) )
     => ( Xa = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_1884_abs__minus__le__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_1885_abs__minus__le__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,abs_abs_rat(A))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_1886_abs__minus__le__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_1887_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = B )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
        & ( ( A = B )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_1888_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat(A) = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
        & ( ( A = B )
          | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_1889_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_1890_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
        & ( ( B = A )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_1891_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = abs_abs_rat(B) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_1892_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_1893_dbl__dec__def,axiom,
    ! [Xa: int] : aa_int_int(neg_nu3811975205180677377ec_int,Xa) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Xa)),one_one_int) ).

% dbl_dec_def
tff(fact_1894_dbl__dec__def,axiom,
    ! [Xa: code_integer] : aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,Xa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Xa),Xa)),one_one_Code_integer) ).

% dbl_dec_def
tff(fact_1895_dbl__dec__def,axiom,
    ! [Xa: rat] : neg_nu3179335615603231917ec_rat(Xa) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Xa),one_one_rat) ).

% dbl_dec_def
tff(fact_1896_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1897_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1898_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1899_field__le__epsilon,axiom,
    ! [Xa: rat,Y3: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y3),E2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y3) ) ).

% field_le_epsilon
tff(fact_1900_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R: int] : aa_Pro6799258434465214121nt_int(adjust_div,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q2),aa_o_int(zero_n2684676970156552555ol_int,R != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_1901_bezw__0,axiom,
    ! [Xa: nat] : aa_nat4424723855478970923nt_int(bezw(Xa),zero_zero_nat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_1902_frac__unique__iff,axiom,
    ! [Xa: rat,A: rat] :
      ( ( archimedean_frac_rat(Xa) = A )
    <=> ( member_rat(minus_minus_rat(Xa,A),ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) ) ) ).

% frac_unique_iff
tff(fact_1903_Nats__altdef2,axiom,
    semiri2551628265256775642nteger = collect_Code_integer(aTP_Lamm_bu) ).

% Nats_altdef2
tff(fact_1904_Nats__altdef2,axiom,
    semiring_1_Nats_rat = collect_rat(aTP_Lamm_bv) ).

% Nats_altdef2
tff(fact_1905_Nats__altdef2,axiom,
    semiring_1_Nats_int = collect_int(aTP_Lamm_bw) ).

% Nats_altdef2
tff(fact_1906_convex__bound__lt,axiom,
    ! [Xa: code_integer,A: code_integer,Y3: code_integer,U: code_integer,V: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y3),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V)
           => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),V) = one_one_Code_integer )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),Xa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V),Y3))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1907_convex__bound__lt,axiom,
    ! [Xa: rat,A: rat,Y3: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y3))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1908_convex__bound__lt,axiom,
    ! [Xa: int,A: int,Y3: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y3))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1909_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% zero_less_nat_eq
tff(fact_1910_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_1911_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_1912_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_1913_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_1914_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_1915_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_1916_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_1917_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_1918_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_1919_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_1920_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_1921_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_1922_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,W) = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( W = Z ) ) ).

% of_int_eq_iff
tff(fact_1923_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,W) = aa_int_int(ring_1_of_int_int,Z) )
    <=> ( W = Z ) ) ).

% of_int_eq_iff
tff(fact_1924_mult__zero__left,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(zero_z2226904508553997617atural),A) = zero_z2226904508553997617atural ).

% mult_zero_left
tff(fact_1925_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_1926_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_1927_mult__zero__left,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).

% mult_zero_left
tff(fact_1928_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_1929_mult__zero__right,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),zero_z2226904508553997617atural) = zero_z2226904508553997617atural ).

% mult_zero_right
tff(fact_1930_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_1931_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_1932_mult__zero__right,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% mult_zero_right
tff(fact_1933_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1934_mult__eq__0__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B) = zero_z2226904508553997617atural )
    <=> ( ( A = zero_z2226904508553997617atural )
        | ( B = zero_z2226904508553997617atural ) ) ) ).

% mult_eq_0_iff
tff(fact_1935_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_1936_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_1937_mult__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = zero_z3403309356797280102nteger )
    <=> ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% mult_eq_0_iff
tff(fact_1938_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_1939_mult__cancel__left,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B) )
    <=> ( ( C = zero_z2226904508553997617atural )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1940_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1941_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1942_mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1943_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1944_mult__cancel__right,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C) )
    <=> ( ( C = zero_z2226904508553997617atural )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1945_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1946_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1947_mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1948_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1949_mult_Oright__neutral,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),one_one_Code_natural) = A ).

% mult.right_neutral
tff(fact_1950_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_1951_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_1952_mult_Oright__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ).

% mult.right_neutral
tff(fact_1953_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_1954_mult__1,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(one_one_Code_natural),A) = A ).

% mult_1
tff(fact_1955_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_1956_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_1957_mult__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A) = A ).

% mult_1
tff(fact_1958_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% mult_1
tff(fact_1959_mult__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_right
tff(fact_1960_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_right
tff(fact_1961_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% mult_minus_right
tff(fact_1962_minus__mult__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),B) ).

% minus_mult_minus
tff(fact_1963_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ).

% minus_mult_minus
tff(fact_1964_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) ).

% minus_mult_minus
tff(fact_1965_mult__minus__left,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_left
tff(fact_1966_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_left
tff(fact_1967_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% mult_minus_left
tff(fact_1968_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_mult
tff(fact_1969_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_mult
tff(fact_1970_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_1971_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_1972_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% nat_0_less_mult_iff
tff(fact_1973_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% mult_less_cancel2
tff(fact_1974_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_1975_abs__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).

% abs_mult_self_eq
tff(fact_1976_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ).

% abs_mult_self_eq
tff(fact_1977_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_mult
tff(fact_1978_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_mult
tff(fact_1979_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_1980_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_1981_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Xa) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) )
    <=> ( Xa = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_1982_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Xa) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W) )
    <=> ( Xa = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_1983_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) = aa_int_rat(ring_1_of_int_rat,Xa) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) = Xa ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_1984_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W) = aa_int_int(ring_1_of_int_int,Xa) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) = Xa ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_1985_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),N)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),N) ).

% of_int_power
tff(fact_1986_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,Z)),N) ).

% of_int_power
tff(fact_1987_nat__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,N)) = N ).

% nat_int
tff(fact_1988_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_1989_mult__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
tff(fact_1990_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1991_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_1992_mult__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = C )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
tff(fact_1993_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1994_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_1995_mult__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
tff(fact_1996_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_1997_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_1998_mult__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = C )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
tff(fact_1999_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_2000_mult__minus1,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1
tff(fact_2001_mult__minus1,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1
tff(fact_2002_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1
tff(fact_2003_mult__minus1__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1_right
tff(fact_2004_mult__minus1__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1_right
tff(fact_2005_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1_right
tff(fact_2006_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% mult_le_cancel2
tff(fact_2007_of__int__0,axiom,
    aa_int_Code_integer(ring_18347121197199848620nteger,zero_zero_int) = zero_z3403309356797280102nteger ).

% of_int_0
tff(fact_2008_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_2009_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_2010_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_z3403309356797280102nteger = aa_int_Code_integer(ring_18347121197199848620nteger,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_2011_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_2012_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int = aa_int_int(ring_1_of_int_int,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_2013_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = zero_z3403309356797280102nteger )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_2014_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = zero_zero_rat )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_2015_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = zero_zero_int )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_2016_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_2017_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_2018_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_2019_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_2020_of__int__1,axiom,
    aa_int_Code_integer(ring_18347121197199848620nteger,one_one_int) = one_one_Code_integer ).

% of_int_1
tff(fact_2021_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_2022_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_2023_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = one_one_Code_integer )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_2024_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = one_one_rat )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_2025_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = one_one_int )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_2026_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_2027_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_add
tff(fact_2028_of__int__minus,axiom,
    ! [Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,Z)) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_minus
tff(fact_2029_of__int__minus,axiom,
    ! [Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,Z)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_minus
tff(fact_2030_of__int__minus,axiom,
    ! [Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,Z)) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_minus
tff(fact_2031_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_diff
tff(fact_2032_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_diff
tff(fact_2033_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_diff
tff(fact_2034_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% of_int_of_nat_eq
tff(fact_2035_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_2036_of__int__abs,axiom,
    ! [Xa: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(abs_abs_int,Xa)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,Xa)) ).

% of_int_abs
tff(fact_2037_of__int__abs,axiom,
    ! [Xa: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(abs_abs_int,Xa)) = abs_abs_rat(aa_int_rat(ring_1_of_int_rat,Xa)) ).

% of_int_abs
tff(fact_2038_of__int__abs,axiom,
    ! [Xa: int] : aa_int_int(ring_1_of_int_int,aa_int_int(abs_abs_int,Xa)) = aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Xa)) ).

% of_int_abs
tff(fact_2039_frac__of__int,axiom,
    ! [Z: int] : archimedean_frac_rat(aa_int_rat(ring_1_of_int_rat,Z)) = zero_zero_rat ).

% frac_of_int
tff(fact_2040_frac__eq__0__iff,axiom,
    ! [Xa: rat] :
      ( ( archimedean_frac_rat(Xa) = zero_zero_rat )
    <=> member_rat(Xa,ring_1_Ints_rat) ) ).

% frac_eq_0_iff
tff(fact_2041_of__int__of__bool,axiom,
    ! [Pa: $o] : aa_int_rat(ring_1_of_int_rat,aa_o_int(zero_n2684676970156552555ol_int,(Pa))) = zero_n2052037380579107095ol_rat((Pa)) ).

% of_int_of_bool
tff(fact_2042_of__int__of__bool,axiom,
    ! [Pa: $o] : aa_int_int(ring_1_of_int_int,aa_o_int(zero_n2684676970156552555ol_int,(Pa))) = aa_o_int(zero_n2684676970156552555ol_int,(Pa)) ).

% of_int_of_bool
tff(fact_2043_nat__of__bool,axiom,
    ! [Pa: $o] : aa_int_nat(nat2,aa_o_int(zero_n2684676970156552555ol_int,(Pa))) = aa_o_nat(zero_n2687167440665602831ol_nat,(Pa)) ).

% nat_of_bool
tff(fact_2044_minus__one__mult__self,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)) = one_one_int ).

% minus_one_mult_self
tff(fact_2045_minus__one__mult__self,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_2046_minus__one__mult__self,axiom,
    ! [N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)) = one_one_rat ).

% minus_one_mult_self
tff(fact_2047_left__minus__one__mult__self,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_2048_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_2049_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_2050_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Xa)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_2051_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Xa)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_2052_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),Xa) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_2053_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),Xa) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_2054_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Xa)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_2055_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Xa)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_2056_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),Xa) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_2057_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),Xa) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_2058_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int) ) ).

% nat_0_iff
tff(fact_2059_nat__le__0,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_2060_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% zless_nat_conj
tff(fact_2061_sgn__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = zero_n356916108424825756nteger(A != zero_z3403309356797280102nteger) ).

% sgn_mult_self_eq
tff(fact_2062_sgn__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,sgn_sgn_rat(A)),sgn_sgn_rat(A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% sgn_mult_self_eq
tff(fact_2063_sgn__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% sgn_mult_self_eq
tff(fact_2064_int__nat__eq,axiom,
    ! [Z: int] :
      aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),Z,zero_zero_int) ).

% int_nat_eq
tff(fact_2065_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_2066_frac__gt__0__iff,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),archimedean_frac_rat(Xa))
    <=> ~ member_rat(Xa,ring_1_Ints_rat) ) ).

% frac_gt_0_iff
tff(fact_2067_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_2068_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_2069_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_2070_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),zero_z3403309356797280102nteger)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_2071_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_2072_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_2073_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_2074_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_2075_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_2076_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),zero_z3403309356797280102nteger)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_2077_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_2078_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_2079_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_2080_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_2081_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_2082_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_2083_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_2084_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_2085_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,Z)) = aa_int_rat(ring_1_of_int_rat,Z) ) ) ).

% of_nat_nat
tff(fact_2086_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = aa_int_int(ring_1_of_int_int,Z) ) ) ).

% of_nat_nat
tff(fact_2087_nat__times__as__int,axiom,
    ! [X: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_times_as_int
tff(fact_2088_nat__mult__distrib,axiom,
    ! [Z: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z2)) ) ) ).

% nat_mult_distrib
tff(fact_2089_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(4)
tff(fact_2090_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(3)
tff(fact_2091_le__cube,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(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_2092_le__refl,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N) ).

% le_refl
tff(fact_2093_nat__mono,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y3)) ) ).

% nat_mono
tff(fact_2094_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ) ).

% le_trans
tff(fact_2095_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% eq_imp_le
tff(fact_2096_le__square,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)) ).

% le_square
tff(fact_2097_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_2098_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_2099_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
       => ( aa_nat_o(aa_nat_fun_nat_o(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))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ) ).

% le_diff_iff
tff(fact_2100_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_2101_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_2102_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(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_2103_diff__le__self,axiom,
    ! [M: nat,N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M) ).

% diff_le_self
tff(fact_2104_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_2105_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_2106_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(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_2107_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(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_2108_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_2109_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ).

% nat_le_linear
tff(fact_2110_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_2111_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_2112_Nat_Oex__has__greatest__nat,axiom,
    ! [Pa: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(Pa,K)
     => ( ! [Y4: nat] :
            ( aa_nat_o(Pa,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => ? [X4: nat] :
            ( aa_nat_o(Pa,X4)
            & ! [Y5: nat] :
                ( aa_nat_o(Pa,Y5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y5),X4) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_2113_verit__la__generic,axiom,
    ! [A: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xa)
      | ( A = Xa )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A) ) ).

% verit_la_generic
tff(fact_2114_pred__subset__eq,axiom,
    ! [R4: set_se7855581050983116737at_nat,S2: set_se7855581050983116737at_nat] :
      ( ord_le290999283863451930_nat_o(aTP_Lamm_a(R4),aTP_Lamm_a(S2))
    <=> ord_le2077887516847798113at_nat(R4,S2) ) ).

% pred_subset_eq
tff(fact_2115_pred__subset__eq,axiom,
    ! [R4: set_se6260736226359567993nt_int,S2: set_se6260736226359567993nt_int] :
      ( ord_le2758740137681283170_int_o(aTP_Lamm_aa(R4),aTP_Lamm_aa(S2))
    <=> ord_le483042692224249369nt_int(R4,S2) ) ).

% pred_subset_eq
tff(fact_2116_pred__subset__eq,axiom,
    ! [R4: set_set_nat,S2: set_set_nat] :
      ( ord_le6468474429079055883_nat_o(aTP_Lamm_ab(R4),aTP_Lamm_ab(S2))
    <=> ord_le6893508408891458716et_nat(R4,S2) ) ).

% pred_subset_eq
tff(fact_2117_pred__subset__eq,axiom,
    ! [R4: set_int,S2: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_ad(R4),aTP_Lamm_ad(S2))
    <=> ord_less_eq_set_int(R4,S2) ) ).

% pred_subset_eq
tff(fact_2118_pred__subset__eq,axiom,
    ! [R4: set_nat,S2: set_nat] :
      ( ord_le7240017405068453697_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ac,R4),aa_set_nat_fun_nat_o(aTP_Lamm_ac,S2))
    <=> aa_set_nat_o(ord_less_eq_set_nat(R4),S2) ) ).

% pred_subset_eq
tff(fact_2119_conj__subset__def,axiom,
    ! [A3: set_se7855581050983116737at_nat,Pa: fun_se4782206882984618170_nat_o,Q: fun_se4782206882984618170_nat_o] :
      ( ord_le2077887516847798113at_nat(A3,collec5514110066124741708at_nat(aa_fun3315923329236548914_nat_o(aTP_Lamm_bx(Pa),Q)))
    <=> ( ord_le2077887516847798113at_nat(A3,collec5514110066124741708at_nat(Pa))
        & ord_le2077887516847798113at_nat(A3,collec5514110066124741708at_nat(Q)) ) ) ).

% conj_subset_def
tff(fact_2120_conj__subset__def,axiom,
    ! [A3: set_se6260736226359567993nt_int,Pa: fun_se7249947736802449410_int_o,Q: fun_se7249947736802449410_int_o] :
      ( ord_le483042692224249369nt_int(A3,collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_by(Pa),Q)))
    <=> ( ord_le483042692224249369nt_int(A3,collec5210948495886036740nt_int(Pa))
        & ord_le483042692224249369nt_int(A3,collec5210948495886036740nt_int(Q)) ) ) ).

% conj_subset_def
tff(fact_2121_conj__subset__def,axiom,
    ! [A3: set_set_nat,Pa: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ord_le6893508408891458716et_nat(A3,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bz(Pa),Q)))
    <=> ( ord_le6893508408891458716et_nat(A3,collect_set_nat(Pa))
        & ord_le6893508408891458716et_nat(A3,collect_set_nat(Q)) ) ) ).

% conj_subset_def
tff(fact_2122_conj__subset__def,axiom,
    ! [A3: set_int,Pa: fun_int_o,Q: fun_int_o] :
      ( ord_less_eq_set_int(A3,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ca(Pa),Q)))
    <=> ( ord_less_eq_set_int(A3,collect_int(Pa))
        & ord_less_eq_set_int(A3,collect_int(Q)) ) ) ).

% conj_subset_def
tff(fact_2123_conj__subset__def,axiom,
    ! [A3: set_nat,Pa: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cb(Pa),Q)))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A3),collect_nat(Pa))
        & aa_set_nat_o(ord_less_eq_set_nat(A3),collect_nat(Q)) ) ) ).

% conj_subset_def
tff(fact_2124_prop__restrict,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Z3: set_se7855581050983116737at_nat,X6: set_se7855581050983116737at_nat,Pa: fun_se4782206882984618170_nat_o] :
      ( member2643936169264416010at_nat(Xa,Z3)
     => ( ord_le2077887516847798113at_nat(Z3,collec5514110066124741708at_nat(aa_fun3315923329236548914_nat_o(aTP_Lamm_cc(X6),Pa)))
       => aa_set3702741264012733747_nat_o(Pa,Xa) ) ) ).

% prop_restrict
tff(fact_2125_prop__restrict,axiom,
    ! [Xa: set_Pr958786334691620121nt_int,Z3: set_se6260736226359567993nt_int,X6: set_se6260736226359567993nt_int,Pa: fun_se7249947736802449410_int_o] :
      ( member2340774599025711042nt_int(Xa,Z3)
     => ( ord_le483042692224249369nt_int(Z3,collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_cd(X6),Pa)))
       => aa_set6170482117830564987_int_o(Pa,Xa) ) ) ).

% prop_restrict
tff(fact_2126_prop__restrict,axiom,
    ! [Xa: set_nat,Z3: set_set_nat,X6: set_set_nat,Pa: fun_set_nat_o] :
      ( member_set_nat(Xa,Z3)
     => ( ord_le6893508408891458716et_nat(Z3,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ce(X6),Pa)))
       => aa_set_nat_o(Pa,Xa) ) ) ).

% prop_restrict
tff(fact_2127_prop__restrict,axiom,
    ! [Xa: int,Z3: set_int,X6: set_int,Pa: fun_int_o] :
      ( member_int(Xa,Z3)
     => ( ord_less_eq_set_int(Z3,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(X6),Pa)))
       => aa_int_o(Pa,Xa) ) ) ).

% prop_restrict
tff(fact_2128_prop__restrict,axiom,
    ! [Xa: nat,Z3: set_nat,X6: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa),Z3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Z3),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(X6),Pa)))
       => aa_nat_o(Pa,Xa) ) ) ).

% prop_restrict
tff(fact_2129_Collect__restrict,axiom,
    ! [X6: set_se7855581050983116737at_nat,Pa: fun_se4782206882984618170_nat_o] : ord_le2077887516847798113at_nat(collec5514110066124741708at_nat(aa_fun3315923329236548914_nat_o(aTP_Lamm_cc(X6),Pa)),X6) ).

% Collect_restrict
tff(fact_2130_Collect__restrict,axiom,
    ! [X6: set_se6260736226359567993nt_int,Pa: fun_se7249947736802449410_int_o] : ord_le483042692224249369nt_int(collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_cd(X6),Pa)),X6) ).

% Collect_restrict
tff(fact_2131_Collect__restrict,axiom,
    ! [X6: set_set_nat,Pa: fun_set_nat_o] : ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ce(X6),Pa)),X6) ).

% Collect_restrict
tff(fact_2132_Collect__restrict,axiom,
    ! [X6: set_int,Pa: fun_int_o] : ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(X6),Pa)),X6) ).

% Collect_restrict
tff(fact_2133_Collect__restrict,axiom,
    ! [X6: set_nat,Pa: fun_nat_o] : aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(X6),Pa))),X6) ).

% Collect_restrict
tff(fact_2134_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.left_commute
tff(fact_2135_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.left_commute
tff(fact_2136_mult_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% mult.left_commute
tff(fact_2137_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% mult.left_commute
tff(fact_2138_mult_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ).

% mult.commute
tff(fact_2139_mult_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ).

% mult.commute
tff(fact_2140_mult_Ocommute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) ).

% mult.commute
tff(fact_2141_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A) ).

% mult.commute
tff(fact_2142_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_2143_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.assoc
tff(fact_2144_mult_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% mult.assoc
tff(fact_2145_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% mult.assoc
tff(fact_2146_mult__of__int__commute,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,Xa)),Y3) = aa_int_int(aa_int_fun_int_int(times_times_int,Y3),aa_int_int(ring_1_of_int_int,Xa)) ).

% mult_of_int_commute
tff(fact_2147_mult__of__int__commute,axiom,
    ! [Xa: int,Y3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Xa)),Y3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),aa_int_Code_integer(ring_18347121197199848620nteger,Xa)) ).

% mult_of_int_commute
tff(fact_2148_mult__of__int__commute,axiom,
    ! [Xa: int,Y3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,Xa)),Y3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),aa_int_rat(ring_1_of_int_rat,Xa)) ).

% mult_of_int_commute
tff(fact_2149_Nats__mult,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),semiring_1_Nats_nat)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B),semiring_1_Nats_nat)
       => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),semiring_1_Nats_nat) ) ) ).

% Nats_mult
tff(fact_2150_Nats__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( member_Code_integer(A,semiri2551628265256775642nteger)
     => ( member_Code_integer(B,semiri2551628265256775642nteger)
       => member_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),semiri2551628265256775642nteger) ) ) ).

% Nats_mult
tff(fact_2151_Nats__mult,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,semiring_1_Nats_rat)
     => ( member_rat(B,semiring_1_Nats_rat)
       => member_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),semiring_1_Nats_rat) ) ) ).

% Nats_mult
tff(fact_2152_Nats__mult,axiom,
    ! [A: int,B: int] :
      ( member_int(A,semiring_1_Nats_int)
     => ( member_int(B,semiring_1_Nats_int)
       => member_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B),semiring_1_Nats_int) ) ) ).

% Nats_mult
tff(fact_2153_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2154_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2155_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2156_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2157_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_2158_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_2159_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z2))) ) ) ).

% nat_mult_distrib_neg
tff(fact_2160_ex__nat,axiom,
    ! [Pa: fun_nat_o] :
      ( ? [X_1: nat] : aa_nat_o(Pa,X_1)
    <=> ? [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
          & aa_nat_o(Pa,aa_int_nat(nat2,X3)) ) ) ).

% ex_nat
tff(fact_2161_all__nat,axiom,
    ! [Pa: fun_nat_o] :
      ( ! [X_1: nat] : aa_nat_o(Pa,X_1)
    <=> ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
         => aa_nat_o(Pa,aa_int_nat(nat2,X3)) ) ) ).

% all_nat
tff(fact_2162_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),N) ) ) ).

% nat_power_eq
tff(fact_2163_eq__nat__nat__iff,axiom,
    ! [Z: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z2) )
        <=> ( Z = Z2 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_2164_nat__diff__distrib,axiom,
    ! [Z2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),Z)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z2)) ) ) ) ).

% nat_diff_distrib
tff(fact_2165_nat__diff__distrib_H,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y3)) ) ) ) ).

% nat_diff_distrib'
tff(fact_2166_nat__le__iff,axiom,
    ! [Xa: int,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xa)),N)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% nat_le_iff
tff(fact_2167_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ).

% nat_abs_int_diff
tff(fact_2168_int__minus,axiom,
    ! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% int_minus
tff(fact_2169_mult__not__zero,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B) != zero_z2226904508553997617atural )
     => ( ( A != zero_z2226904508553997617atural )
        & ( B != zero_z2226904508553997617atural ) ) ) ).

% mult_not_zero
tff(fact_2170_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_2171_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_2172_mult__not__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) != zero_z3403309356797280102nteger )
     => ( ( A != zero_z3403309356797280102nteger )
        & ( B != zero_z3403309356797280102nteger ) ) ) ).

% mult_not_zero
tff(fact_2173_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_2174_divisors__zero,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B) = zero_z2226904508553997617atural )
     => ( ( A = zero_z2226904508553997617atural )
        | ( B = zero_z2226904508553997617atural ) ) ) ).

% divisors_zero
tff(fact_2175_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_2176_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_2177_divisors__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = zero_z3403309356797280102nteger )
     => ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% divisors_zero
tff(fact_2178_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_2179_no__zero__divisors,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( B != zero_z2226904508553997617atural )
       => ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B) != zero_z2226904508553997617atural ) ) ) ).

% no_zero_divisors
tff(fact_2180_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_2181_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_2182_no__zero__divisors,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( B != zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) != zero_z3403309356797280102nteger ) ) ) ).

% no_zero_divisors
tff(fact_2183_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_2184_mult__left__cancel,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2185_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2186_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2187_mult__left__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2188_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2189_mult__right__cancel,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2190_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2191_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2192_mult__right__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2193_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2194_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2195_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2196_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2197_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2198_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2199_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2200_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2201_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2202_comm__semiring__class_Odistrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2203_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2204_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_2205_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% distrib_left
tff(fact_2206_distrib__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).

% distrib_left
tff(fact_2207_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% distrib_left
tff(fact_2208_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_2209_distrib__right,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% distrib_right
tff(fact_2210_distrib__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% distrib_right
tff(fact_2211_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% distrib_right
tff(fact_2212_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_2213_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E)),C) ).

% combine_common_factor
tff(fact_2214_combine__common__factor,axiom,
    ! [A: code_integer,E: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),E)),C) ).

% combine_common_factor
tff(fact_2215_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_2216_crossproduct__eq,axiom,
    ! [W: nat,Y3: nat,Xa: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),Y3)) )
    <=> ( ( W = Xa )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
tff(fact_2217_crossproduct__eq,axiom,
    ! [W: int,Y3: int,Xa: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y3)),aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Y3)) )
    <=> ( ( W = Xa )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
tff(fact_2218_crossproduct__eq,axiom,
    ! [W: code_integer,Y3: code_integer,Xa: code_integer,Z: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W),Y3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Y3)) )
    <=> ( ( W = Xa )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
tff(fact_2219_crossproduct__eq,axiom,
    ! [W: rat,Y3: rat,Xa: rat,Z: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Y3)) )
    <=> ( ( W = Xa )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
tff(fact_2220_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2221_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2222_crossproduct__noteq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2223_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),D)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2224_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(one_one_Code_natural),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2225_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_2226_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_2227_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2228_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2229_mult_Ocomm__neutral,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),one_one_Code_natural) = A ).

% mult.comm_neutral
tff(fact_2230_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_2231_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_2232_mult_Ocomm__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ).

% mult.comm_neutral
tff(fact_2233_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_2234_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_2235_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib'
tff(fact_2236_right__diff__distrib_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).

% right_diff_distrib'
tff(fact_2237_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% right_diff_distrib'
tff(fact_2238_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_2239_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ).

% left_diff_distrib'
tff(fact_2240_left__diff__distrib_H,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)) ).

% left_diff_distrib'
tff(fact_2241_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B,C)),A) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)) ).

% left_diff_distrib'
tff(fact_2242_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib
tff(fact_2243_right__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).

% right_diff_distrib
tff(fact_2244_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% right_diff_distrib
tff(fact_2245_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% left_diff_distrib
tff(fact_2246_left__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% left_diff_distrib
tff(fact_2247_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% left_diff_distrib
tff(fact_2248_minus__mult__commute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_2249_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_mult_commute
tff(fact_2250_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_mult_commute
tff(fact_2251_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),A) = aa_int_int(aa_int_fun_int_int(times_times_int,B),B) )
    <=> ( ( A = B )
        | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_2252_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
    <=> ( ( A = B )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_2253_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),B) )
    <=> ( ( A = B )
        | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_2254_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_2255_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),Y3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)) ).

% mult_of_nat_commute
tff(fact_2256_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),Y3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),aa_nat_rat(semiri681578069525770553at_rat,Xa)) ).

% mult_of_nat_commute
tff(fact_2257_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),Y3) = aa_int_int(aa_int_fun_int_int(times_times_int,Y3),aa_nat_int(semiri1314217659103216013at_int,Xa)) ).

% mult_of_nat_commute
tff(fact_2258_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),Y3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y3),aa_nat_nat(semiri1316708129612266289at_nat,Xa)) ).

% mult_of_nat_commute
tff(fact_2259_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_2260_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_2261_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_2262_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_2263_abs__mult,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_mult
tff(fact_2264_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ).

% abs_mult
tff(fact_2265_abs__mult,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(B)) ).

% abs_mult
tff(fact_2266_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_2267_Ints__of__int,axiom,
    ! [Z: int] : member_rat(aa_int_rat(ring_1_of_int_rat,Z),ring_1_Ints_rat) ).

% Ints_of_int
tff(fact_2268_Ints__of__int,axiom,
    ! [Z: int] : member_int(aa_int_int(ring_1_of_int_int,Z),ring_1_Ints_int) ).

% Ints_of_int
tff(fact_2269_Ints__induct,axiom,
    ! [Q2: rat,Pa: fun_rat_o] :
      ( member_rat(Q2,ring_1_Ints_rat)
     => ( ! [Z4: int] : aa_rat_o(Pa,aa_int_rat(ring_1_of_int_rat,Z4))
       => aa_rat_o(Pa,Q2) ) ) ).

% Ints_induct
tff(fact_2270_Ints__induct,axiom,
    ! [Q2: int,Pa: fun_int_o] :
      ( member_int(Q2,ring_1_Ints_int)
     => ( ! [Z4: int] : aa_int_o(Pa,aa_int_int(ring_1_of_int_int,Z4))
       => aa_int_o(Pa,Q2) ) ) ).

% Ints_induct
tff(fact_2271_Ints__cases,axiom,
    ! [Q2: rat] :
      ( member_rat(Q2,ring_1_Ints_rat)
     => ~ ! [Z4: int] : Q2 != aa_int_rat(ring_1_of_int_rat,Z4) ) ).

% Ints_cases
tff(fact_2272_Ints__cases,axiom,
    ! [Q2: int] :
      ( member_int(Q2,ring_1_Ints_int)
     => ~ ! [Z4: int] : Q2 != aa_int_int(ring_1_of_int_int,Z4) ) ).

% Ints_cases
tff(fact_2273_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(2)
tff(fact_2274_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(1)
tff(fact_2275_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_2276_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_2277_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_2278_sgn__mult,axiom,
    ! [A: int,B: int] : aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B)) ).

% sgn_mult
tff(fact_2279_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ).

% sgn_mult
tff(fact_2280_sgn__mult,axiom,
    ! [A: rat,B: rat] : sgn_sgn_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,sgn_sgn_rat(A)),sgn_sgn_rat(B)) ).

% sgn_mult
tff(fact_2281_of__bool__conj,axiom,
    ! [Pa: $o,Q: $o] :
      zero_n356916108424825756nteger(( (Pa)
        & (Q) )) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((Pa))),zero_n356916108424825756nteger((Q))) ).

% of_bool_conj
tff(fact_2282_of__bool__conj,axiom,
    ! [Pa: $o,Q: $o] :
      zero_n2052037380579107095ol_rat(( (Pa)
        & (Q) )) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_n2052037380579107095ol_rat((Pa))),zero_n2052037380579107095ol_rat((Q))) ).

% of_bool_conj
tff(fact_2283_of__bool__conj,axiom,
    ! [Pa: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (Pa)
        & (Q) )) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,(Pa))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_conj
tff(fact_2284_of__bool__conj,axiom,
    ! [Pa: $o,Q: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (Pa)
        & (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ).

% of_bool_conj
tff(fact_2285_Ints__mult,axiom,
    ! [A: int,B: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B),ring_1_Ints_int) ) ) ).

% Ints_mult
tff(fact_2286_Ints__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( member_Code_integer(A,ring_11222124179247155820nteger)
     => ( member_Code_integer(B,ring_11222124179247155820nteger)
       => member_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),ring_11222124179247155820nteger) ) ) ).

% Ints_mult
tff(fact_2287_Ints__mult,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),ring_1_Ints_rat) ) ) ).

% Ints_mult
tff(fact_2288_nat__minus__as__int,axiom,
    ! [X: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_minus_as_int
tff(fact_2289_lambda__zero,axiom,
    aTP_Lamm_ch = times_2397367101498566445atural(zero_z2226904508553997617atural) ).

% lambda_zero
tff(fact_2290_lambda__zero,axiom,
    aTP_Lamm_ci = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_2291_lambda__zero,axiom,
    aTP_Lamm_cj = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_2292_lambda__zero,axiom,
    aTP_Lamm_ck = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger) ).

% lambda_zero
tff(fact_2293_lambda__zero,axiom,
    aTP_Lamm_cl = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).

% lambda_zero
tff(fact_2294_lambda__one,axiom,
    aTP_Lamm_cm = times_2397367101498566445atural(one_one_Code_natural) ).

% lambda_one
tff(fact_2295_lambda__one,axiom,
    aTP_Lamm_av = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_2296_lambda__one,axiom,
    aTP_Lamm_cn = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_2297_lambda__one,axiom,
    aTP_Lamm_co = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer) ).

% lambda_one
tff(fact_2298_lambda__one,axiom,
    aTP_Lamm_cp = aa_rat_fun_rat_rat(times_times_rat,one_one_rat) ).

% lambda_one
tff(fact_2299_frac__ge__0,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),archimedean_frac_rat(Xa)) ).

% frac_ge_0
tff(fact_2300_frac__lt__1,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,archimedean_frac_rat(Xa)),one_one_rat) ).

% frac_lt_1
tff(fact_2301_frac__1__eq,axiom,
    ! [Xa: rat] : archimedean_frac_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),one_one_rat)) = archimedean_frac_rat(Xa) ).

% frac_1_eq
tff(fact_2302_Nats__0,axiom,
    aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),semiring_1_Nats_nat) ).

% Nats_0
tff(fact_2303_Nats__0,axiom,
    member_Code_integer(zero_z3403309356797280102nteger,semiri2551628265256775642nteger) ).

% Nats_0
tff(fact_2304_Nats__0,axiom,
    member_rat(zero_zero_rat,semiring_1_Nats_rat) ).

% Nats_0
tff(fact_2305_Nats__0,axiom,
    member_Code_natural(zero_z2226904508553997617atural,semiri1375223417013493157atural) ).

% Nats_0
tff(fact_2306_Nats__0,axiom,
    member_int(zero_zero_int,semiring_1_Nats_int) ).

% Nats_0
tff(fact_2307_Nats__add,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,semiring_1_Nats_rat)
     => ( member_rat(B,semiring_1_Nats_rat)
       => member_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),semiring_1_Nats_rat) ) ) ).

% Nats_add
tff(fact_2308_Nats__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),semiring_1_Nats_nat)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B),semiring_1_Nats_nat)
       => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),semiring_1_Nats_nat) ) ) ).

% Nats_add
tff(fact_2309_Nats__add,axiom,
    ! [A: int,B: int] :
      ( member_int(A,semiring_1_Nats_int)
     => ( member_int(B,semiring_1_Nats_int)
       => member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B),semiring_1_Nats_int) ) ) ).

% Nats_add
tff(fact_2310_Nats__1,axiom,
    aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,one_one_nat),semiring_1_Nats_nat) ).

% Nats_1
tff(fact_2311_Nats__1,axiom,
    member_Code_integer(one_one_Code_integer,semiri2551628265256775642nteger) ).

% Nats_1
tff(fact_2312_Nats__1,axiom,
    member_rat(one_one_rat,semiring_1_Nats_rat) ).

% Nats_1
tff(fact_2313_Nats__1,axiom,
    member_Code_natural(one_one_Code_natural,semiri1375223417013493157atural) ).

% Nats_1
tff(fact_2314_Nats__1,axiom,
    member_int(one_one_int,semiring_1_Nats_int) ).

% Nats_1
tff(fact_2315_of__nat__in__Nats,axiom,
    ! [N: nat] : member_int(aa_nat_int(semiri1314217659103216013at_int,N),semiring_1_Nats_int) ).

% of_nat_in_Nats
tff(fact_2316_of__nat__in__Nats,axiom,
    ! [N: nat] : aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),semiring_1_Nats_nat) ).

% of_nat_in_Nats
tff(fact_2317_Nats__induct,axiom,
    ! [Xa: int,Pa: fun_int_o] :
      ( member_int(Xa,semiring_1_Nats_int)
     => ( ! [N2: nat] : aa_int_o(Pa,aa_nat_int(semiri1314217659103216013at_int,N2))
       => aa_int_o(Pa,Xa) ) ) ).

% Nats_induct
tff(fact_2318_Nats__induct,axiom,
    ! [Xa: nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa),semiring_1_Nats_nat)
     => ( ! [N2: nat] : aa_nat_o(Pa,aa_nat_nat(semiri1316708129612266289at_nat,N2))
       => aa_nat_o(Pa,Xa) ) ) ).

% Nats_induct
tff(fact_2319_Nats__cases,axiom,
    ! [Xa: int] :
      ( member_int(Xa,semiring_1_Nats_int)
     => ~ ! [N2: nat] : Xa != aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% Nats_cases
tff(fact_2320_Nats__cases,axiom,
    ! [Xa: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa),semiring_1_Nats_nat)
     => ~ ! [N2: nat] : Xa != aa_nat_nat(semiri1316708129612266289at_nat,N2) ) ).

% Nats_cases
tff(fact_2321_Nats__subset__Ints,axiom,
    ord_less_eq_set_int(semiring_1_Nats_int,ring_1_Ints_int) ).

% Nats_subset_Ints
tff(fact_2322_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_2323_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_2324_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),K) ) ) ).

% le_nat_iff
tff(fact_2325_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_2326_nat__0__le,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_2327_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_2328_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_2329_ex__less__of__nat__mult,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),Xa)) ) ).

% ex_less_of_nat_mult
tff(fact_2330_mult__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,C),D)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2331_mult__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2332_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2333_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2334_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2335_mult__mono_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,C),D)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2336_mult__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2337_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2338_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2339_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2340_zero__le__square,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A)) ).

% zero_le_square
tff(fact_2341_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)) ).

% zero_le_square
tff(fact_2342_zero__le__square,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A)) ).

% zero_le_square
tff(fact_2343_split__mult__pos__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ).

% split_mult_pos_le
tff(fact_2344_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ).

% split_mult_pos_le
tff(fact_2345_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% split_mult_pos_le
tff(fact_2346_mult__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2347_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2348_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2349_mult__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2350_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2351_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2352_mult__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)) ) ) ).

% mult_left_mono
tff(fact_2353_mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_left_mono
tff(fact_2354_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono
tff(fact_2355_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_left_mono
tff(fact_2356_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono
tff(fact_2357_mult__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2358_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2359_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2360_mult__right__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)) ) ) ).

% mult_right_mono
tff(fact_2361_mult__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_right_mono
tff(fact_2362_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono
tff(fact_2363_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_right_mono
tff(fact_2364_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono
tff(fact_2365_mult__le__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2366_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2367_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2368_split__mult__neg__le,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
          & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),zero_z2226904508553997617atural) )
        | ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural)
          & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B) ) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),zero_z2226904508553997617atural) ) ).

% split_mult_neg_le
tff(fact_2369_split__mult__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ).

% split_mult_neg_le
tff(fact_2370_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_2371_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_2372_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_2373_mult__nonneg__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2374_mult__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2375_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2376_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2377_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2378_mult__nonneg__nonpos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),zero_z2226904508553997617atural) ) ) ).

% mult_nonneg_nonpos
tff(fact_2379_mult__nonneg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonneg_nonpos
tff(fact_2380_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2381_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2382_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_2383_mult__nonpos__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),zero_z2226904508553997617atural) ) ) ).

% mult_nonpos_nonneg
tff(fact_2384_mult__nonpos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonpos_nonneg
tff(fact_2385_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2386_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2387_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_2388_mult__nonneg__nonpos2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),A)),zero_z2226904508553997617atural) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2389_mult__nonneg__nonpos2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2390_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2391_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2392_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2393_zero__le__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) ) ) ).

% zero_le_mult_iff
tff(fact_2394_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_2395_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_2396_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2397_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2398_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2399_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2400_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2401_mult__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).

% mult_neg_neg
tff(fact_2402_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_neg_neg
tff(fact_2403_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_neg_neg
tff(fact_2404_not__square__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A)),zero_z3403309356797280102nteger) ).

% not_square_less_zero
tff(fact_2405_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_2406_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_2407_mult__less__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2408_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2409_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2410_mult__neg__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),zero_z2226904508553997617atural) ) ) ).

% mult_neg_pos
tff(fact_2411_mult__neg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_neg_pos
tff(fact_2412_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_2413_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_2414_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_2415_mult__pos__neg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),zero_z2226904508553997617atural) ) ) ).

% mult_pos_neg
tff(fact_2416_mult__pos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_pos_neg
tff(fact_2417_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_2418_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_2419_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_2420_mult__pos__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)) ) ) ).

% mult_pos_pos
tff(fact_2421_mult__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2422_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2423_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2424_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2425_mult__pos__neg2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),A)),zero_z2226904508553997617atural) ) ) ).

% mult_pos_neg2
tff(fact_2426_mult__pos__neg2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),zero_z3403309356797280102nteger) ) ) ).

% mult_pos_neg2
tff(fact_2427_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_2428_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_2429_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_2430_zero__less__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger) ) ) ) ).

% zero_less_mult_iff
tff(fact_2431_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_2432_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_2433_zero__less__mult__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B))
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B) ) ) ).

% zero_less_mult_pos
tff(fact_2434_zero__less__mult__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ).

% zero_less_mult_pos
tff(fact_2435_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_2436_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_2437_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_2438_zero__less__mult__pos2,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),A))
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2439_zero__less__mult__pos2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2440_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2441_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2442_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2443_mult__less__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2444_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2445_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2446_mult__less__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2447_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2448_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2449_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2450_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2451_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2452_mult__strict__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2453_mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2454_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2455_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2456_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2457_mult__less__cancel__left__disj,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2458_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2459_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2460_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2461_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2462_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2463_mult__strict__right__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2464_mult__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2465_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2466_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2467_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2468_mult__less__cancel__right__disj,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2469_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2470_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2471_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2472_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2473_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2474_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2475_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2476_add__scale__eq__noteq,axiom,
    ! [R: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2477_add__scale__eq__noteq,axiom,
    ! [R: int,A: int,B: int,C: int,D: int] :
      ( ( R != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2478_add__scale__eq__noteq,axiom,
    ! [R: code_integer,A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( R != zero_z3403309356797280102nteger )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R),C)) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2479_add__scale__eq__noteq,axiom,
    ! [R: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2480_sum__squares__eq__zero__iff,axiom,
    ! [Xa: int,Y3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Y3)) = zero_zero_int )
    <=> ( ( Xa = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2481_sum__squares__eq__zero__iff,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Xa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),Y3)) = zero_z3403309356797280102nteger )
    <=> ( ( Xa = zero_z3403309356797280102nteger )
        & ( Y3 = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2482_sum__squares__eq__zero__iff,axiom,
    ! [Xa: rat,Y3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Y3)) = zero_zero_rat )
    <=> ( ( Xa = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2483_less__1__mult,axiom,
    ! [M: code_natural,N: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),M)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),N)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(M),N)) ) ) ).

% less_1_mult
tff(fact_2484_less__1__mult,axiom,
    ! [M: code_integer,N: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),N)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,M),N)) ) ) ).

% less_1_mult
tff(fact_2485_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),N)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),N)) ) ) ).

% less_1_mult
tff(fact_2486_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ) ) ).

% less_1_mult
tff(fact_2487_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),N)) ) ) ).

% less_1_mult
tff(fact_2488_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_2489_eq__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_2490_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_2491_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_2492_eq__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D) )
    <=> ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_2493_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D) )
    <=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B,A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_2494_square__diff__square__factored,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Y3)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y3)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y3)) ).

% square_diff_square_factored
tff(fact_2495_square__diff__square__factored,axiom,
    ! [Xa: code_integer,Y3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Xa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),Y3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Xa),Y3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xa),Y3)) ).

% square_diff_square_factored
tff(fact_2496_square__diff__square__factored,axiom,
    ! [Xa: rat,Y3: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Y3)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y3)),minus_minus_rat(Xa,Y3)) ).

% square_diff_square_factored
tff(fact_2497_square__eq__1__iff,axiom,
    ! [Xa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa) = one_one_int )
    <=> ( ( Xa = one_one_int )
        | ( Xa = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_2498_square__eq__1__iff,axiom,
    ! [Xa: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Xa) = one_one_Code_integer )
    <=> ( ( Xa = one_one_Code_integer )
        | ( Xa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_2499_square__eq__1__iff,axiom,
    ! [Xa: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa) = one_one_rat )
    <=> ( ( Xa = one_one_rat )
        | ( Xa = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_2500_left__right__inverse__power,axiom,
    ! [Xa: code_natural,Y3: code_natural,N: nat] :
      ( ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(Xa),Y3) = one_one_Code_natural )
     => ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(power_7079662738309270450atural(Xa),N)),aa_nat_Code_natural(power_7079662738309270450atural(Y3),N)) = one_one_Code_natural ) ) ).

% left_right_inverse_power
tff(fact_2501_left__right__inverse__power,axiom,
    ! [Xa: nat,Y3: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),Y3) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),N)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_2502_left__right__inverse__power,axiom,
    ! [Xa: int,Y3: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Y3) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),N)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_2503_left__right__inverse__power,axiom,
    ! [Xa: code_integer,Y3: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Y3) = one_one_Code_integer )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),N)) = one_one_Code_integer ) ) ).

% left_right_inverse_power
tff(fact_2504_left__right__inverse__power,axiom,
    ! [Xa: rat,Y3: rat,N: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Y3) = one_one_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xa),N)),aa_nat_rat(power_power_rat(Y3),N)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_2505_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D)) ) ) ).

% abs_mult_less
tff(fact_2506_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(B)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),D)) ) ) ).

% abs_mult_less
tff(fact_2507_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D)) ) ) ).

% abs_mult_less
tff(fact_2508_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_2509_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_2510_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_2511_power__add,axiom,
    ! [A: int,M: nat,N: nat] : aa_nat_int(aa_int_fun_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(aa_int_fun_nat_int(power_power_int,A),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ).

% power_add
tff(fact_2512_power__add,axiom,
    ! [A: code_integer,M: nat,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% power_add
tff(fact_2513_power__add,axiom,
    ! [A: rat,M: nat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)) ).

% power_add
tff(fact_2514_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_2515_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_2516_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_2517_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_2518_mult__sgn__abs,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Xa)),aa_int_int(abs_abs_int,Xa)) = Xa ).

% mult_sgn_abs
tff(fact_2519_mult__sgn__abs,axiom,
    ! [Xa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,Xa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xa)) = Xa ).

% mult_sgn_abs
tff(fact_2520_mult__sgn__abs,axiom,
    ! [Xa: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,sgn_sgn_rat(Xa)),abs_abs_rat(Xa)) = Xa ).

% mult_sgn_abs
tff(fact_2521_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_2522_sgn__mult__abs,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = A ).

% sgn_mult_abs
tff(fact_2523_sgn__mult__abs,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,sgn_sgn_rat(A)),abs_abs_rat(A)) = A ).

% sgn_mult_abs
tff(fact_2524_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_2525_abs__mult__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = A ).

% abs_mult_sgn
tff(fact_2526_abs__mult__sgn,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),sgn_sgn_rat(A)) = A ).

% abs_mult_sgn
tff(fact_2527_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_2528_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_2529_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: rat] : abs_abs_rat(K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),sgn_sgn_rat(K)) ).

% linordered_idom_class.abs_sgn
tff(fact_2530_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_Code_integer(ring_18347121197199848620nteger,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_2531_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_rat(ring_1_of_int_rat,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_2532_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_2533_field__le__mult__one__interval,axiom,
    ! [Xa: rat,Y3: rat] :
      ( ! [Z4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z4),Xa)),Y3) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y3) ) ).

% field_le_mult_one_interval
tff(fact_2534_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ) ).

% of_int_nonneg
tff(fact_2535_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_nonneg
tff(fact_2536_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_nonneg
tff(fact_2537_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_mono_iff
tff(fact_2538_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_2539_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa_int_nat(nat2,W) = M )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_2540_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = aa_int_nat(nat2,W) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_2541_nat__add__distrib,axiom,
    ! [Z: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z2)) ) ) ) ).

% nat_add_distrib
tff(fact_2542_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_2543_nat__int__add,axiom,
    ! [A: nat,B: nat] : aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) ).

% nat_int_add
tff(fact_2544_Nats__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( member_Code_integer(A,semiri2551628265256775642nteger)
     => ( member_Code_integer(B,semiri2551628265256775642nteger)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
         => member_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B),semiri2551628265256775642nteger) ) ) ) ).

% Nats_diff
tff(fact_2545_Nats__diff,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,semiring_1_Nats_rat)
     => ( member_rat(B,semiring_1_Nats_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
         => member_rat(minus_minus_rat(A,B),semiring_1_Nats_rat) ) ) ) ).

% Nats_diff
tff(fact_2546_Nats__diff,axiom,
    ! [A: int,B: int] :
      ( member_int(A,semiring_1_Nats_int)
     => ( member_int(B,semiring_1_Nats_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B),semiring_1_Nats_int) ) ) ) ).

% Nats_diff
tff(fact_2547_mult__less__le__imp__less,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,C),D)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2548_mult__less__le__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2549_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2550_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2551_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2552_mult__le__less__imp__less,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,C),D)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2553_mult__le__less__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2554_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2555_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2556_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2557_mult__right__le__imp__le,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C))
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2558_mult__right__le__imp__le,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2559_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2560_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2561_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2562_mult__left__le__imp__le,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B))
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2563_mult__left__le__imp__le,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2564_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2565_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2566_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2567_mult__le__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2568_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2569_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2570_mult__le__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2571_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2572_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2573_mult__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2574_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2575_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2576_mult__strict__mono_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,C),D)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2577_mult__strict__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2578_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2579_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2580_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2581_mult__right__less__imp__less,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C))
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2582_mult__right__less__imp__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2583_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2584_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2585_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2586_mult__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2587_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2588_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2589_mult__strict__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,C),D)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2590_mult__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2591_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2592_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2593_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2594_mult__left__less__imp__less,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B))
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2595_mult__left__less__imp__less,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2596_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2597_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2598_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2599_mult__le__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2600_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2601_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2602_mult__le__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2603_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2604_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2605_sum__squares__ge__zero,axiom,
    ! [Xa: code_integer,Y3: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Xa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),Y3))) ).

% sum_squares_ge_zero
tff(fact_2606_sum__squares__ge__zero,axiom,
    ! [Xa: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Y3))) ).

% sum_squares_ge_zero
tff(fact_2607_sum__squares__ge__zero,axiom,
    ! [Xa: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Y3))) ).

% sum_squares_ge_zero
tff(fact_2608_sum__squares__le__zero__iff,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Xa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),Y3))),zero_z3403309356797280102nteger)
    <=> ( ( Xa = zero_z3403309356797280102nteger )
        & ( Y3 = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2609_sum__squares__le__zero__iff,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Y3))),zero_zero_rat)
    <=> ( ( Xa = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2610_sum__squares__le__zero__iff,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Y3))),zero_zero_int)
    <=> ( ( Xa = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2611_mult__left__le__one__le,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y3)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y3),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_2612_mult__left__le__one__le,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y3)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_2613_mult__left__le__one__le,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_2614_mult__right__le__one__le,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y3)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y3),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Y3)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_2615_mult__right__le__one__le,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y3)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Y3)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_2616_mult__right__le__one__le,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Y3)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_2617_mult__le__one,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),one_one_Code_natural)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),one_one_Code_natural) ) ) ) ).

% mult_le_one
tff(fact_2618_mult__le__one,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ) ).

% mult_le_one
tff(fact_2619_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_2620_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_2621_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_2622_mult__left__le,axiom,
    ! [C: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,C),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),A) ) ) ).

% mult_left_le
tff(fact_2623_mult__left__le,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2624_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2625_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2626_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2627_not__sum__squares__lt__zero,axiom,
    ! [Xa: code_integer,Y3: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Xa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),Y3))),zero_z3403309356797280102nteger) ).

% not_sum_squares_lt_zero
tff(fact_2628_not__sum__squares__lt__zero,axiom,
    ! [Xa: rat,Y3: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Y3))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_2629_not__sum__squares__lt__zero,axiom,
    ! [Xa: int,Y3: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Y3))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_2630_sum__squares__gt__zero__iff,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Xa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),Y3)))
    <=> ( ( Xa != zero_z3403309356797280102nteger )
        | ( Y3 != zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2631_sum__squares__gt__zero__iff,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Y3)))
    <=> ( ( Xa != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2632_sum__squares__gt__zero__iff,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Y3)))
    <=> ( ( Xa != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2633_le__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A)),E)),D)) ) ).

% le_add_iff2
tff(fact_2634_le__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B,A)),E)),D)) ) ).

% le_add_iff2
tff(fact_2635_le__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% le_add_iff2
tff(fact_2636_le__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),E)),C)),D) ) ).

% le_add_iff1
tff(fact_2637_le__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),E)),C)),D) ) ).

% le_add_iff1
tff(fact_2638_le__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% le_add_iff1
tff(fact_2639_less__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_2640_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B,A)),E)),D)) ) ).

% less_add_iff2
tff(fact_2641_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_2642_less__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_2643_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_2644_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_2645_square__diff__one__factored,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),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,Xa),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),one_one_int)) ).

% square_diff_one_factored
tff(fact_2646_square__diff__one__factored,axiom,
    ! [Xa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Xa)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Xa),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xa),one_one_Code_integer)) ).

% square_diff_one_factored
tff(fact_2647_square__diff__one__factored,axiom,
    ! [Xa: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),one_one_rat)),minus_minus_rat(Xa,one_one_rat)) ).

% square_diff_one_factored
tff(fact_2648_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ) ) ).

% abs_eq_mult
tff(fact_2649_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => ( abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(B)) ) ) ).

% abs_eq_mult
tff(fact_2650_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_2651_abs__mult__pos,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y3)),Xa) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),Xa)) ) ) ).

% abs_mult_pos
tff(fact_2652_abs__mult__pos,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(Y3)),Xa) = abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Xa)) ) ) ).

% abs_mult_pos
tff(fact_2653_abs__mult__pos,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y3)),Xa) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Xa)) ) ) ).

% abs_mult_pos
tff(fact_2654_power__gt1__lemma,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_nat_Code_natural(power_7079662738309270450atural(A),N))) ) ).

% power_gt1_lemma
tff(fact_2655_power__gt1__lemma,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).

% power_gt1_lemma
tff(fact_2656_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N))) ) ).

% power_gt1_lemma
tff(fact_2657_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% power_gt1_lemma
tff(fact_2658_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))) ) ).

% power_gt1_lemma
tff(fact_2659_power__less__power__Suc,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A),N)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_nat_Code_natural(power_7079662738309270450atural(A),N))) ) ).

% power_less_power_Suc
tff(fact_2660_power__less__power__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).

% power_less_power_Suc
tff(fact_2661_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N))) ) ).

% power_less_power_Suc
tff(fact_2662_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),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_2663_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))) ) ).

% power_less_power_Suc
tff(fact_2664_power__minus,axiom,
    ! [A: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ).

% power_minus
tff(fact_2665_power__minus,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% power_minus
tff(fact_2666_power__minus,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(A),N)) ).

% power_minus
tff(fact_2667_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = $ite(M = zero_zero_nat,zero_zero_nat,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_2668_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(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_2669_diff__nat__eq__if,axiom,
    ! [Z: int,Z2: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z2)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z2),
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_2670_frac__eq,axiom,
    ! [Xa: rat] :
      ( ( archimedean_frac_rat(Xa) = Xa )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),one_one_rat) ) ) ).

% frac_eq
tff(fact_2671_nat__plus__as__int,axiom,
    ! [X: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_plus_as_int
tff(fact_2672_frac__add,axiom,
    ! [Xa: rat,Y3: rat] :
      archimedean_frac_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y3)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,archimedean_frac_rat(Xa)),archimedean_frac_rat(Y3))),one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,archimedean_frac_rat(Xa)),archimedean_frac_rat(Y3)),minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,archimedean_frac_rat(Xa)),archimedean_frac_rat(Y3)),one_one_rat)) ).

% frac_add
tff(fact_2673_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ) ).

% of_int_pos
tff(fact_2674_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_pos
tff(fact_2675_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_pos
tff(fact_2676_of__int__leD,axiom,
    ! [N: int,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,N))),Xa)
     => ( ( N = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xa) ) ) ).

% of_int_leD
tff(fact_2677_of__int__leD,axiom,
    ! [N: int,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,N))),Xa)
     => ( ( N = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xa) ) ) ).

% of_int_leD
tff(fact_2678_of__int__leD,axiom,
    ! [N: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),Xa)
     => ( ( N = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xa) ) ) ).

% of_int_leD
tff(fact_2679_of__int__lessD,axiom,
    ! [N: int,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,N))),Xa)
     => ( ( N = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xa) ) ) ).

% of_int_lessD
tff(fact_2680_of__int__lessD,axiom,
    ! [N: int,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,N))),Xa)
     => ( ( N = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xa) ) ) ).

% of_int_lessD
tff(fact_2681_of__int__lessD,axiom,
    ! [N: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),Xa)
     => ( ( N = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xa) ) ) ).

% of_int_lessD
tff(fact_2682_of__nat__less__of__int__iff,axiom,
    ! [N: nat,Xa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_int_rat(ring_1_of_int_rat,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),Xa) ) ).

% of_nat_less_of_int_iff
tff(fact_2683_of__nat__less__of__int__iff,axiom,
    ! [N: nat,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(ring_1_of_int_int,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),Xa) ) ).

% of_nat_less_of_int_iff
tff(fact_2684_split__nat,axiom,
    ! [Pa: fun_nat_o,I: int] :
      ( aa_nat_o(Pa,aa_int_nat(nat2,I))
    <=> ( ! [N6: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N6) )
           => aa_nat_o(Pa,N6) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int)
         => aa_nat_o(Pa,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_2685_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),M)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M)) ) ) ).

% nat_less_iff
tff(fact_2686_mult__le__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2687_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2688_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2689_mult__le__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2690_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2691_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2692_mult__le__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2693_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2694_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2695_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2696_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2697_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2698_mult__less__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2699_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2700_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2701_mult__less__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2702_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2703_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2704_mult__less__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2705_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2706_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2707_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2708_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2709_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2710_convex__bound__le,axiom,
    ! [Xa: code_integer,A: code_integer,Y3: code_integer,U: code_integer,V: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y3),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V)
           => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),V) = one_one_Code_integer )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),Xa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V),Y3))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2711_convex__bound__le,axiom,
    ! [Xa: rat,A: rat,Y3: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y3))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2712_convex__bound__le,axiom,
    ! [Xa: int,A: int,Y3: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y3))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2713_power__Suc__less,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),one_one_Code_natural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_nat_Code_natural(power_7079662738309270450atural(A),N))),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) ) ) ).

% power_Suc_less
tff(fact_2714_power__Suc__less,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).

% power_Suc_less
tff(fact_2715_power__Suc__less,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N))),aa_nat_rat(power_power_rat(A),N)) ) ) ).

% power_Suc_less
tff(fact_2716_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).

% power_Suc_less
tff(fact_2717_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ).

% power_Suc_less
tff(fact_2718_power__eq__if,axiom,
    ! [P: code_natural,M: nat] :
      aa_nat_Code_natural(power_7079662738309270450atural(P),M) = $ite(M = zero_zero_nat,one_one_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(P),aa_nat_Code_natural(power_7079662738309270450atural(P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2719_power__eq__if,axiom,
    ! [P: nat,M: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P),M) = $ite(M = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2720_power__eq__if,axiom,
    ! [P: int,M: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,P),M) = $ite(M = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P),aa_nat_int(aa_int_fun_nat_int(power_power_int,P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2721_power__eq__if,axiom,
    ! [P: code_integer,M: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(P),M) = $ite(M = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,P),aa_nat_Code_integer(power_8256067586552552935nteger(P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2722_power__eq__if,axiom,
    ! [P: rat,M: nat] :
      aa_nat_rat(power_power_rat(P),M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P),aa_nat_rat(power_power_rat(P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2723_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2724_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N) ) ) ).

% power_minus_mult
tff(fact_2725_power__minus__mult,axiom,
    ! [N: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).

% power_minus_mult
tff(fact_2726_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% power_minus_mult
tff(fact_2727_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),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_2728_frac__neg,axiom,
    ! [Xa: rat] :
      archimedean_frac_rat(aa_rat_rat(uminus_uminus_rat,Xa)) = $ite(member_rat(Xa,ring_1_Ints_rat),zero_zero_rat,minus_minus_rat(one_one_rat,archimedean_frac_rat(Xa))) ).

% frac_neg
tff(fact_2729_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2730_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_2731_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N) ) ) ).

% nat_less_add_iff1
tff(fact_2732_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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_o(aa_nat_fun_nat_o(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_2733_mult__le__cancel__iff1,axiom,
    ! [Z: code_integer,Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),Z))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y3) ) ) ).

% mult_le_cancel_iff1
tff(fact_2734_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y3) ) ) ).

% mult_le_cancel_iff1
tff(fact_2735_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y3) ) ) ).

% mult_le_cancel_iff1
tff(fact_2736_mult__le__cancel__iff2,axiom,
    ! [Z: code_integer,Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),Xa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),Y3))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y3) ) ) ).

% mult_le_cancel_iff2
tff(fact_2737_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y3))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y3) ) ) ).

% mult_le_cancel_iff2
tff(fact_2738_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y3))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y3) ) ) ).

% mult_le_cancel_iff2
tff(fact_2739_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2740_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2741_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N) ) ) ).

% nat_le_add_iff1
tff(fact_2742_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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_o(aa_nat_fun_nat_o(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_2743_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_2744_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_2745_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% nat_mult_less_cancel1
tff(fact_2746_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_2747_mult__less__iff1,axiom,
    ! [Z: code_integer,Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),Z))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y3) ) ) ).

% mult_less_iff1
tff(fact_2748_mult__less__iff1,axiom,
    ! [Z: rat,Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y3) ) ) ).

% mult_less_iff1
tff(fact_2749_mult__less__iff1,axiom,
    ! [Z: int,Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y3) ) ) ).

% mult_less_iff1
tff(fact_2750_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% nat_mult_le_cancel1
tff(fact_2751_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2752_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2753_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K3) ) )
        | ? [L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_2754_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22) ) ) )
         => ~ ! [R2: int,Q3: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2) )
               => ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22)),R2) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_2755_pochhammer__code,axiom,
    ! [A: code_natural,N: nat] :
      aa_nat_Code_natural(comm_s7406298101470620109atural(A),N) = $ite(N = zero_zero_nat,one_one_Code_natural,set_fo9131927060563008058atural(aTP_Lamm_cq(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),one_one_Code_natural)) ).

% pochhammer_code
tff(fact_2756_pochhammer__code,axiom,
    ! [A: code_integer,N: nat] :
      aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N) = $ite(N = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_cr(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_Code_integer)) ).

% pochhammer_code
tff(fact_2757_pochhammer__code,axiom,
    ! [A: rat,N: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = $ite(N = zero_zero_nat,one_one_rat,set_fo1949268297981939178at_rat(aTP_Lamm_cs(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),one_one_rat)) ).

% pochhammer_code
tff(fact_2758_pochhammer__code,axiom,
    ! [A: int,N: nat] :
      aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N) = $ite(N = zero_zero_nat,one_one_int,aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_ct(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_2759_pochhammer__code,axiom,
    ! [A: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = $ite(N = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_cu(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_2760_less__eq__set__def,axiom,
    ! [A3: set_se7855581050983116737at_nat,B3: set_se7855581050983116737at_nat] :
      ( ord_le2077887516847798113at_nat(A3,B3)
    <=> ord_le290999283863451930_nat_o(aTP_Lamm_a(A3),aTP_Lamm_a(B3)) ) ).

% less_eq_set_def
tff(fact_2761_less__eq__set__def,axiom,
    ! [A3: set_se6260736226359567993nt_int,B3: set_se6260736226359567993nt_int] :
      ( ord_le483042692224249369nt_int(A3,B3)
    <=> ord_le2758740137681283170_int_o(aTP_Lamm_aa(A3),aTP_Lamm_aa(B3)) ) ).

% less_eq_set_def
tff(fact_2762_less__eq__set__def,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A3,B3)
    <=> ord_le6468474429079055883_nat_o(aTP_Lamm_ab(A3),aTP_Lamm_ab(B3)) ) ).

% less_eq_set_def
tff(fact_2763_less__eq__set__def,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A3,B3)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_ad(A3),aTP_Lamm_ad(B3)) ) ).

% less_eq_set_def
tff(fact_2764_less__eq__set__def,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
    <=> ord_le7240017405068453697_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ac,A3),aa_set_nat_fun_nat_o(aTP_Lamm_ac,B3)) ) ).

% less_eq_set_def
tff(fact_2765_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q2: int] :
      ( ( aa_int_int(sgn_sgn_int,R) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R)),aa_int_int(abs_abs_int,L))
       => ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L)),R) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_2766_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q2)),R) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int) ),
              Q2 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_2767_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
     => ( ( B != zero_zero_int )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,
                $ite(R = zero_zero_int,aa_int_int(uminus_uminus_int,Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q2)),one_one_int))),
              $ite(R = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R)))) ) ) ).

% zminus1_lemma
tff(fact_2768_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),one_one_Code_integer)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K)) ).

% pochhammer_minus'
tff(fact_2769_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(B,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K)) ).

% pochhammer_minus'
tff(fact_2770_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,B)),K)) ).

% pochhammer_minus'
tff(fact_2771_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_2772_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_2773_pochhammer__0,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),zero_zero_nat) = one_one_Code_integer ).

% pochhammer_0
tff(fact_2774_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_2775_pochhammer__0,axiom,
    ! [A: code_natural] : aa_nat_Code_natural(comm_s7406298101470620109atural(A),zero_zero_nat) = one_one_Code_natural ).

% pochhammer_0
tff(fact_2776_unique__quotient,axiom,
    ! [A: int,B: int,Q2: int,R: int,Q5: int,R3: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R3))
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
tff(fact_2777_unique__remainder,axiom,
    ! [A: int,B: int,Q2: int,R: int,Q5: int,R3: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R3))
       => ( R = R3 ) ) ) ).

% unique_remainder
tff(fact_2778_pochhammer__pos,axiom,
    ! [Xa: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Xa)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(comm_s8582702949713902594nteger(Xa),N)) ) ).

% pochhammer_pos
tff(fact_2779_pochhammer__pos,axiom,
    ! [Xa: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),Xa)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_nat_Code_natural(comm_s7406298101470620109atural(Xa),N)) ) ).

% pochhammer_pos
tff(fact_2780_pochhammer__pos,axiom,
    ! [Xa: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xa),N)) ) ).

% pochhammer_pos
tff(fact_2781_pochhammer__pos,axiom,
    ! [Xa: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xa),N)) ) ).

% pochhammer_pos
tff(fact_2782_pochhammer__pos,axiom,
    ! [Xa: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Xa),N)) ) ).

% pochhammer_pos
tff(fact_2783_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_2784_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_2785_eucl__rel__int__by0,axiom,
    ! [K: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K)) ).

% eucl_rel_int_by0
tff(fact_2786_pochhammer__nonneg,axiom,
    ! [Xa: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Xa)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(comm_s8582702949713902594nteger(Xa),N)) ) ).

% pochhammer_nonneg
tff(fact_2787_pochhammer__nonneg,axiom,
    ! [Xa: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),Xa)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(comm_s7406298101470620109atural(Xa),N)) ) ).

% pochhammer_nonneg
tff(fact_2788_pochhammer__nonneg,axiom,
    ! [Xa: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xa),N)) ) ).

% pochhammer_nonneg
tff(fact_2789_pochhammer__nonneg,axiom,
    ! [Xa: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xa),N)) ) ).

% pochhammer_nonneg
tff(fact_2790_pochhammer__nonneg,axiom,
    ! [Xa: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Xa),N)) ) ).

% pochhammer_nonneg
tff(fact_2791_pochhammer__0__left,axiom,
    ! [N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = $ite(N = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% pochhammer_0_left
tff(fact_2792_pochhammer__0__left,axiom,
    ! [N: nat] :
      aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,zero_zero_int),N) = $ite(N = zero_zero_nat,one_one_int,zero_zero_int) ).

% pochhammer_0_left
tff(fact_2793_pochhammer__0__left,axiom,
    ! [N: nat] :
      aa_nat_Code_integer(comm_s8582702949713902594nteger(zero_z3403309356797280102nteger),N) = $ite(N = zero_zero_nat,one_one_Code_integer,zero_z3403309356797280102nteger) ).

% pochhammer_0_left
tff(fact_2794_pochhammer__0__left,axiom,
    ! [N: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = $ite(N = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% pochhammer_0_left
tff(fact_2795_pochhammer__0__left,axiom,
    ! [N: nat] :
      aa_nat_Code_natural(comm_s7406298101470620109atural(zero_z2226904508553997617atural),N) = $ite(N = zero_zero_nat,one_one_Code_natural,zero_z2226904508553997617atural) ).

% pochhammer_0_left
tff(fact_2796_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),N)
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_2797_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2798_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2799_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2800_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_2801_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_2802_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K)
     => ( aa_nat_int(aa_int_fun_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_2803_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_2804_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_2805_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_int(aa_int_fun_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_2806_pochhammer__product_H,axiom,
    ! [Z: code_integer,N: nat,M: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,N))),M)) ).

% pochhammer_product'
tff(fact_2807_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,N))),M)) ).

% pochhammer_product'
tff(fact_2808_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),N)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),M)) ).

% pochhammer_product'
tff(fact_2809_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),M)) ).

% pochhammer_product'
tff(fact_2810_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_2811_Collect__subset,axiom,
    ! [A3: set_se7855581050983116737at_nat,Pa: fun_se4782206882984618170_nat_o] : ord_le2077887516847798113at_nat(collec5514110066124741708at_nat(aa_fun3315923329236548914_nat_o(aTP_Lamm_cc(A3),Pa)),A3) ).

% Collect_subset
tff(fact_2812_Collect__subset,axiom,
    ! [A3: set_se6260736226359567993nt_int,Pa: fun_se7249947736802449410_int_o] : ord_le483042692224249369nt_int(collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_cd(A3),Pa)),A3) ).

% Collect_subset
tff(fact_2813_Collect__subset,axiom,
    ! [A3: set_set_nat,Pa: fun_set_nat_o] : ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ce(A3),Pa)),A3) ).

% Collect_subset
tff(fact_2814_Collect__subset,axiom,
    ! [A3: set_int,Pa: fun_int_o] : ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(A3),Pa)),A3) ).

% Collect_subset
tff(fact_2815_Collect__subset,axiom,
    ! [A3: set_nat,Pa: fun_nat_o] : aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(A3),Pa))),A3) ).

% Collect_subset
tff(fact_2816_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),M)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_2817_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_2818_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),M)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_2819_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),M)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_2820_pochhammer__absorb__comp,axiom,
    ! [R: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,R),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R)),one_one_Code_integer)),K)) ).

% pochhammer_absorb_comp
tff(fact_2821_pochhammer__absorb__comp,axiom,
    ! [R: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(R,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,R)),one_one_rat)),K)) ).

% pochhammer_absorb_comp
tff(fact_2822_pochhammer__absorb__comp,axiom,
    ! [R: 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,R),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,R)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_2823_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),one_one_Code_integer)),K)) ).

% pochhammer_minus
tff(fact_2824_pochhammer__minus,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(B,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% pochhammer_minus
tff(fact_2825_pochhammer__minus,axiom,
    ! [B: int,K: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,B)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).

% pochhammer_minus
tff(fact_2826_nat__ivt__aux,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,N))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2827_power__minus_H,axiom,
    ! [Xa: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,one_one_int),Xa)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,Xa)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),N)) ) ) ).

% power_minus'
tff(fact_2828_power__minus_H,axiom,
    ! [Xa: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(nO_MAT8252062027627875367nteger,one_one_Code_integer),Xa)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xa)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Xa),N)) ) ) ).

% power_minus'
tff(fact_2829_power__minus_H,axiom,
    ! [Xa: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(nO_MATCH_rat_rat,one_one_rat),Xa)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xa)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(Xa),N)) ) ) ).

% power_minus'
tff(fact_2830_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% one_less_nat_eq
tff(fact_2831_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,M)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,N))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N)
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2832_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).

% pochhammer_same
tff(fact_2833_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% pochhammer_same
tff(fact_2834_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(semiri1406184849735516958ct_int,N)) ).

% pochhammer_same
tff(fact_2835_fact__reduce,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_Code_integer(semiri3624122377584611663nteger,N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_2836_fact__reduce,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_2837_fact__reduce,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2838_fact__reduce,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2839_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_Code_natural(semiri2447717529341329178atural,M) = $ite(M = zero_zero_nat,one_one_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri2447717529341329178atural,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_2840_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_Code_integer(semiri3624122377584611663nteger,M) = $ite(M = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_2841_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_rat(semiri773545260158071498ct_rat,M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_2842_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_int(semiri1406184849735516958ct_int,M) = $ite(M = zero_zero_nat,one_one_int,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_2843_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_nat(semiri1408675320244567234ct_nat,M) = $ite(M = zero_zero_nat,one_one_nat,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_2844_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_2845_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_2846_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_2847_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_2848_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% Suc_less_eq
tff(fact_2849_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) ) ).

% Suc_mono
tff(fact_2850_lessI,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(suc,N)) ).

% lessI
tff(fact_2851_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_nat_nat(suc,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ).

% Suc_le_mono
tff(fact_2852_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_2853_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_2854_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_2855_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_nat_nat(suc,N)) = zero_z3403309356797280102nteger ).

% power_0_Suc
tff(fact_2856_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,N)) = zero_zero_rat ).

% power_0_Suc
tff(fact_2857_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_Code_natural(power_7079662738309270450atural(zero_z2226904508553997617atural),aa_nat_nat(suc,N)) = zero_z2226904508553997617atural ).

% power_0_Suc
tff(fact_2858_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_2859_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_nat_nat(suc,N)) = zero_zero_int ).

% power_0_Suc
tff(fact_2860_less__Suc0,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(suc,zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_2861_zero__less__Suc,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N)) ).

% zero_less_Suc
tff(fact_2862_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_2863_power__Suc0__right,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2864_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_2865_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_2866_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_2867_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_2868_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_2869_fact__0,axiom,
    aa_nat_Code_integer(semiri3624122377584611663nteger,zero_zero_nat) = one_one_Code_integer ).

% fact_0
tff(fact_2870_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_2871_fact__0,axiom,
    aa_nat_Code_natural(semiri2447717529341329178atural,zero_zero_nat) = one_one_Code_natural ).

% fact_0
tff(fact_2872_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_2873_ceiling__zero,axiom,
    archim2889992004027027881ng_rat(zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_2874_nat__power__eq__Suc__0__iff,axiom,
    ! [Xa: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( Xa = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2875_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_2876_ceiling__one,axiom,
    archim2889992004027027881ng_rat(one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_2877_fact__1,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).

% fact_1
tff(fact_2878_fact__1,axiom,
    aa_nat_Code_integer(semiri3624122377584611663nteger,one_one_nat) = one_one_Code_integer ).

% fact_1
tff(fact_2879_fact__1,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).

% fact_1
tff(fact_2880_fact__1,axiom,
    aa_nat_Code_natural(semiri2447717529341329178atural,one_one_nat) = one_one_Code_natural ).

% fact_1
tff(fact_2881_fact__1,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).

% fact_1
tff(fact_2882_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_nat_Code_integer(semiri4939895301339042750nteger,M)) ).

% of_nat_Suc
tff(fact_2883_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(suc,M)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_nat_Code_natural(semiri3763490453095760265atural,M)) ).

% of_nat_Suc
tff(fact_2884_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_2885_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_2886_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_2887_Suc__pred,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2888_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N) ) ) ).

% one_le_mult_iff
tff(fact_2889_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),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_2890_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,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_2891_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_2892_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_2893_fact__Suc__0,axiom,
    aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,zero_zero_nat)) = one_one_Code_integer ).

% fact_Suc_0
tff(fact_2894_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_2895_fact__Suc__0,axiom,
    aa_nat_Code_natural(semiri2447717529341329178atural,aa_nat_nat(suc,zero_zero_nat)) = one_one_Code_natural ).

% fact_Suc_0
tff(fact_2896_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_2897_negative__zless,axiom,
    ! [N: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zless
tff(fact_2898_ceiling__add__of__int,axiom,
    ! [Xa: rat,Z: int] : archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xa)),Z) ).

% ceiling_add_of_int
tff(fact_2899_Suc__diff__1,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2900_ceiling__le__zero,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xa)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_2901_zero__less__ceiling,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),archim2889992004027027881ng_rat(Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa) ) ).

% zero_less_ceiling
tff(fact_2902_ceiling__less__one,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xa)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_2903_one__le__ceiling,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),archim2889992004027027881ng_rat(Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa) ) ).

% one_le_ceiling
tff(fact_2904_ceiling__le__one,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xa)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),one_one_rat) ) ).

% ceiling_le_one
tff(fact_2905_one__less__ceiling,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),archim2889992004027027881ng_rat(Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xa) ) ).

% one_less_ceiling
tff(fact_2906_ceiling__add__one,axiom,
    ! [Xa: rat] : archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xa)),one_one_int) ).

% ceiling_add_one
tff(fact_2907_ceiling__diff__one,axiom,
    ! [Xa: rat] : archim2889992004027027881ng_rat(minus_minus_rat(Xa,one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(Xa)),one_one_int) ).

% ceiling_diff_one
tff(fact_2908_ceiling__less__zero,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xa)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_2909_zero__le__ceiling,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),archim2889992004027027881ng_rat(Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xa) ) ).

% zero_le_ceiling
tff(fact_2910_Suc__inject,axiom,
    ! [Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(suc,Xa) = aa_nat_nat(suc,Y3) )
     => ( Xa = Y3 ) ) ).

% Suc_inject
tff(fact_2911_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_2912_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).

% fact_ge_Suc_0_nat
tff(fact_2913_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) != zero_zero_int ).

% fact_nonzero
tff(fact_2914_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,N) != zero_z3403309356797280102nteger ).

% fact_nonzero
tff(fact_2915_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) != zero_zero_rat ).

% fact_nonzero
tff(fact_2916_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_Code_natural(semiri2447717529341329178atural,N) != zero_z2226904508553997617atural ).

% fact_nonzero
tff(fact_2917_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) != zero_zero_nat ).

% fact_nonzero
tff(fact_2918_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] : N = aa_nat_nat(suc,M4) ) ).

% not0_implies_Suc
tff(fact_2919_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_2920_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_2921_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_2922_zero__induct,axiom,
    ! [Pa: fun_nat_o,K: nat] :
      ( aa_nat_o(Pa,K)
     => ( ! [N2: nat] :
            ( aa_nat_o(Pa,aa_nat_nat(suc,N2))
           => aa_nat_o(Pa,N2) )
       => aa_nat_o(Pa,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_2923_diff__induct,axiom,
    ! [Pa: fun_nat_fun_nat_o,M: nat,N: nat] :
      ( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,X4),zero_zero_nat)
     => ( ! [Y4: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,zero_zero_nat),aa_nat_nat(suc,Y4))
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(Pa,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(Pa,aa_nat_nat(suc,X4)),aa_nat_nat(suc,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(Pa,M),N) ) ) ) ).

% diff_induct
tff(fact_2924_nat__induct,axiom,
    ! [Pa: fun_nat_o,N: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(Pa,N2)
           => aa_nat_o(Pa,aa_nat_nat(suc,N2)) )
       => aa_nat_o(Pa,N) ) ) ).

% nat_induct
tff(fact_2925_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y3 != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_2926_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_2927_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_2928_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_2929_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_2930_exists__least__lemma,axiom,
    ! [Pa: fun_nat_o] :
      ( ~ aa_nat_o(Pa,zero_zero_nat)
     => ( ? [X_12: nat] : aa_nat_o(Pa,X_12)
       => ? [N2: nat] :
            ( ~ aa_nat_o(Pa,N2)
            & aa_nat_o(Pa,aa_nat_nat(suc,N2)) ) ) ) ).

% exists_least_lemma
tff(fact_2931_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(suc,M))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_2932_strict__inc__induct,axiom,
    ! [I: nat,J: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => aa_nat_o(Pa,I2) )
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
             => ( aa_nat_o(Pa,aa_nat_nat(suc,I2))
               => aa_nat_o(Pa,I2) ) )
         => aa_nat_o(Pa,I) ) ) ) ).

% strict_inc_induct
tff(fact_2933_less__Suc__induct,axiom,
    ! [I: nat,J: nat,Pa: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,I2),aa_nat_nat(suc,I2))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K2)
               => ( aa_nat_o(aa_nat_fun_nat_o(Pa,I2),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(Pa,J2),K2)
                   => aa_nat_o(aa_nat_fun_nat_o(Pa,I2),K2) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(Pa,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_2934_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_2935_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% Suc_less_SucD
tff(fact_2936_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(suc,M))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_2937_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M)
    <=> ? [M5: nat] :
          ( ( M = aa_nat_nat(suc,M5) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M5) ) ) ).

% Suc_less_eq2
tff(fact_2938_All__less__Suc,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,N))
         => aa_nat_o(Pa,I4) )
    <=> ( aa_nat_o(Pa,N)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N)
           => aa_nat_o(Pa,I4) ) ) ) ).

% All_less_Suc
tff(fact_2939_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(suc,M)) ) ).

% not_less_eq
tff(fact_2940_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_2941_Ex__less__Suc,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,N))
          & aa_nat_o(Pa,I4) )
    <=> ( aa_nat_o(Pa,N)
        | ? [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N)
            & aa_nat_o(Pa,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_2942_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N)) ) ).

% less_SucI
tff(fact_2943_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_2944_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( ( aa_nat_nat(suc,M) != N )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),N) ) ) ).

% Suc_lessI
tff(fact_2945_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_2946_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% Suc_lessD
tff(fact_2947_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_2948_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R4: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(R4,X4),X4)
       => ( ! [X4: nat,Y4: nat,Z4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R4,X4),Y4)
             => ( aa_nat_o(aa_nat_fun_nat_o(R4,Y4),Z4)
               => aa_nat_o(aa_nat_fun_nat_o(R4,X4),Z4) ) )
         => ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(R4,N2),aa_nat_nat(suc,N2))
           => aa_nat_o(aa_nat_fun_nat_o(R4,M),N) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_2949_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(Pa,M)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
             => ( aa_nat_o(Pa,N2)
               => aa_nat_o(Pa,aa_nat_nat(suc,N2)) ) )
         => aa_nat_o(Pa,N) ) ) ) ).

% nat_induct_at_least
tff(fact_2950_full__nat__induct,axiom,
    ! [Pa: fun_nat_o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M3)),N2)
             => aa_nat_o(Pa,M3) )
         => aa_nat_o(Pa,N2) )
     => aa_nat_o(Pa,N) ) ).

% full_nat_induct
tff(fact_2951_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),M) ) ).

% not_less_eq_eq
tff(fact_2952_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),N) ).

% Suc_n_not_le_n
tff(fact_2953_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
        | ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_2954_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),M6)
     => ? [M4: nat] : M6 = aa_nat_nat(suc,M4) ) ).

% Suc_le_D
tff(fact_2955_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N)) ) ).

% le_SucI
tff(fact_2956_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_SucE
tff(fact_2957_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% Suc_leD
tff(fact_2958_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_2959_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_2960_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_2961_zero__induct__lemma,axiom,
    ! [Pa: fun_nat_o,K: nat,I: nat] :
      ( aa_nat_o(Pa,K)
     => ( ! [N2: nat] :
            ( aa_nat_o(Pa,aa_nat_nat(suc,N2))
           => aa_nat_o(Pa,N2) )
       => aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_2962_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_2963_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ) ) ).

% fact_less_mono_nat
tff(fact_2964_fact__ge__zero,axiom,
    ! [N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).

% fact_ge_zero
tff(fact_2965_fact__ge__zero,axiom,
    ! [N: nat] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(semiri2447717529341329178atural,N)) ).

% fact_ge_zero
tff(fact_2966_fact__ge__zero,axiom,
    ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% fact_ge_zero
tff(fact_2967_fact__ge__zero,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N)) ).

% fact_ge_zero
tff(fact_2968_fact__ge__zero,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).

% fact_ge_zero
tff(fact_2969_fact__not__neg,axiom,
    ! [N: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,N)),zero_z3403309356797280102nteger) ).

% fact_not_neg
tff(fact_2970_fact__not__neg,axiom,
    ! [N: nat] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(semiri2447717529341329178atural,N)),zero_z2226904508553997617atural) ).

% fact_not_neg
tff(fact_2971_fact__not__neg,axiom,
    ! [N: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),zero_zero_rat) ).

% fact_not_neg
tff(fact_2972_fact__not__neg,axiom,
    ! [N: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,N)),zero_zero_int) ).

% fact_not_neg
tff(fact_2973_fact__not__neg,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),zero_zero_nat) ).

% fact_not_neg
tff(fact_2974_fact__gt__zero,axiom,
    ! [N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).

% fact_gt_zero
tff(fact_2975_fact__gt__zero,axiom,
    ! [N: nat] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_nat_Code_natural(semiri2447717529341329178atural,N)) ).

% fact_gt_zero
tff(fact_2976_fact__gt__zero,axiom,
    ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% fact_gt_zero
tff(fact_2977_fact__gt__zero,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N)) ).

% fact_gt_zero
tff(fact_2978_fact__gt__zero,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).

% fact_gt_zero
tff(fact_2979_fact__ge__1,axiom,
    ! [N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).

% fact_ge_1
tff(fact_2980_fact__ge__1,axiom,
    ! [N: nat] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),aa_nat_Code_natural(semiri2447717529341329178atural,N)) ).

% fact_ge_1
tff(fact_2981_fact__ge__1,axiom,
    ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% fact_ge_1
tff(fact_2982_fact__ge__1,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,N)) ).

% fact_ge_1
tff(fact_2983_fact__ge__1,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).

% fact_ge_1
tff(fact_2984_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N2)),aa_nat_set_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_2985_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_2986_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_num_o(ord_less_eq_num(aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
       => aa_num_o(ord_less_eq_num(aa_nat_num(F,N)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_2987_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_2988_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_2989_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N2))),aa_nat_set_nat(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N3)),aa_nat_set_nat(F,N)) ) ) ).

% lift_Suc_antimono_le
tff(fact_2990_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,N)) ) ) ).

% lift_Suc_antimono_le
tff(fact_2991_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_num_o(ord_less_eq_num(aa_nat_num(F,aa_nat_nat(suc,N2))),aa_nat_num(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
       => aa_num_o(ord_less_eq_num(aa_nat_num(F,N3)),aa_nat_num(F,N)) ) ) ).

% lift_Suc_antimono_le
tff(fact_2992_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,N)) ) ) ).

% lift_Suc_antimono_le
tff(fact_2993_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(F,N)) ) ) ).

% lift_Suc_antimono_le
tff(fact_2994_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_literal,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,N2)),aa_nat_literal(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
       => aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,N)),aa_nat_literal(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_2995_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_2996_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_num_o(ord_less_num(aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
       => aa_num_o(ord_less_num(aa_nat_num(F,N)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_2997_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_2998_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_2999_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_literal,N: nat,M: nat] :
      ( ! [N2: nat] : aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,N2)),aa_nat_literal(F,aa_nat_nat(suc,N2)))
     => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_nat_literal(F,N)),aa_nat_literal(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3000_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,N: nat,M: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3001_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,N: nat,M: nat] :
      ( ! [N2: nat] : aa_num_o(ord_less_num(aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_num_o(ord_less_num(aa_nat_num(F,N)),aa_nat_num(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3002_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,N: nat,M: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3003_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,N: nat,M: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3004_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,N)) != zero_z3403309356797280102nteger ).

% of_nat_neq_0
tff(fact_3005_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_3006_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(suc,N)) != zero_z2226904508553997617atural ).

% of_nat_neq_0
tff(fact_3007_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_3008_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_3009_Ex__less__Suc2,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,N))
          & aa_nat_o(Pa,I4) )
    <=> ( aa_nat_o(Pa,zero_zero_nat)
        | ? [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N)
            & aa_nat_o(Pa,aa_nat_nat(suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_3010_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
    <=> ? [M7: nat] : N = aa_nat_nat(suc,M7) ) ).

% gr0_conv_Suc
tff(fact_3011_All__less__Suc2,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,N))
         => aa_nat_o(Pa,I4) )
    <=> ( aa_nat_o(Pa,zero_zero_nat)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N)
           => aa_nat_o(Pa,aa_nat_nat(suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_3012_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ? [M4: nat] : N = aa_nat_nat(suc,M4) ) ).

% gr0_implies_Suc
tff(fact_3013_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),N) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_3014_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_3015_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_3016_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N)) ) ).

% le_imp_less_Suc
tff(fact_3017_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),M) ) ).

% less_eq_Suc_le
tff(fact_3018_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% less_Suc_eq_le
tff(fact_3019_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(suc,M))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_3020_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% Suc_le_lessD
tff(fact_3021_inc__induct,axiom,
    ! [I: nat,J: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(Pa,J)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),J)
               => ( aa_nat_o(Pa,aa_nat_nat(suc,N2))
                 => aa_nat_o(Pa,N2) ) ) )
         => aa_nat_o(Pa,I) ) ) ) ).

% inc_induct
tff(fact_3022_dec__induct,axiom,
    ! [I: nat,J: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(Pa,I)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),J)
               => ( aa_nat_o(Pa,N2)
                 => aa_nat_o(Pa,aa_nat_nat(suc,N2)) ) ) )
         => aa_nat_o(Pa,J) ) ) ) ).

% dec_induct
tff(fact_3023_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% Suc_le_eq
tff(fact_3024_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N) ) ).

% Suc_leI
tff(fact_3025_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ~ ! [Q3: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ) ).

% less_natE
tff(fact_3026_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(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_3027_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(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_3028_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_iff_Suc_add
tff(fact_3029_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_imp_Suc_add
tff(fact_3030_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_o(aa_nat_fun_nat_o(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_3031_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3032_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% Suc_mult_less_cancel1
tff(fact_3033_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3034_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_3035_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% Suc_mult_le_cancel1
tff(fact_3036_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_3037_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_3038_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_3039_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_3040_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_3041_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% int_cases
tff(fact_3042_int__of__nat__induct,axiom,
    ! [Pa: fun_int_o,Z: int] :
      ( ! [N2: nat] : aa_int_o(Pa,aa_nat_int(semiri1314217659103216013at_int,N2))
     => ( ! [N2: nat] : aa_int_o(Pa,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))))
       => aa_int_o(Pa,Z) ) ) ).

% int_of_nat_induct
tff(fact_3043_pochhammer__fact,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,one_one_int),N) ).

% pochhammer_fact
tff(fact_3044_pochhammer__fact,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,N) = aa_nat_Code_integer(comm_s8582702949713902594nteger(one_one_Code_integer),N) ).

% pochhammer_fact
tff(fact_3045_pochhammer__fact,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),N) ).

% pochhammer_fact
tff(fact_3046_pochhammer__fact,axiom,
    ! [N: nat] : aa_nat_Code_natural(semiri2447717529341329178atural,N) = aa_nat_Code_natural(comm_s7406298101470620109atural(one_one_Code_natural),N) ).

% pochhammer_fact
tff(fact_3047_pochhammer__fact,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),N) ).

% pochhammer_fact
tff(fact_3048_power__le__imp__le__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,N)))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3049_power__le__imp__le__base,axiom,
    ! [A: code_natural,N: nat,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(suc,N))),aa_nat_Code_natural(power_7079662738309270450atural(B),aa_nat_nat(suc,N)))
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3050_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3051_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3052_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N))),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3053_power__inject__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N)) = aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,N)) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3054_power__inject__base,axiom,
    ! [A: code_natural,N: nat,B: code_natural] :
      ( ( aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(suc,N)) = aa_nat_Code_natural(power_7079662738309270450atural(B),aa_nat_nat(suc,N)) )
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3055_power__inject__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3056_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3057_power__inject__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),aa_nat_nat(suc,N)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3058_power__gt1,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_3059_power__gt1,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_3060_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_3061_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_3062_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_3063_ceiling__add__le,axiom,
    ! [Xa: rat,Y3: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y3))),aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xa)),archim2889992004027027881ng_rat(Y3))) ).

% ceiling_add_le
tff(fact_3064_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N)) ) ) ).

% fact_less_mono
tff(fact_3065_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N)) ) ) ).

% fact_less_mono
tff(fact_3066_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ) ) ).

% fact_less_mono
tff(fact_3067_ex__least__nat__less,axiom,
    ! [Pa: fun_nat_o,N: nat] :
      ( aa_nat_o(Pa,N)
     => ( ~ aa_nat_o(Pa,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),N)
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),K2)
               => ~ aa_nat_o(Pa,I3) )
            & aa_nat_o(Pa,aa_nat_nat(suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_3068_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(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_3069_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)) ) ) ).

% n_less_n_mult_m
tff(fact_3070_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ) ) ).

% n_less_m_mult_n
tff(fact_3071_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ) ) ).

% one_less_mult
tff(fact_3072_nat__induct__non__zero,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(Pa,one_one_nat)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => ( aa_nat_o(Pa,N2)
               => aa_nat_o(Pa,aa_nat_nat(suc,N2)) ) )
         => aa_nat_o(Pa,N) ) ) ) ).

% nat_induct_non_zero
tff(fact_3073_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K)) ) ).

% power_gt_expt
tff(fact_3074_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)) ) ).

% nat_one_le_power
tff(fact_3075_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
    <=> ? [N6: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N6))) ) ).

% zless_iff_Suc_zadd
tff(fact_3076_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_3077_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_3078_power__Suc__le__self,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_3079_power__Suc__le__self,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),one_one_Code_natural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(suc,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_3080_power__Suc__le__self,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_3081_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_3082_power__Suc__le__self,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_3083_power__Suc__less__one,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),one_one_Code_integer) ) ) ).

% power_Suc_less_one
tff(fact_3084_power__Suc__less__one,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),one_one_Code_natural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(suc,N))),one_one_Code_natural) ) ) ).

% power_Suc_less_one
tff(fact_3085_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_3086_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_3087_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_3088_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3089_Suc__pred_H,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3090_pochhammer__rec,axiom,
    ! [A: code_natural,N: nat] : aa_nat_Code_natural(comm_s7406298101470620109atural(A),aa_nat_nat(suc,N)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_nat_Code_natural(comm_s7406298101470620109atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),one_one_Code_natural)),N)) ).

% pochhammer_rec
tff(fact_3091_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_3092_pochhammer__rec,axiom,
    ! [A: int,N: nat] : aa_nat_int(aa_int_fun_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(aa_int_fun_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_3093_pochhammer__rec,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),N)) ).

% pochhammer_rec
tff(fact_3094_pochhammer__rec,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),N)) ).

% pochhammer_rec
tff(fact_3095_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = $ite(M = zero_zero_nat,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_3096_pochhammer__rec_H,axiom,
    ! [Z: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,N))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N)) ).

% pochhammer_rec'
tff(fact_3097_pochhammer__rec_H,axiom,
    ! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,N))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)) ).

% pochhammer_rec'
tff(fact_3098_pochhammer__rec_H,axiom,
    ! [Z: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),N)) ).

% pochhammer_rec'
tff(fact_3099_pochhammer__rec_H,axiom,
    ! [Z: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)) ).

% pochhammer_rec'
tff(fact_3100_pochhammer__Suc,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) ).

% pochhammer_Suc
tff(fact_3101_pochhammer__Suc,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(A),N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% pochhammer_Suc
tff(fact_3102_pochhammer__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(aa_int_fun_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(aa_int_fun_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_3103_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_3104_not__zle__0__negative,axiom,
    ! [N: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))) ).

% not_zle_0_negative
tff(fact_3105_negative__zless__0,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),zero_zero_int) ).

% negative_zless_0
tff(fact_3106_negD,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),zero_zero_int)
     => ? [N2: nat] : Xa = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% negD
tff(fact_3107_Suc__as__int,axiom,
    ! [X: nat] : aa_nat_nat(suc,X) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),one_one_int)) ).

% Suc_as_int
tff(fact_3108_ceiling__correct,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(Xa)),one_one_rat)),Xa)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(Xa))) ) ).

% ceiling_correct
tff(fact_3109_ceiling__unique,axiom,
    ! [Z: int,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z))
       => ( archim2889992004027027881ng_rat(Xa) = Z ) ) ) ).

% ceiling_unique
tff(fact_3110_ceiling__eq__iff,axiom,
    ! [Xa: rat,A: int] :
      ( ( archim2889992004027027881ng_rat(Xa) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A),one_one_rat)),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).

% ceiling_eq_iff
tff(fact_3111_ceiling__split,axiom,
    ! [Pa: fun_int_o,T: rat] :
      ( aa_int_o(Pa,archim2889992004027027881ng_rat(T))
    <=> ! [I4: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I4),one_one_rat)),T)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),aa_int_rat(ring_1_of_int_rat,I4)) )
         => aa_int_o(Pa,I4) ) ) ).

% ceiling_split
tff(fact_3112_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B))) ) ) ).

% mult_ceiling_le
tff(fact_3113_ceiling__less__iff,axiom,
    ! [Xa: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xa)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_3114_le__ceiling__iff,axiom,
    ! [Z: int,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),archim2889992004027027881ng_rat(Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),Xa) ) ).

% le_ceiling_iff
tff(fact_3115_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_3116_option_Osize__gen_I2_J,axiom,
    ! [Xa: fun_Pr367382672041210069at_nat,X2: produc8664842809031399944it_nat] : size_o29782932136985253it_nat(Xa,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro5228051289423594006at_nat(Xa,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_3117_option_Osize__gen_I2_J,axiom,
    ! [Xa: fun_num_nat,X2: num] : aa_option_num_nat(size_option_num(Xa),aa_num_option_num(some_num,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(Xa,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_3118_Heap_Osize__gen,axiom,
    ! [Xa: fun_Product_unit_nat,Xb: fun_he5234484606919320760it_nat] : heap_T4142866422068808505t_unit(Xa,heap_T6183433275982383450t_unit(Xb)) = aa_nat_nat(suc,zero_zero_nat) ).

% Heap.size_gen
tff(fact_3119_option_Osize__gen_I1_J,axiom,
    ! [Xa: fun_num_nat] : aa_option_num_nat(size_option_num(Xa),none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_3120_option_Osize__gen_I1_J,axiom,
    ! [Xa: fun_Pr4403862266400233414at_nat] : size_o1731029420616209814it_nat(Xa,none_P2651198173097904984it_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_3121_Heap_Osize_I2_J,axiom,
    ! [Xa: fun_he5234484606919320760it_nat] : size_s1564113455978345259t_unit(heap_T6183433275982383450t_unit(Xa)) = aa_nat_nat(suc,zero_zero_nat) ).

% Heap.size(2)
tff(fact_3122_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,K),M)),K) = aa_nat_int(aa_int_fun_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_3123_ceiling__divide__lower,axiom,
    ! [Q2: rat,P: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P,Q2))),one_one_rat)),Q2)),P) ) ).

% ceiling_divide_lower
tff(fact_3124_option_Osize_I4_J,axiom,
    ! [X2: produc8664842809031399944it_nat] : size_s8766407808098229740it_nat(aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_3125_option_Osize_I4_J,axiom,
    ! [X2: num] : aa_option_num_nat(size_size_option_num,aa_num_option_num(some_num,X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_3126_option_Osize_I3_J,axiom,
    aa_option_num_nat(size_size_option_num,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3127_option_Osize_I3_J,axiom,
    size_s2363601347547812957it_nat(none_P2651198173097904984it_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3128_list__decode_Ocases,axiom,
    ! [Xa: nat] :
      ( ( Xa != zero_zero_nat )
     => ~ ! [N2: nat] : Xa != aa_nat_nat(suc,N2) ) ).

% list_decode.cases
tff(fact_3129_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_3130_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( divide_divide_rat(C,A) = divide_divide_rat(C,B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_3131_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_3132_division__ring__divide__zero,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_3133_div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_3134_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_3135_div__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).

% div_0
tff(fact_3136_div__0,axiom,
    ! [A: rat] : divide_divide_rat(zero_zero_rat,A) = zero_zero_rat ).

% div_0
tff(fact_3137_div__0,axiom,
    ! [A: code_natural] : divide5121882707175180666atural(zero_z2226904508553997617atural,A) = zero_z2226904508553997617atural ).

% div_0
tff(fact_3138_div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_3139_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_3140_div__by__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% div_by_0
tff(fact_3141_div__by__0,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_3142_div__by__0,axiom,
    ! [A: code_natural] : divide5121882707175180666atural(A,zero_z2226904508553997617atural) = zero_z2226904508553997617atural ).

% div_by_0
tff(fact_3143_div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% div_by_1
tff(fact_3144_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_3145_div__by__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),one_one_Code_integer) = A ).

% div_by_1
tff(fact_3146_div__by__1,axiom,
    ! [A: rat] : divide_divide_rat(A,one_one_rat) = A ).

% div_by_1
tff(fact_3147_div__by__1,axiom,
    ! [A: code_natural] : divide5121882707175180666atural(A,one_one_Code_natural) = A ).

% div_by_1
tff(fact_3148_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: 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)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ).

% nat_mult_div_cancel_disj
tff(fact_3149_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_3150_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_3151_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_3152_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_3153_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,divide_divide_rat(A,B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_3154_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_3155_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_3156_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_3157_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_3158_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_3159_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_3160_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_3161_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_3162_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_3163_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_3164_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_3165_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = divide_divide_rat(A,B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_3166_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% divide_self
tff(fact_3167_divide__self__if,axiom,
    ! [A: rat] :
      divide_divide_rat(A,A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_3168_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( divide_divide_rat(B,A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_3169_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = divide_divide_rat(B,A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_3170_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat(one_one_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_3171_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = divide_divide_rat(one_one_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_3172_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).

% div_self
tff(fact_3173_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_3174_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),A) = one_one_Code_integer ) ) ).

% div_self
tff(fact_3175_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% div_self
tff(fact_3176_div__self,axiom,
    ! [A: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(A,A) = one_one_Code_natural ) ) ).

% div_self
tff(fact_3177_divide__minus1,axiom,
    ! [Xa: rat] : divide_divide_rat(Xa,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Xa) ).

% divide_minus1
tff(fact_3178_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_3179_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_divide_1_iff
tff(fact_3180_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(one_one_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_3181_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_3182_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_3183_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_3184_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_3185_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_divide_1_iff
tff(fact_3186_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = divide_divide_rat(one_one_rat,B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_3187_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(B,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = divide_divide_rat(one_one_rat,A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_3188_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,abs_abs_rat(B))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_3189_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A,abs_abs_rat(B)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_3190_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_3191_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_3192_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_3193_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_3194_size__neq__size__imp__neq,axiom,
    ! [Xa: literal,Y3: literal] :
      ( ( size_size_literal(Xa) != size_size_literal(Y3) )
     => ( Xa != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_3195_size__neq__size__imp__neq,axiom,
    ! [Xa: char,Y3: char] :
      ( ( size_size_char(Xa) != size_size_char(Y3) )
     => ( Xa != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_3196_size__neq__size__imp__neq,axiom,
    ! [Xa: list_char,Y3: list_char] :
      ( ( aa_list_char_nat(size_size_list_char,Xa) != aa_list_char_nat(size_size_list_char,Y3) )
     => ( Xa != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_3197_size__neq__size__imp__neq,axiom,
    ! [Xa: list_nat,Y3: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xa) != aa_list_nat_nat(size_size_list_nat,Y3) )
     => ( Xa != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_3198_size__neq__size__imp__neq,axiom,
    ! [Xa: num,Y3: num] :
      ( ( aa_num_nat(size_size_num,Xa) != aa_num_nat(size_size_num,Y3) )
     => ( Xa != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_3199_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ).

% add_divide_distrib
tff(fact_3200_nat__div__as__int,axiom,
    ! [X: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_div_as_int
tff(fact_3201_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% divide_le_0_iff
tff(fact_3202_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_right_mono
tff(fact_3203_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_3204_divide__nonneg__nonneg,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xa,Y3)) ) ) ).

% divide_nonneg_nonneg
tff(fact_3205_divide__nonneg__nonpos,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xa,Y3)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_3206_divide__nonpos__nonneg,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xa,Y3)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_3207_divide__nonpos__nonpos,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xa,Y3)) ) ) ).

% divide_nonpos_nonpos
tff(fact_3208_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),divide_divide_rat(A,C)) ) ) ).

% divide_right_mono_neg
tff(fact_3209_divide__neg__neg,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(Xa,Y3)) ) ) ).

% divide_neg_neg
tff(fact_3210_divide__neg__pos,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xa,Y3)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_3211_divide__pos__neg,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xa,Y3)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_3212_divide__pos__pos,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(Xa,Y3)) ) ) ).

% divide_pos_pos
tff(fact_3213_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% divide_less_0_iff
tff(fact_3214_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_3215_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_3216_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_strict_right_mono
tff(fact_3217_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_3218_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = divide_divide_rat(B,C) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3219_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat(B,C) = A )
      <=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3220_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B )
       => ( A = divide_divide_rat(B,C) ) ) ) ).

% eq_divide_imp
tff(fact_3221_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) )
       => ( divide_divide_rat(B,C) = A ) ) ) ).

% divide_eq_imp
tff(fact_3222_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_3223_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( divide_divide_rat(B,C) = A )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_3224_frac__eq__eq,axiom,
    ! [Y3: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( divide_divide_rat(Xa,Y3) = divide_divide_rat(W,Z) )
        <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y3) ) ) ) ) ).

% frac_eq_eq
tff(fact_3225_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat(A,B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_3226_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,divide_divide_rat(A,B)) = divide_divide_rat(A,aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_3227_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_3228_power__one__over,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(one_one_rat,A)),N) = divide_divide_rat(one_one_rat,aa_nat_rat(power_power_rat(A),N)) ).

% power_one_over
tff(fact_3229_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( abs_abs_rat(divide_divide_rat(A,B)) = divide_divide_rat(abs_abs_rat(A),abs_abs_rat(B)) ) ) ).

% nonzero_abs_divide
tff(fact_3230_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_3231_frac__le,axiom,
    ! [Y3: rat,Xa: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y3)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xa,Z)),divide_divide_rat(Y3,W)) ) ) ) ) ).

% frac_le
tff(fact_3232_frac__less,axiom,
    ! [Xa: rat,Y3: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y3)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xa,Z)),divide_divide_rat(Y3,W)) ) ) ) ) ).

% frac_less
tff(fact_3233_frac__less2,axiom,
    ! [Xa: rat,Y3: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y3)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xa,Z)),divide_divide_rat(Y3,W)) ) ) ) ) ).

% frac_less2
tff(fact_3234_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_3235_divide__nonneg__neg,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xa,Y3)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_3236_divide__nonneg__pos,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xa,Y3)) ) ) ).

% divide_nonneg_pos
tff(fact_3237_divide__nonpos__neg,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xa,Y3)) ) ) ).

% divide_nonpos_neg
tff(fact_3238_divide__nonpos__pos,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xa,Y3)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_3239_div__positive,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).

% div_positive
tff(fact_3240_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_positive
tff(fact_3241_div__positive,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_positive
tff(fact_3242_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) = zero_z3403309356797280102nteger ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3243_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3244_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3245_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3246_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3247_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3248_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_3249_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_3250_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_3251_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_3252_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_3253_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_3254_mult__imp__div__pos__less,axiom,
    ! [Y3: rat,Xa: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y3))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xa,Y3)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_3255_mult__imp__less__div__pos,axiom,
    ! [Y3: rat,Z: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y3)),Xa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),divide_divide_rat(Xa,Y3)) ) ) ).

% mult_imp_less_div_pos
tff(fact_3256_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_3257_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3258_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_3259_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B,A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_3260_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(A,Z)),B) = $ite(Z = zero_zero_rat,B,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z)),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3261_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),divide_divide_rat(B,Z)) = $ite(Z = zero_zero_rat,A,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3262_add__frac__eq,axiom,
    ! [Y3: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(Xa,Y3)),divide_divide_rat(W,Z)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3263_add__frac__num,axiom,
    ! [Y3: rat,Xa: rat,Z: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(Xa,Y3)),Z) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y3)),Y3) ) ) ).

% add_frac_num
tff(fact_3264_add__num__frac,axiom,
    ! [Y3: rat,Z: rat,Xa: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),divide_divide_rat(Xa,Y3)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y3)),Y3) ) ) ).

% add_num_frac
tff(fact_3265_add__divide__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),divide_divide_rat(Y3,Z)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),Y3),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3266_divide__add__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(Xa,Z)),Y3) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Z)),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3267_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      minus_minus_rat(A,divide_divide_rat(B,Z)) = $ite(Z = zero_zero_rat,A,divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z),B),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3268_diff__frac__eq,axiom,
    ! [Y3: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(divide_divide_rat(Xa,Y3),divide_divide_rat(W,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3269_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(Xa,divide_divide_rat(Y3,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z),Y3),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3270_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(divide_divide_rat(Xa,Z),Y3) = divide_divide_rat(minus_minus_rat(Xa,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Z)),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3271_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_3272_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_3273_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(uminus_uminus_rat,B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_3274_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)) = A )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_3275_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,divide_divide_rat(A,B)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_3276_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,divide_divide_rat(A,B)) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_3277_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_3278_abs__div__pos,axiom,
    ! [Y3: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
     => ( divide_divide_rat(abs_abs_rat(Xa),Y3) = abs_abs_rat(divide_divide_rat(Xa,Y3)) ) ) ).

% abs_div_pos
tff(fact_3279_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),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_3280_distrib__left__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(nO_MATCH_int_nat,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3281_distrib__left__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3282_distrib__left__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3283_distrib__left__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,A: rat,B: rat,C: rat] :
      ( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3284_distrib__left__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(nO_MATCH_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3285_distrib__left__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,A: int,B: int,C: int] :
      ( aa_int_o(aa_nat_fun_int_o(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3286_distrib__left__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3287_distrib__left__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,A: rat,B: rat,C: rat] :
      ( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3288_distrib__left__NO__MATCH,axiom,
    ! [Xa: code_integer,Y3: code_integer,A: nat,B: nat,C: nat] :
      ( aa_nat_o(nO_MAT8430403764537989018er_nat(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Xa),Y3)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3289_distrib__left__NO__MATCH,axiom,
    ! [Xa: code_integer,Y3: code_integer,A: int,B: int,C: int] :
      ( aa_int_o(nO_MAT8427913294028938742er_int(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Xa),Y3)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3290_distrib__right__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(nO_MATCH_int_nat,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3291_distrib__right__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3292_distrib__right__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3293_distrib__right__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,C: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),C)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3294_distrib__right__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(nO_MATCH_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3295_distrib__right__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,C: int,A: int,B: int] :
      ( aa_int_o(aa_nat_fun_int_o(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3296_distrib__right__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3297_distrib__right__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,C: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),C)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3298_distrib__right__NO__MATCH,axiom,
    ! [Xa: code_integer,Y3: code_integer,C: nat,A: nat,B: nat] :
      ( aa_nat_o(nO_MAT8430403764537989018er_nat(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Xa),Y3)),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3299_distrib__right__NO__MATCH,axiom,
    ! [Xa: code_integer,Y3: code_integer,C: int,A: int,B: int] :
      ( aa_int_o(nO_MAT8427913294028938742er_int(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Xa),Y3)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3300_left__diff__distrib__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3301_left__diff__distrib__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3302_left__diff__distrib__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,C: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),C)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3303_left__diff__distrib__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,C: int,A: int,B: int] :
      ( aa_int_o(aa_nat_fun_int_o(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3304_left__diff__distrib__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3305_left__diff__distrib__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,C: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),C)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3306_left__diff__distrib__NO__MATCH,axiom,
    ! [Xa: code_integer,Y3: code_integer,C: int,A: int,B: int] :
      ( aa_int_o(nO_MAT8427913294028938742er_int(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Xa),Y3)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3307_left__diff__distrib__NO__MATCH,axiom,
    ! [Xa: code_integer,Y3: code_integer,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(nO_MAT8252062027627875367nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Xa),Y3)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3308_left__diff__distrib__NO__MATCH,axiom,
    ! [Xa: code_integer,Y3: code_integer,C: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MAT7795273704451493282er_rat(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Xa),Y3)),C)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3309_left__diff__distrib__NO__MATCH,axiom,
    ! [Xa: rat,Y3: rat,C: int,A: int,B: int] :
      ( aa_int_o(nO_MATCH_rat_int(divide_divide_rat(Xa,Y3)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3310_right__diff__distrib__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3311_right__diff__distrib__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3312_right__diff__distrib__NO__MATCH,axiom,
    ! [Xa: int,Y3: int,A: rat,B: rat,C: rat] :
      ( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y3)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3313_right__diff__distrib__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,A: int,B: int,C: int] :
      ( aa_int_o(aa_nat_fun_int_o(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3314_right__diff__distrib__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3315_right__diff__distrib__NO__MATCH,axiom,
    ! [Xa: nat,Y3: nat,A: rat,B: rat,C: rat] :
      ( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3316_right__diff__distrib__NO__MATCH,axiom,
    ! [Xa: code_integer,Y3: code_integer,A: int,B: int,C: int] :
      ( aa_int_o(nO_MAT8427913294028938742er_int(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Xa),Y3)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3317_right__diff__distrib__NO__MATCH,axiom,
    ! [Xa: code_integer,Y3: code_integer,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(nO_MAT8252062027627875367nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Xa),Y3)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3318_right__diff__distrib__NO__MATCH,axiom,
    ! [Xa: code_integer,Y3: code_integer,A: rat,B: rat,C: rat] :
      ( aa_rat_o(nO_MAT7795273704451493282er_rat(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Xa),Y3)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3319_right__diff__distrib__NO__MATCH,axiom,
    ! [Xa: rat,Y3: rat,A: int,B: int,C: int] :
      ( aa_int_o(nO_MATCH_rat_int(divide_divide_rat(Xa,Y3)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3320_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_3321_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_3322_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_left_mono
tff(fact_3323_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3324_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3325_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3326_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3327_mult__imp__div__pos__le,axiom,
    ! [Y3: rat,Xa: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y3))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xa,Y3)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3328_mult__imp__le__div__pos,axiom,
    ! [Y3: rat,Z: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y3)),Xa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),divide_divide_rat(Xa,Y3)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3329_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3330_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3331_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_3332_frac__le__eq,axiom,
    ! [Y3: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xa,Y3)),divide_divide_rat(W,Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_3333_frac__less__eq,axiom,
    ! [Y3: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xa,Y3)),divide_divide_rat(W,Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_3334_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_3335_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_less_eq
tff(fact_3336_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_3337_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_3338_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_3339_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_3340_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,divide_divide_rat(Xa,Z))),Y3) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Z)),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_3341_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,divide_divide_rat(A,Z))),B) = $ite(Z = zero_zero_rat,B,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z)),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_3342_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      minus_minus_rat(aa_rat_rat(uminus_uminus_rat,divide_divide_rat(A,Z)),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),divide_divide_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z)),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_3343_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      minus_minus_rat(divide_divide_rat(A,Z),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),divide_divide_rat(minus_minus_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z)),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_3344_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(aa_rat_rat(uminus_uminus_rat,divide_divide_rat(Xa,Z)),Y3) = divide_divide_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y3),Z)),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_3345_power__diff,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ) ) ).

% power_diff
tff(fact_3346_power__diff,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(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_3347_power__diff,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).

% power_diff
tff(fact_3348_power__diff,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = divide_divide_rat(aa_nat_rat(power_power_rat(A),M),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% power_diff
tff(fact_3349_power__diff,axiom,
    ! [A: code_natural,N: nat,M: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = divide5121882707175180666atural(aa_nat_Code_natural(power_7079662738309270450atural(A),M),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) ) ) ) ).

% power_diff
tff(fact_3350_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(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_3351_verit__less__mono__div__int2,axiom,
    ! [A3: int,B3: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N)) ) ) ).

% verit_less_mono_div_int2
tff(fact_3352_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_3353_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_le_eq
tff(fact_3354_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_3355_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_3356_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_3357_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_3358_scaling__mono,axiom,
    ! [U: rat,V: rat,R: rat,S: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),minus_minus_rat(V,U)),S))),V) ) ) ) ).

% scaling_mono
tff(fact_3359_nat__approx__posE,axiom,
    ! [E: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
     => ~ ! [N2: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),E) ) ).

% nat_approx_posE
tff(fact_3360_ceiling__divide__upper,axiom,
    ! [Q2: rat,P: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P,Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_3361_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_3362_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_3363_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).

% div_mult_self1
tff(fact_3364_div__mult__self1,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self1
tff(fact_3365_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_3366_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_3367_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).

% div_mult_self2
tff(fact_3368_div__mult__self2,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self2
tff(fact_3369_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_3370_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_3371_div__mult__self3,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).

% div_mult_self3
tff(fact_3372_div__mult__self3,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)),A),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self3
tff(fact_3373_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_3374_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_3375_div__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).

% div_mult_self4
tff(fact_3376_div__mult__self4,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)),A),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self4
tff(fact_3377_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3378_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3379_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult1
tff(fact_3380_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_3381_div__mult__mult1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) ) ) ).

% div_mult_mult1
tff(fact_3382_div__mult__mult1,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)) = divide5121882707175180666atural(A,B) ) ) ).

% div_mult_mult1
tff(fact_3383_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult2
tff(fact_3384_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_3385_div__mult__mult2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) ) ) ).

% div_mult_mult2
tff(fact_3386_div__mult__mult2,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)) = divide5121882707175180666atural(A,B) ) ) ).

% div_mult_mult2
tff(fact_3387_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% div_mult_mult1_if
tff(fact_3388_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% div_mult_mult1_if
tff(fact_3389_div__mult__mult1__if,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = $ite(C = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ).

% div_mult_mult1_if
tff(fact_3390_div__mult__mult1__if,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)) = $ite(C = zero_z2226904508553997617atural,zero_z2226904508553997617atural,divide5121882707175180666atural(A,B)) ).

% div_mult_mult1_if
tff(fact_3391_div__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).

% div_minus1_right
tff(fact_3392_div__minus1__right,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% div_minus1_right
tff(fact_3393_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_3394_div__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3395_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 )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_3396_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(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_3397_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% div_greater_zero_iff
tff(fact_3398_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q2)),N)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_3399_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3400_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M) ) ) ).

% div_less_dividend
tff(fact_3401_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_3402_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_3403_div__add__self2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),one_one_Code_integer) ) ) ).

% div_add_self2
tff(fact_3404_div__add__self2,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,B)),one_one_Code_natural) ) ) ).

% div_add_self2
tff(fact_3405_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_3406_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_3407_div__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),one_one_Code_integer) ) ) ).

% div_add_self1
tff(fact_3408_div__add__self1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,B)),one_one_Code_natural) ) ) ).

% div_add_self1
tff(fact_3409_div__if,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | ( N = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ).

% div_if
tff(fact_3410_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Q2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),N) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3411_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(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_3412_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(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_3413_split__div,axiom,
    ! [Pa: fun_nat_o,M: nat,N: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))
    <=> ( ( ( N = zero_zero_nat )
         => aa_nat_o(Pa,zero_zero_nat) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),N)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
               => aa_nat_o(Pa,I4) ) ) ) ) ) ).

% split_div
tff(fact_3414_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_3415_split__div_H,axiom,
    ! [Pa: fun_nat_o,M: nat,N: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))
    <=> ( ( ( N = zero_zero_nat )
          & aa_nat_o(Pa,zero_zero_nat) )
        | ? [Q4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)),M)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q4)))
            & aa_nat_o(Pa,Q4) ) ) ) ).

% split_div'
tff(fact_3416_power__diff__power__eq,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3417_power__diff__power__eq,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(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,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_3418_power__diff__power__eq,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3419_power__diff__power__eq,axiom,
    ! [A: code_natural,M: nat,N: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_nat_Code_natural(power_7079662738309270450atural(A),M),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M),aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),divide5121882707175180666atural(one_one_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3420_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% bits_div_by_1
tff(fact_3421_bits__div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% bits_div_by_1
tff(fact_3422_bits__div__by__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),one_one_Code_integer) = A ).

% bits_div_by_1
tff(fact_3423_bits__div__by__1,axiom,
    ! [A: code_natural] : divide5121882707175180666atural(A,one_one_Code_natural) = A ).

% bits_div_by_1
tff(fact_3424_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_3425_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_3426_bits__div__by__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% bits_div_by_0
tff(fact_3427_bits__div__by__0,axiom,
    ! [A: code_natural] : divide5121882707175180666atural(A,zero_z2226904508553997617atural) = zero_z2226904508553997617atural ).

% bits_div_by_0
tff(fact_3428_bits__div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_3429_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_3430_bits__div__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).

% bits_div_0
tff(fact_3431_bits__div__0,axiom,
    ! [A: code_natural] : divide5121882707175180666atural(zero_z2226904508553997617atural,A) = zero_z2226904508553997617atural ).

% bits_div_0
tff(fact_3432_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).

% divide_int_def
tff(fact_3433_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,N: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,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))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( N = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),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)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N),M)))))) ) ).

% divide_int_unfold
tff(fact_3434_normalize__negative,axiom,
    ! [Q2: int,P: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q2),zero_zero_int)
     => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q2)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,P)),aa_int_int(uminus_uminus_int,Q2))) ) ) ).

% normalize_negative
tff(fact_3435_gbinomial__code,axiom,
    ! [A: rat,K: nat] :
      aa_nat_rat(gbinomial_rat(A),K) = $ite(K = zero_zero_nat,one_one_rat,divide_divide_rat(set_fo1949268297981939178at_rat(aTP_Lamm_cv(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat),one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,K))) ).

% gbinomial_code
tff(fact_3436_dvd__0__right,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% dvd_0_right
tff(fact_3437_dvd__0__right,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),zero_zero_int) ).

% dvd_0_right
tff(fact_3438_dvd__0__right,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),zero_z3403309356797280102nteger) ).

% dvd_0_right
tff(fact_3439_dvd__0__right,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),zero_zero_rat) ).

% dvd_0_right
tff(fact_3440_dvd__0__right,axiom,
    ! [A: code_natural] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),zero_z2226904508553997617atural) ).

% dvd_0_right
tff(fact_3441_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_3442_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_3443_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
tff(fact_3444_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_3445_dvd__0__left__iff,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,zero_z2226904508553997617atural),A)
    <=> ( A = zero_z2226904508553997617atural ) ) ).

% dvd_0_left_iff
tff(fact_3446_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3447_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3448_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3449_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3450_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3451_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3452_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3453_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3454_minus__dvd__iff,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,Xa)),Y3)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),Y3) ) ).

% minus_dvd_iff
tff(fact_3455_minus__dvd__iff,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xa)),Y3)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),Y3) ) ).

% minus_dvd_iff
tff(fact_3456_minus__dvd__iff,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(uminus_uminus_rat,Xa)),Y3)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xa),Y3) ) ).

% minus_dvd_iff
tff(fact_3457_dvd__minus__iff,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),aa_int_int(uminus_uminus_int,Y3))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),Y3) ) ).

% dvd_minus_iff
tff(fact_3458_dvd__minus__iff,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y3))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),Y3) ) ).

% dvd_minus_iff
tff(fact_3459_dvd__minus__iff,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xa),aa_rat_rat(uminus_uminus_rat,Y3))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xa),Y3) ) ).

% dvd_minus_iff
tff(fact_3460_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ) ).

% div_dvd_div
tff(fact_3461_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ) ).

% div_dvd_div
tff(fact_3462_div__dvd__div,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,C),A))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ) ).

% div_dvd_div
tff(fact_3463_div__dvd__div,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),C)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,divide5121882707175180666atural(B,A)),divide5121882707175180666atural(C,A))
        <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),C) ) ) ) ).

% div_dvd_div
tff(fact_3464_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),aa_int_int(abs_abs_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),K) ) ).

% dvd_abs_iff
tff(fact_3465_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),K) ) ).

% dvd_abs_iff
tff(fact_3466_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),abs_abs_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),K) ) ).

% dvd_abs_iff
tff(fact_3467_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(abs_abs_int,M)),K)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),K) ) ).

% abs_dvd_iff
tff(fact_3468_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,M)),K)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),K) ) ).

% abs_dvd_iff
tff(fact_3469_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,abs_abs_rat(M)),K)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),K) ) ).

% abs_dvd_iff
tff(fact_3470_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),one_one_nat)
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_3471_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3472_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3473_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3474_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3475_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3476_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3477_dvd__times__left__cancel__iff,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C))
      <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3478_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3479_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3480_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3481_dvd__times__right__cancel__iff,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A))
      <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3482_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3483_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3484_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3485_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3486_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3487_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3488_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3489_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3490_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3491_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3492_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3493_unit__prod,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),one_one_Code_natural) ) ) ).

% unit_prod
tff(fact_3494_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ).

% unit_prod
tff(fact_3495_unit__prod,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ).

% unit_prod
tff(fact_3496_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_3497_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3498_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3499_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3500_dvd__div__mult__self,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),B)
     => ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(divide5121882707175180666atural(B,A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3501_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3502_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3503_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3504_dvd__mult__div__cancel,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),B)
     => ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),divide5121882707175180666atural(B,A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3505_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_add
tff(fact_3506_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_3507_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C)) ) ) ) ).

% div_add
tff(fact_3508_div__add,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),B)
       => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,C)),divide5121882707175180666atural(B,C)) ) ) ) ).

% div_add
tff(fact_3509_unit__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% unit_div
tff(fact_3510_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% unit_div
tff(fact_3511_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_3512_unit__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,divide5121882707175180666atural(A,B)),one_one_Code_natural) ) ) ).

% unit_div
tff(fact_3513_unit__div__1__unit,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_3514_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_3515_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),A)),one_one_Code_integer) ) ).

% unit_div_1_unit
tff(fact_3516_unit__div__1__unit,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,divide5121882707175180666atural(one_one_Code_natural,A)),one_one_Code_natural) ) ).

% unit_div_1_unit
tff(fact_3517_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3518_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3519_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3520_unit__div__1__div__1,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
     => ( divide5121882707175180666atural(one_one_Code_natural,divide5121882707175180666atural(one_one_Code_natural,A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3521_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_diff
tff(fact_3522_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C)) ) ) ) ).

% div_diff
tff(fact_3523_dvd__1__left,axiom,
    ! [K: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K) ).

% dvd_1_left
tff(fact_3524_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3525_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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 )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_3526_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_3527_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_int(aa_int_fun_nat_int(gbinomial_int,zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_3528_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_Code_integer(gbinom8545251970709558553nteger(zero_z3403309356797280102nteger),aa_nat_nat(suc,K)) = zero_z3403309356797280102nteger ).

% gbinomial_0(2)
tff(fact_3529_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_3530_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_Code_natural(gbinom7368847122466276068atural(zero_z2226904508553997617atural),aa_nat_nat(suc,K)) = zero_z2226904508553997617atural ).

% gbinomial_0(2)
tff(fact_3531_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_3532_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(gbinomial_int,A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_3533_gbinomial__0_I1_J,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(gbinom8545251970709558553nteger(A),zero_zero_nat) = one_one_Code_integer ).

% gbinomial_0(1)
tff(fact_3534_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_3535_gbinomial__0_I1_J,axiom,
    ! [A: code_natural] : aa_nat_Code_natural(gbinom7368847122466276068atural(A),zero_zero_nat) = one_one_Code_natural ).

% gbinomial_0(1)
tff(fact_3536_int__dvd__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N) ) ).

% int_dvd_int_iff
tff(fact_3537_zdvd1__eq,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),one_one_int)
    <=> ( aa_int_int(abs_abs_int,Xa) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_3538_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3539_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3540_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3541_unit__div__mult__self,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
     => ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(divide5121882707175180666atural(B,A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3542_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% unit_mult_div_div
tff(fact_3543_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).

% unit_mult_div_div
tff(fact_3544_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),A) ) ) ).

% unit_mult_div_div
tff(fact_3545_unit__mult__div__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
     => ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),divide5121882707175180666atural(one_one_Code_natural,A)) = divide5121882707175180666atural(B,A) ) ) ).

% unit_mult_div_div
tff(fact_3546_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_3547_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_3548_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R)),L)),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_3549_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R))),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_3550_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R)),K))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_3551_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,R)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_3552_normalize__denom__zero,axiom,
    ! [P: int] : aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),zero_zero_int)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_3553_nat__abs__dvd__iff,axiom,
    ! [K: int,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% nat_abs_dvd_iff
tff(fact_3554_dvd__nat__abs__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,N)),K) ) ).

% dvd_nat_abs_iff
tff(fact_3555_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_cw,A)),collect_int(aa_int_fun_int_o(aTP_Lamm_cw,B)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3556_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cx,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cx,B)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3557_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_cy,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_cy,B)))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3558_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
          & ( A != B ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
tff(fact_3559_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_trans
tff(fact_3560_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_trans
tff(fact_3561_dvd__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_trans
tff(fact_3562_dvd__refl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),A) ).

% dvd_refl
tff(fact_3563_dvd__refl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A) ).

% dvd_refl
tff(fact_3564_dvd__refl,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),A) ).

% dvd_refl
tff(fact_3565_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ( A != B ) ) ).

% gcd_nat.strict_implies_not_eq
tff(fact_3566_gcd__nat_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% gcd_nat.strict_implies_order
tff(fact_3567_gcd__nat_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) ) ) ).

% gcd_nat.strict_iff_order
tff(fact_3568_gcd__nat_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
          & ( A != B ) )
        | ( A = B ) ) ) ).

% gcd_nat.order_iff_strict
tff(fact_3569_gcd__nat_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% gcd_nat.strict_iff_not
tff(fact_3570_gcd__nat_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans2
tff(fact_3571_gcd__nat_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
          & ( B != C ) )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans1
tff(fact_3572_gcd__nat_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
          & ( B != C ) )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans
tff(fact_3573_gcd__nat_Oantisym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( A = B ) ) ) ).

% gcd_nat.antisym
tff(fact_3574_gcd__nat_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A)
        & ( A != A ) ) ).

% gcd_nat.irrefl
tff(fact_3575_gcd__nat_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% gcd_nat.eq_iff
tff(fact_3576_gcd__nat_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% gcd_nat.trans
tff(fact_3577_gcd__nat_Orefl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A) ).

% gcd_nat.refl
tff(fact_3578_gcd__nat_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
          & ( B != A ) ) ) ).

% gcd_nat.asym
tff(fact_3579_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N),M)
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_3580_dvd__field__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
    <=> ( ( A = zero_zero_rat )
       => ( B = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_3581_dvd__0__left,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_3582_dvd__0__left,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_3583_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
tff(fact_3584_dvd__0__left,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_3585_dvd__0__left,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,zero_z2226904508553997617atural),A)
     => ( A = zero_z2226904508553997617atural ) ) ).

% dvd_0_left
tff(fact_3586_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) ).

% dvd_triv_right
tff(fact_3587_dvd__triv__right,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) ).

% dvd_triv_right
tff(fact_3588_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) ).

% dvd_triv_right
tff(fact_3589_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)) ).

% dvd_triv_right
tff(fact_3590_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_3591_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_3592_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ).

% dvd_mult_right
tff(fact_3593_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),C) ) ).

% dvd_mult_right
tff(fact_3594_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3595_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3596_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3597_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3598_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ).

% dvd_triv_left
tff(fact_3599_dvd__triv__left,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% dvd_triv_left
tff(fact_3600_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% dvd_triv_left
tff(fact_3601_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% dvd_triv_left
tff(fact_3602_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_mult_left
tff(fact_3603_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_mult_left
tff(fact_3604_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ).

% dvd_mult_left
tff(fact_3605_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ).

% dvd_mult_left
tff(fact_3606_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_3607_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult2
tff(fact_3608_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult2
tff(fact_3609_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% dvd_mult2
tff(fact_3610_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ? [B5: nat,C4: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B5),C4) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B5),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C4),C) ) ) ).

% division_decomp
tff(fact_3611_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ? [B5: int,C4: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B5),C4) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B5),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C4),C) ) ) ).

% division_decomp
tff(fact_3612_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_3613_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult
tff(fact_3614_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult
tff(fact_3615_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% dvd_mult
tff(fact_3616_dvd__productE,axiom,
    ! [P: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ~ ! [X4: nat,Y4: nat] :
            ( ( P = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X4),Y4) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X4),A)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y4),B) ) ) ) ).

% dvd_productE
tff(fact_3617_dvd__productE,axiom,
    ! [P: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ~ ! [X4: int,Y4: int] :
            ( ( P = aa_int_int(aa_int_fun_int_int(times_times_int,X4),Y4) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X4),A)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y4),B) ) ) ) ).

% dvd_productE
tff(fact_3618_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ? [K3: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvd_def
tff(fact_3619_dvd__def,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
    <=> ? [K3: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K3) ) ).

% dvd_def
tff(fact_3620_dvd__def,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
    <=> ? [K3: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K3) ) ).

% dvd_def
tff(fact_3621_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
    <=> ? [K3: rat] : A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K3) ) ).

% dvd_def
tff(fact_3622_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% dvdI
tff(fact_3623_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% dvdI
tff(fact_3624_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% dvdI
tff(fact_3625_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A) ) ).

% dvdI
tff(fact_3626_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ~ ! [K2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvdE
tff(fact_3627_dvdE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ~ ! [K2: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) ) ).

% dvdE
tff(fact_3628_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ~ ! [K2: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) ) ).

% dvdE
tff(fact_3629_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ~ ! [K2: rat] : A != aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K2) ) ).

% dvdE
tff(fact_3630_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ) ) ).

% dvd_add
tff(fact_3631_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ) ).

% dvd_add
tff(fact_3632_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ) ).

% dvd_add
tff(fact_3633_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ) ).

% dvd_add
tff(fact_3634_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3635_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3636_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3637_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3638_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3639_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3640_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3641_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3642_one__dvd,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).

% one_dvd
tff(fact_3643_one__dvd,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).

% one_dvd
tff(fact_3644_one__dvd,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),A) ).

% one_dvd
tff(fact_3645_one__dvd,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A) ).

% one_dvd
tff(fact_3646_one__dvd,axiom,
    ! [A: code_natural] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,one_one_Code_natural),A) ).

% one_dvd
tff(fact_3647_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% unit_imp_dvd
tff(fact_3648_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% unit_imp_dvd
tff(fact_3649_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% unit_imp_dvd
tff(fact_3650_unit__imp__dvd,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A) ) ).

% unit_imp_dvd
tff(fact_3651_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_3652_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_3653_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_3654_dvd__unit__imp__unit,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural) ) ) ).

% dvd_unit_imp_unit
tff(fact_3655_dvd__diff,axiom,
    ! [Xa: int,Y3: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),Y3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),Z)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y3),Z)) ) ) ).

% dvd_diff
tff(fact_3656_dvd__diff,axiom,
    ! [Xa: code_integer,Y3: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),Y3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),Z)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Y3),Z)) ) ) ).

% dvd_diff
tff(fact_3657_dvd__diff,axiom,
    ! [Xa: rat,Y3: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xa),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xa),Z)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xa),minus_minus_rat(Y3,Z)) ) ) ).

% dvd_diff
tff(fact_3658_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),D)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% div_div_div_same
tff(fact_3659_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_div_div_same
tff(fact_3660_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),D)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),D)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) ) ) ) ).

% div_div_div_same
tff(fact_3661_div__div__div__same,axiom,
    ! [D: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,D),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A)
       => ( divide5121882707175180666atural(divide5121882707175180666atural(A,D),divide5121882707175180666atural(B,D)) = divide5121882707175180666atural(A,B) ) ) ) ).

% div_div_div_same
tff(fact_3662_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3663_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3664_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3665_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3666_dvd__div__eq__cancel,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( divide5121882707175180666atural(A,C) = divide5121882707175180666atural(B,C) )
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),A)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3667_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3668_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3669_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3670_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
       => ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3671_dvd__div__eq__iff,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),B)
       => ( ( divide5121882707175180666atural(A,C) = divide5121882707175180666atural(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3672_gcd__nat_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% gcd_nat.extremum
tff(fact_3673_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_3674_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_3675_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat)
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_3676_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_3677_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K) ) ).

% dvd_if_abs_eq
tff(fact_3678_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),K) ) ).

% dvd_if_abs_eq
tff(fact_3679_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( abs_abs_rat(L) = abs_abs_rat(K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),K) ) ).

% dvd_if_abs_eq
tff(fact_3680_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

% dvd_diff_nat
tff(fact_3681_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),N)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),M) ) ) ).

% zdvd_zdiffD
tff(fact_3682_zdvd__antisym__abs,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) ) ) ) ).

% zdvd_antisym_abs
tff(fact_3683_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_cw,A)),collect_int(aa_int_fun_int_o(aTP_Lamm_cw,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_3684_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_cy,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_cy,B)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% subset_divisors_dvd
tff(fact_3685_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cx,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cx,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_3686_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_3687_not__is__unit__0,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),one_one_nat) ).

% not_is_unit_0
tff(fact_3688_not__is__unit__0,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),one_one_int) ).

% not_is_unit_0
tff(fact_3689_not__is__unit__0,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% not_is_unit_0
tff(fact_3690_not__is__unit__0,axiom,
    ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,zero_z2226904508553997617atural),one_one_Code_natural) ).

% not_is_unit_0
tff(fact_3691_unit__mult__right__cancel,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
     => ( ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),A) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3692_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3693_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3694_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3695_unit__mult__left__cancel,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
     => ( ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3696_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3697_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3698_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3699_mult__unit__dvd__iff_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),C)
      <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3700_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3701_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3702_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3703_dvd__mult__unit__iff_H,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C))
      <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3704_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3705_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3706_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3707_mult__unit__dvd__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),C)
      <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_3708_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_3709_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_3710_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_3711_dvd__mult__unit__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B))
      <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_3712_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_3713_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_3714_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_3715_is__unit__mult__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),one_one_Code_natural)
    <=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
        & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural) ) ) ).

% is_unit_mult_iff
tff(fact_3716_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_3717_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_3718_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_3719_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3720_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3721_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3722_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( ( divide_divide_rat(A,B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3723_dvd__div__eq__0__iff,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A)
     => ( ( divide5121882707175180666atural(A,B) = zero_z2226904508553997617atural )
      <=> ( A = zero_z2226904508553997617atural ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3724_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),C)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3725_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),C)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3726_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,C),D)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3727_div__mult__div__if__dvd,axiom,
    ! [B: code_natural,A: code_natural,D: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,D),C)
       => ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(divide5121882707175180666atural(A,B)),divide5121882707175180666atural(C,D)) = divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3728_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_3729_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_3730_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_3731_dvd__mult__imp__div,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),B)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),divide5121882707175180666atural(B,C)) ) ).

% dvd_mult_imp_div
tff(fact_3732_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3733_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3734_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3735_dvd__div__mult2__eq,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)),A)
     => ( divide5121882707175180666atural(A,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)) = divide5121882707175180666atural(divide5121882707175180666atural(A,B),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3736_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3737_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3738_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3739_div__div__eq__right,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A)
       => ( divide5121882707175180666atural(A,divide5121882707175180666atural(B,C)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(divide5121882707175180666atural(A,B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3740_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_3741_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_3742_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_3743_div__mult__swap,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),B)
     => ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),divide5121882707175180666atural(B,C)) = divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B),C) ) ) ).

% div_mult_swap
tff(fact_3744_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_3745_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_3746_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_3747_dvd__div__mult,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),B)
     => ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(divide5121882707175180666atural(B,C)),A) = divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),A),C) ) ) ).

% dvd_div_mult
tff(fact_3748_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3749_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3750_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3751_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),B)
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,C)),divide5121882707175180666atural(B,C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3752_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3753_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3754_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3755_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),A)
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,C)),divide5121882707175180666atural(B,C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3756_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3757_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3758_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3759_unit__div__cancel,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
     => ( ( divide5121882707175180666atural(B,A) = divide5121882707175180666atural(C,A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3760_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3761_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3762_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3763_div__unit__dvd__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,divide5121882707175180666atural(A,B)),C)
      <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3764_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3765_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3766_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3767_dvd__div__unit__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),divide5121882707175180666atural(C,B))
      <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3768_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_div_neg
tff(fact_3769_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).

% dvd_div_neg
tff(fact_3770_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( divide_divide_rat(A,aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,divide_divide_rat(A,B)) ) ) ).

% dvd_div_neg
tff(fact_3771_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_neg_div
tff(fact_3772_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).

% dvd_neg_div
tff(fact_3773_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A),B) = aa_rat_rat(uminus_uminus_rat,divide_divide_rat(A,B)) ) ) ).

% dvd_neg_div
tff(fact_3774_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N),M) ) ) ).

% nat_dvd_not_less
tff(fact_3775_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M) ) ) ).

% dvd_pos_nat
tff(fact_3776_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),N)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),N)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,N),M)
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_3777_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,M),N)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,N),M) ) ) ).

% zdvd_not_zless
tff(fact_3778_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,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_3779_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,N))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N))) ).

% fact_fact_dvd_fact
tff(fact_3780_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,N))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N))) ).

% fact_fact_dvd_fact
tff(fact_3781_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,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_3782_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N) ) ) ).

% dvd_minus_self
tff(fact_3783_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(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 )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),N) ) ) ).

% zdvd_mult_cancel
tff(fact_3784_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_3785_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N) ) ) ) ).

% dvd_diffD1
tff(fact_3786_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_3787_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X4: nat,Y4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)),D3) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)),D3) ) ) ) ).

% bezout_add_nat
tff(fact_3788_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,Xa: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Xa) = 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) ) )
         => ? [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y4)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_3789_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X4: nat,Y4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)) = D3 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)) = D3 ) ) ) ).

% bezout1_nat
tff(fact_3790_zdvd__period,axiom,
    ! [A: int,D: int,Xa: int,T: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),T))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),T)) ) ) ).

% zdvd_period
tff(fact_3791_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(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)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),N) ) ).

% zdvd_reduce
tff(fact_3792_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),M)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M)),M = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_3793_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ).

% gbinomial_addition_formula
tff(fact_3794_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ).

% gbinomial_absorb_comp
tff(fact_3795_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_3796_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_3797_unit__dvdE,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
     => ~ ( ( A != zero_z2226904508553997617atural )
         => ! [C2: code_natural] : B != aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C2) ) ) ).

% unit_dvdE
tff(fact_3798_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) ) ) ).

% unit_dvdE
tff(fact_3799_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) ) ) ).

% unit_dvdE
tff(fact_3800_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2) ) ) ).

% unit_dvdE
tff(fact_3801_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = C )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3802_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3803_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),A) = C )
        <=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3804_dvd__div__eq__mult,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),B)
       => ( ( divide5121882707175180666atural(B,A) = C )
        <=> ( B = aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3805_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3806_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3807_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3808_div__dvd__iff__mult,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,divide5121882707175180666atural(A,B)),C)
        <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3809_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3810_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3811_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3812_dvd__div__iff__mult,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),B)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),divide5121882707175180666atural(B,C))
        <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3813_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3814_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3815_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
           => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,D),C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3816_dvd__div__div__eq__mult,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural,D: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( C != zero_z2226904508553997617atural )
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),B)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),D)
           => ( ( divide5121882707175180666atural(B,A) = divide5121882707175180666atural(D,C) )
            <=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3817_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3818_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3819_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3820_unit__div__eq__0__iff,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( ( divide5121882707175180666atural(A,B) = zero_z2226904508553997617atural )
      <=> ( A = zero_z2226904508553997617atural ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3821_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3822_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3823_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3824_is__unit__div__mult2__eq,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),one_one_Code_natural)
       => ( divide5121882707175180666atural(A,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)) = divide5121882707175180666atural(divide5121882707175180666atural(A,B),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3825_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_3826_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_3827_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_3828_unit__div__mult__swap,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),one_one_Code_natural)
     => ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),divide5121882707175180666atural(B,C)) = divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B),C) ) ) ).

% unit_div_mult_swap
tff(fact_3829_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_3830_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_3831_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_3832_unit__div__commute,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(divide5121882707175180666atural(A,B)),C) = divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C),B) ) ) ).

% unit_div_commute
tff(fact_3833_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_3834_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_3835_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_3836_div__mult__unit2,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A)
       => ( divide5121882707175180666atural(A,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)) = divide5121882707175180666atural(divide5121882707175180666atural(A,B),C) ) ) ) ).

% div_mult_unit2
tff(fact_3837_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( A = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3838_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3839_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,C),B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3840_unit__eq__div2,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( ( A = divide5121882707175180666atural(C,B) )
      <=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3841_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = C )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3842_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3843_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) = C )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3844_unit__eq__div1,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( ( divide5121882707175180666atural(A,B) = C )
      <=> ( A = aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3845_is__unit__power__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3846_is__unit__power__iff,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(A),N)),one_one_Code_natural)
    <=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3847_is__unit__power__iff,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3848_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3849_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) ) ) ).

% dvd_imp_le
tff(fact_3850_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N) ) ) ).

% dvd_mult_cancel
tff(fact_3851_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_3852_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X4: nat,Y4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),B)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3853_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),N) ) ) ).

% zdvd_imp_le
tff(fact_3854_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I)) ) ) ).

% dvd_imp_le_int
tff(fact_3855_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => member_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(ring_1_of_int_int,A)),aa_int_int(ring_1_of_int_int,B)),ring_1_Ints_int) ) ).

% of_int_divide_in_Ints
tff(fact_3856_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => member_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_int_Code_integer(ring_18347121197199848620nteger,B)),ring_11222124179247155820nteger) ) ).

% of_int_divide_in_Ints
tff(fact_3857_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => member_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A),aa_int_rat(ring_1_of_int_rat,B)),ring_1_Ints_rat) ) ).

% of_int_divide_in_Ints
tff(fact_3858_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ) ) ).

% dvd_fact
tff(fact_3859_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).

% Suc_times_gbinomial
tff(fact_3860_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ).

% gbinomial_absorption
tff(fact_3861_is__unitE,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [B2: int] :
              ( ( B2 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B2 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3862_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [B2: nat] :
              ( ( B2 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B2 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3863_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B2: code_integer] :
              ( ( B2 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
               => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),A) = B2 )
                 => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),B2) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2) = one_one_Code_integer )
                     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,C),A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3864_is__unitE,axiom,
    ! [A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
     => ~ ( ( A != zero_z2226904508553997617atural )
         => ! [B2: code_natural] :
              ( ( B2 != zero_z2226904508553997617atural )
             => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B2),one_one_Code_natural)
               => ( ( divide5121882707175180666atural(one_one_Code_natural,A) = B2 )
                 => ( ( divide5121882707175180666atural(one_one_Code_natural,B2) = A )
                   => ( ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B2) = one_one_Code_natural )
                     => ( divide5121882707175180666atural(C,A) != aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3865_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3866_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3867_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3868_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
       => ( divide5121882707175180666atural(A,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)) = divide5121882707175180666atural(one_one_Code_natural,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3869_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3870_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3871_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3872_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
       => ( divide5121882707175180666atural(A,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),A)) = divide5121882707175180666atural(one_one_Code_natural,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3873_dvd__power__iff,axiom,
    ! [Xa: code_integer,M: nat,N: nat] :
      ( ( Xa != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),M)),aa_nat_Code_integer(power_8256067586552552935nteger(Xa),N))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ) ).

% dvd_power_iff
tff(fact_3874_dvd__power__iff,axiom,
    ! [Xa: code_natural,M: nat,N: nat] :
      ( ( Xa != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(Xa),M)),aa_nat_Code_natural(power_7079662738309270450atural(Xa),N))
      <=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,Xa),one_one_Code_natural)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ) ).

% dvd_power_iff
tff(fact_3875_dvd__power__iff,axiom,
    ! [Xa: nat,M: nat,N: nat] :
      ( ( Xa != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),N))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ) ).

% dvd_power_iff
tff(fact_3876_dvd__power__iff,axiom,
    ! [Xa: int,M: nat,N: nat] :
      ( ( Xa != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),N))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ) ).

% dvd_power_iff
tff(fact_3877_dvd__power,axiom,
    ! [N: nat,Xa: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( Xa = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),aa_nat_Code_integer(power_8256067586552552935nteger(Xa),N)) ) ).

% dvd_power
tff(fact_3878_dvd__power,axiom,
    ! [N: nat,Xa: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( Xa = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xa),aa_nat_rat(power_power_rat(Xa),N)) ) ).

% dvd_power
tff(fact_3879_dvd__power,axiom,
    ! [N: nat,Xa: code_natural] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( Xa = one_one_Code_natural ) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,Xa),aa_nat_Code_natural(power_7079662738309270450atural(Xa),N)) ) ).

% dvd_power
tff(fact_3880_dvd__power,axiom,
    ! [N: nat,Xa: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( Xa = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),N)) ) ).

% dvd_power
tff(fact_3881_dvd__power,axiom,
    ! [N: nat,Xa: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( Xa = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),N)) ) ).

% dvd_power
tff(fact_3882_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_3883_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_3884_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),Q2))
        <=> aa_nat_o(aa_nat_fun_nat_o(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,R),M)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_3885_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)),M)
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_3886_gbinomial__factors,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(A),K)) ).

% gbinomial_factors
tff(fact_3887_gbinomial__rec,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_3888_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_dvd_imp_le
tff(fact_3889_gbinomial__negated__upper,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K),A),one_one_rat)),K)) ).

% gbinomial_negated_upper
tff(fact_3890_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(gbinomial_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),N)) ).

% gbinomial_index_swap
tff(fact_3891_normalize__denom__pos,axiom,
    ! [R: product_prod_int_int,P: int,Q2: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% normalize_denom_pos
tff(fact_3892_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P: int,R: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q2)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R),S)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P),S) = aa_int_int(aa_int_fun_int_int(times_times_int,R),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_3893_gbinomial__minus,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),K)) ).

% gbinomial_minus
tff(fact_3894_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_3895_gbinomial__pochhammer,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,A)),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer
tff(fact_3896_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer'
tff(fact_3897_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_3898_inf__period_I4_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int,K4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),T))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),T)) ) ) ).

% inf_period(4)
tff(fact_3899_inf__period_I4_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D4)
     => ! [X: code_integer,K4: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),T))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D4))),T)) ) ) ).

% inf_period(4)
tff(fact_3900_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D4)
     => ! [X: rat,K4: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),T))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(X,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4))),T)) ) ) ).

% inf_period(4)
tff(fact_3901_inf__period_I3_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int,K4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),T))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),T)) ) ) ).

% inf_period(3)
tff(fact_3902_inf__period_I3_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D4)
     => ! [X: code_integer,K4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),T))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D4))),T)) ) ) ).

% inf_period(3)
tff(fact_3903_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D4)
     => ! [X: rat,K4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),T))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(X,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4))),T)) ) ) ).

% inf_period(3)
tff(fact_3904_unity__coeff__ex,axiom,
    ! [Pa: fun_Code_natural_o,L: code_natural] :
      ( ? [X3: code_natural] : aa_Code_natural_o(Pa,aa_Cod4272041906116450446atural(times_2397367101498566445atural(L),X3))
    <=> ? [X3: code_natural] :
          ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,L),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X3),zero_z2226904508553997617atural))
          & aa_Code_natural_o(Pa,X3) ) ) ).

% unity_coeff_ex
tff(fact_3905_unity__coeff__ex,axiom,
    ! [Pa: fun_nat_o,L: nat] :
      ( ? [X3: nat] : aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X3))
    <=> ? [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),zero_zero_nat))
          & aa_nat_o(Pa,X3) ) ) ).

% unity_coeff_ex
tff(fact_3906_unity__coeff__ex,axiom,
    ! [Pa: fun_int_o,L: int] :
      ( ? [X3: int] : aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(times_times_int,L),X3))
    <=> ? [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),zero_zero_int))
          & aa_int_o(Pa,X3) ) ) ).

% unity_coeff_ex
tff(fact_3907_unity__coeff__ex,axiom,
    ! [Pa: fun_Code_integer_o,L: code_integer] :
      ( ? [X3: code_integer] : aa_Code_integer_o(Pa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X3))
    <=> ? [X3: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),zero_z3403309356797280102nteger))
          & aa_Code_integer_o(Pa,X3) ) ) ).

% unity_coeff_ex
tff(fact_3908_unity__coeff__ex,axiom,
    ! [Pa: fun_rat_o,L: rat] :
      ( ? [X3: rat] : aa_rat_o(Pa,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,L),X3))
    <=> ? [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),zero_zero_rat))
          & aa_rat_o(Pa,X3) ) ) ).

% unity_coeff_ex
tff(fact_3909_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z4)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),S)) ) ) ).

% minf(10)
tff(fact_3910_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S)) ) ) ).

% minf(10)
tff(fact_3911_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S)) ) ) ).

% minf(10)
tff(fact_3912_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S)) ) ) ).

% minf(10)
tff(fact_3913_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),S)) ) ) ).

% minf(9)
tff(fact_3914_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S)) ) ) ).

% minf(9)
tff(fact_3915_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S)) ) ) ).

% minf(9)
tff(fact_3916_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S)) ) ) ).

% minf(9)
tff(fact_3917_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z4),X)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),S)) ) ) ).

% pinf(10)
tff(fact_3918_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S)) ) ) ).

% pinf(10)
tff(fact_3919_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S)) ) ) ).

% pinf(10)
tff(fact_3920_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S)) ) ) ).

% pinf(10)
tff(fact_3921_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z4),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),S)) ) ) ).

% pinf(9)
tff(fact_3922_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S)) ) ) ).

% pinf(9)
tff(fact_3923_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S)) ) ) ).

% pinf(9)
tff(fact_3924_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S)) ) ) ).

% pinf(9)
tff(fact_3925_less__set__def,axiom,
    ! [A3: set_se7855581050983116737at_nat,B3: set_se7855581050983116737at_nat] :
      ( ord_le3519150613208228973at_nat(A3,B3)
    <=> ord_le6502550848784664846_nat_o(aTP_Lamm_a(A3),aTP_Lamm_a(B3)) ) ).

% less_set_def
tff(fact_3926_less__set__def,axiom,
    ! [A3: set_se6260736226359567993nt_int,B3: set_se6260736226359567993nt_int] :
      ( ord_le1924305788584680229nt_int(A3,B3)
    <=> ord_le8970291702602496086_int_o(aTP_Lamm_aa(A3),aTP_Lamm_aa(B3)) ) ).

% less_set_def
tff(fact_3927_less__set__def,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ord_less_set_set_nat(A3,B3)
    <=> ord_le921202756137246487_nat_o(aTP_Lamm_ab(A3),aTP_Lamm_ab(B3)) ) ).

% less_set_def
tff(fact_3928_less__set__def,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B3)
    <=> ord_less_fun_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ac,A3),aa_set_nat_fun_nat_o(aTP_Lamm_ac,B3)) ) ).

% less_set_def
tff(fact_3929_less__set__def,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ord_less_set_int(A3,B3)
    <=> ord_less_fun_int_o(aTP_Lamm_ad(A3),aTP_Lamm_ad(B3)) ) ).

% less_set_def
tff(fact_3930_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_3931_floor__divide__upper,axiom,
    ! [Q2: rat,P: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P,Q2)))),one_one_rat)),Q2)) ) ).

% floor_divide_upper
tff(fact_3932_verit__le__mono__div__int,axiom,
    ! [A3: int,B3: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),N) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),N)) ) ) ).

% verit_le_mono_div_int
tff(fact_3933_verit__le__mono__div,axiom,
    ! [A3: nat,B3: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),N)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),N) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),N)) ) ) ).

% verit_le_mono_div
tff(fact_3934_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_3935_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_3936_floor__add,axiom,
    ! [Xa: rat,Y3: rat] :
      archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y3)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,archimedean_frac_rat(Xa)),archimedean_frac_rat(Y3))),one_one_rat),aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xa)),archim3151403230148437115or_rat(Y3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xa)),archim3151403230148437115or_rat(Y3))),one_one_int)) ).

% floor_add
tff(fact_3937_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_3938_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_numeral
tff(fact_3939_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_num_nat(numeral_numeral_nat,K)) = zero_z3403309356797280102nteger ).

% power_zero_numeral
tff(fact_3940_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_3941_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_Code_natural(power_7079662738309270450atural(zero_z2226904508553997617atural),aa_num_nat(numeral_numeral_nat,K)) = zero_z2226904508553997617atural ).

% power_zero_numeral
tff(fact_3942_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_3943_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_3944_bits__mod__0,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(zero_z2226904508553997617atural,A) = zero_z2226904508553997617atural ).

% bits_mod_0
tff(fact_3945_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_3946_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_3947_bits__mod__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_3948_mod__0,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(zero_z2226904508553997617atural,A) = zero_z2226904508553997617atural ).

% mod_0
tff(fact_3949_mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_3950_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_3951_mod__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_3952_mod__by__0,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,zero_z2226904508553997617atural) = A ).

% mod_by_0
tff(fact_3953_mod__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),zero_zero_int) = A ).

% mod_by_0
tff(fact_3954_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_3955_mod__by__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_3956_mod__self,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,A) = zero_z2226904508553997617atural ).

% mod_self
tff(fact_3957_mod__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),A) = zero_zero_int ).

% mod_self
tff(fact_3958_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_3959_mod__self,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_3960_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self2
tff(fact_3961_mod__add__self2,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self2
tff(fact_3962_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).

% mod_add_self2
tff(fact_3963_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self1
tff(fact_3964_mod__add__self1,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self1
tff(fact_3965_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).

% mod_add_self1
tff(fact_3966_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).

% of_int_numeral
tff(fact_3967_of__int__numeral,axiom,
    ! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_num_int(numeral_numeral_int,K)) = aa_num_Code_integer(numera6620942414471956472nteger,K) ).

% of_int_numeral
tff(fact_3968_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_rat(K) ).

% of_int_numeral
tff(fact_3969_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_3970_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = aa_num_Code_integer(numera6620942414471956472nteger,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_3971_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = numeral_numeral_rat(N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_3972_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_3973_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_3974_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C)) ).

% distrib_left_numeral
tff(fact_3975_distrib__left__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),C)) ).

% distrib_left_numeral
tff(fact_3976_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),C)) ).

% distrib_left_numeral
tff(fact_3977_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_3978_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_3979_distrib__right__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_num_Code_integer(numera6620942414471956472nteger,V))) ).

% distrib_right_numeral
tff(fact_3980_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),numeral_numeral_rat(V)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),numeral_numeral_rat(V))) ).

% distrib_right_numeral
tff(fact_3981_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% add_neg_numeral_simps(3)
tff(fact_3982_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% add_neg_numeral_simps(3)
tff(fact_3983_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(M)),numeral_numeral_rat(N))) ).

% add_neg_numeral_simps(3)
tff(fact_3984_mod__mult__self2__is__0,axiom,
    ! [A: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B),B) = zero_z2226904508553997617atural ).

% mod_mult_self2_is_0
tff(fact_3985_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_3986_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_3987_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_3988_mod__mult__self1__is__0,axiom,
    ! [B: code_natural,A: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),A),B) = zero_z2226904508553997617atural ).

% mod_mult_self1_is_0
tff(fact_3989_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_3990_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),B) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_3991_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_3992_bits__mod__by__1,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,one_one_Code_natural) = zero_z2226904508553997617atural ).

% bits_mod_by_1
tff(fact_3993_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_3994_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_3995_bits__mod__by__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_3996_mod__by__1,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,one_one_Code_natural) = zero_z2226904508553997617atural ).

% mod_by_1
tff(fact_3997_mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_3998_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_3999_mod__by__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_4000_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self4
tff(fact_4001_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_4002_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).

% mod_mult_self4
tff(fact_4003_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self3
tff(fact_4004_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_4005_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).

% mod_mult_self3
tff(fact_4006_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self2
tff(fact_4007_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_4008_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).

% mod_mult_self2
tff(fact_4009_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self1
tff(fact_4010_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_4011_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).

% mod_mult_self1
tff(fact_4012_dvd__imp__mod__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),B)
     => ( modulo8411746178871703098atural(B,A) = zero_z2226904508553997617atural ) ) ).

% dvd_imp_mod_0
tff(fact_4013_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_4014_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_4015_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_4016_bits__mod__div__trivial,axiom,
    ! [A: code_natural,B: code_natural] : divide5121882707175180666atural(modulo8411746178871703098atural(A,B),B) = zero_z2226904508553997617atural ).

% bits_mod_div_trivial
tff(fact_4017_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_4018_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_4019_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_4020_mod__div__trivial,axiom,
    ! [A: code_natural,B: code_natural] : divide5121882707175180666atural(modulo8411746178871703098atural(A,B),B) = zero_z2226904508553997617atural ).

% mod_div_trivial
tff(fact_4021_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_4022_mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% mod_div_trivial
tff(fact_4023_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_4024_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_4025_floor__zero,axiom,
    archim3151403230148437115or_rat(zero_zero_rat) = zero_zero_int ).

% floor_zero
tff(fact_4026_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_4027_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_4028_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_4029_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(M)),numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_4030_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_4031_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_4032_add__numeral__left,axiom,
    ! [V: num,W: num,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_4033_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(W)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_4034_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_4035_floor__one,axiom,
    archim3151403230148437115or_rat(one_one_rat) = one_one_int ).

% floor_one
tff(fact_4036_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_4037_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xa: num,N: nat] :
      ( ( aa_int_nat(nat2,Y3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),N) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_4038_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xa: num,N: nat,Y3: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),N) = aa_int_nat(nat2,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N) = Y3 ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_4039_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( divide_divide_rat(B,numeral_numeral_rat(W)) = A )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_4040_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = divide_divide_rat(B,numeral_numeral_rat(W)) )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_4041_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_4042_mod__minus1__right,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_4043_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),aa_num_Code_integer(numera6620942414471956472nteger,N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N)) ) ).

% of_int_le_numeral_iff
tff(fact_4044_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N)) ) ).

% of_int_le_numeral_iff
tff(fact_4045_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N)) ) ).

% of_int_le_numeral_iff
tff(fact_4046_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4047_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(N)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4048_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4049_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4050_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4051_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(N)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4052_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,N)) ) ).

% of_int_less_numeral_iff
tff(fact_4053_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),aa_num_Code_integer(numera6620942414471956472nteger,N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,N)) ) ).

% of_int_less_numeral_iff
tff(fact_4054_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,N)) ) ).

% of_int_less_numeral_iff
tff(fact_4055_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W)))),Y3) ).

% semiring_norm(168)
tff(fact_4056_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(V),W)))),Y3) ).

% semiring_norm(168)
tff(fact_4057_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),Y3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(plus_plus_num(V),W)))),Y3) ).

% semiring_norm(168)
tff(fact_4058_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_4059_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_4060_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_4061_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_4062_ceiling__add__numeral,axiom,
    ! [Xa: rat,V: num] : archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xa)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_4063_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,Y3: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N) = aa_int_int(ring_1_of_int_int,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N) = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4064_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,Y3: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N) = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4065_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,Y3: int] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(Xa)),N) = aa_int_rat(ring_1_of_int_rat,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N) = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4066_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xa: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4067_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xa: num,N: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y3) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),N) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4068_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xa: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y3) = aa_nat_rat(power_power_rat(numeral_numeral_rat(Xa)),N) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4069_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_4070_floor__add2,axiom,
    ! [Xa: rat,Y3: rat] :
      ( ( member_rat(Xa,ring_1_Ints_rat)
        | member_rat(Y3,ring_1_Ints_rat) )
     => ( archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xa)),archim3151403230148437115or_rat(Y3)) ) ) ).

% floor_add2
tff(fact_4071_Suc__0__mod__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_o_nat(zero_n2687167440665602831ol_nat,N != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_4072_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = divide_divide_rat(B,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4073_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( divide_divide_rat(B,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) = A )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4074_zero__le__floor,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),archim3151403230148437115or_rat(Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa) ) ).

% zero_le_floor
tff(fact_4075_floor__less__zero,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xa)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat) ) ).

% floor_less_zero
tff(fact_4076_zero__less__floor,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),archim3151403230148437115or_rat(Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xa) ) ).

% zero_less_floor
tff(fact_4077_floor__le__zero,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xa)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),one_one_rat) ) ).

% floor_le_zero
tff(fact_4078_one__le__floor,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),archim3151403230148437115or_rat(Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xa) ) ).

% one_le_floor
tff(fact_4079_floor__less__one,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xa)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),one_one_rat) ) ).

% floor_less_one
tff(fact_4080_floor__diff__one,axiom,
    ! [Xa: rat] : archim3151403230148437115or_rat(minus_minus_rat(Xa,one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim3151403230148437115or_rat(Xa)),one_one_int) ).

% floor_diff_one
tff(fact_4081_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_4082_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),N)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_4083_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_4084_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_4085_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),N)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_4086_numeral__less__floor,axiom,
    ! [V: num,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),archim3151403230148437115or_rat(Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),one_one_rat)),Xa) ) ).

% numeral_less_floor
tff(fact_4087_floor__le__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),one_one_rat)) ) ).

% floor_le_numeral
tff(fact_4088_ceiling__less__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),minus_minus_rat(numeral_numeral_rat(V),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_4089_numeral__le__ceiling,axiom,
    ! [V: num,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),archim2889992004027027881ng_rat(Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(numeral_numeral_rat(V),one_one_rat)),Xa) ) ).

% numeral_le_ceiling
tff(fact_4090_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4091_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(Xa)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4092_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4093_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4094_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(Xa)),N)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4095_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4096_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4097_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4098_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(Xa)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4099_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4100_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4101_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(Xa)),N)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xa)),N)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4102_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xa: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4103_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xa: num,N: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y3) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa))),N) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4104_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xa: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y3) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(Xa))),N) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4105_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,Y3: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N) = aa_int_int(ring_1_of_int_int,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N) = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4106_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,Y3: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa))),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N) = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4107_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,Y3: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(Xa))),N) = aa_int_rat(ring_1_of_int_rat,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N) = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4108_neg__numeral__less__floor,axiom,
    ! [V: num,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim3151403230148437115or_rat(Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),one_one_rat)),Xa) ) ).

% neg_numeral_less_floor
tff(fact_4109_floor__le__neg__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xa)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),one_one_rat)) ) ).

% floor_le_neg_numeral
tff(fact_4110_ceiling__less__neg__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xa)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),minus_minus_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V)),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_4111_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim2889992004027027881ng_rat(Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V)),one_one_rat)),Xa) ) ).

% neg_numeral_le_ceiling
tff(fact_4112_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4113_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(Xa))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4114_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4115_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4116_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(Xa))),N)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4117_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4118_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4119_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4120_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(Xa))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4121_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4122_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4123_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,N: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(Xa))),N)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),N)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4124_int__ops_I3_J,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% int_ops(3)
tff(fact_4125_nat__numeral__as__int,axiom,
    ! [X: num] : aa_num_nat(numeral_numeral_nat,X) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X)) ).

% nat_numeral_as_int
tff(fact_4126_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_right_eq
tff(fact_4127_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_right_eq
tff(fact_4128_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),C))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ).

% mod_add_right_eq
tff(fact_4129_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_left_eq
tff(fact_4130_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_left_eq
tff(fact_4131_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),C)),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ).

% mod_add_left_eq
tff(fact_4132_mod__add__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B4)),C) ) ) ) ).

% mod_add_cong
tff(fact_4133_mod__add__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B4: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B4)),C) ) ) ) ).

% mod_add_cong
tff(fact_4134_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A4),C) )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B4),C) )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A4),B4)),C) ) ) ) ).

% mod_add_cong
tff(fact_4135_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_eq
tff(fact_4136_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_eq
tff(fact_4137_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),C))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ).

% mod_add_eq
tff(fact_4138_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_4139_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_4140_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_4141_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_iff
tff(fact_4142_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_iff
tff(fact_4143_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_iff
tff(fact_4144_zero__neq__numeral,axiom,
    ! [N: num] : zero_z2226904508553997617atural != aa_num_Code_natural(numera5444537566228673987atural,N) ).

% zero_neq_numeral
tff(fact_4145_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).

% zero_neq_numeral
tff(fact_4146_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).

% zero_neq_numeral
tff(fact_4147_zero__neq__numeral,axiom,
    ! [N: num] : zero_z3403309356797280102nteger != aa_num_Code_integer(numera6620942414471956472nteger,N) ).

% zero_neq_numeral
tff(fact_4148_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_rat != numeral_numeral_rat(N) ).

% zero_neq_numeral
tff(fact_4149_nat__mod__as__int,axiom,
    ! [X: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_mod_as_int
tff(fact_4150_Ints__numeral,axiom,
    ! [N: num] : member_int(aa_num_int(numeral_numeral_int,N),ring_1_Ints_int) ).

% Ints_numeral
tff(fact_4151_Ints__numeral,axiom,
    ! [N: num] : member_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,N),ring_11222124179247155820nteger) ).

% Ints_numeral
tff(fact_4152_Ints__numeral,axiom,
    ! [N: num] : member_rat(numeral_numeral_rat(N),ring_1_Ints_rat) ).

% Ints_numeral
tff(fact_4153_Nats__numeral,axiom,
    ! [W: num] : member_int(aa_num_int(numeral_numeral_int,W),semiring_1_Nats_int) ).

% Nats_numeral
tff(fact_4154_Nats__numeral,axiom,
    ! [W: num] : aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_num_nat(numeral_numeral_nat,W)),semiring_1_Nats_nat) ).

% Nats_numeral
tff(fact_4155_Nats__numeral,axiom,
    ! [W: num] : member_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,W),semiri2551628265256775642nteger) ).

% Nats_numeral
tff(fact_4156_Nats__numeral,axiom,
    ! [W: num] : member_rat(numeral_numeral_rat(W),semiring_1_Nats_rat) ).

% Nats_numeral
tff(fact_4157_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_neg_numeral
tff(fact_4158_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K)) ).

% of_int_neg_numeral
tff(fact_4159_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(K)) ).

% of_int_neg_numeral
tff(fact_4160_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4161_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4162_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4163_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4164_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4165_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4166_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
     => ~ ! [D3: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D3)) ) ).

% mod_eqE
tff(fact_4167_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),C) )
     => ~ ! [D3: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D3)) ) ).

% mod_eqE
tff(fact_4168_mod__eq__0__iff__dvd,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural(A,B) = zero_z2226904508553997617atural )
    <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_4169_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_4170_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_4171_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_4172_dvd__eq__mod__eq__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),B)
    <=> ( modulo8411746178871703098atural(B,A) = zero_z2226904508553997617atural ) ) ).

% dvd_eq_mod_eq_0
tff(fact_4173_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_4174_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_4175_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_4176_mod__0__imp__dvd,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural(A,B) = zero_z2226904508553997617atural )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A) ) ).

% mod_0_imp_dvd
tff(fact_4177_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_0_imp_dvd
tff(fact_4178_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_0_imp_dvd
tff(fact_4179_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_0_imp_dvd
tff(fact_4180_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural(A,B) = A )
    <=> ( divide5121882707175180666atural(A,B) = zero_z2226904508553997617atural ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4181_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4182_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4183_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) = A )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4184_div__add1__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,C)),divide5121882707175180666atural(B,C))),divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,C)),modulo8411746178871703098atural(B,C)),C)) ).

% div_add1_eq
tff(fact_4185_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_add1_eq
tff(fact_4186_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_add1_eq
tff(fact_4187_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),C))),C)) ).

% div_add1_eq
tff(fact_4188_dvd__minus__mod,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ).

% dvd_minus_mod
tff(fact_4189_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))) ).

% dvd_minus_mod
tff(fact_4190_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B))) ).

% dvd_minus_mod
tff(fact_4191_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = N,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ).

% mod_Suc
tff(fact_4192_gcd__nat__induct,axiom,
    ! [Pa: fun_nat_fun_nat_o,M: nat,N: nat] :
      ( ! [M4: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,M4),zero_zero_nat)
     => ( ! [M4: nat,N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( aa_nat_o(aa_nat_fun_nat_o(Pa,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M4),N2))
             => aa_nat_o(aa_nat_fun_nat_o(Pa,M4),N2) ) )
       => aa_nat_o(aa_nat_fun_nat_o(Pa,M),N) ) ) ).

% gcd_nat_induct
tff(fact_4193_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N) ) ).

% mod_less_divisor
tff(fact_4194_zero__le__numeral,axiom,
    ! [N: num] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_num_Code_natural(numera5444537566228673987atural,N)) ).

% zero_le_numeral
tff(fact_4195_zero__le__numeral,axiom,
    ! [N: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% zero_le_numeral
tff(fact_4196_zero__le__numeral,axiom,
    ! [N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(N)) ).

% zero_le_numeral
tff(fact_4197_zero__le__numeral,axiom,
    ! [N: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N)) ).

% zero_le_numeral
tff(fact_4198_zero__le__numeral,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,N)) ).

% zero_le_numeral
tff(fact_4199_not__numeral__le__zero,axiom,
    ! [N: num] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_num_Code_natural(numera5444537566228673987atural,N)),zero_z2226904508553997617atural) ).

% not_numeral_le_zero
tff(fact_4200_not__numeral__le__zero,axiom,
    ! [N: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),zero_z3403309356797280102nteger) ).

% not_numeral_le_zero
tff(fact_4201_not__numeral__le__zero,axiom,
    ! [N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(N)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_4202_not__numeral__le__zero,axiom,
    ! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_4203_not__numeral__le__zero,axiom,
    ! [N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_4204_zero__less__numeral,axiom,
    ! [N: num] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_num_Code_natural(numera5444537566228673987atural,N)) ).

% zero_less_numeral
tff(fact_4205_zero__less__numeral,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,N)) ).

% zero_less_numeral
tff(fact_4206_zero__less__numeral,axiom,
    ! [N: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N)) ).

% zero_less_numeral
tff(fact_4207_zero__less__numeral,axiom,
    ! [N: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% zero_less_numeral
tff(fact_4208_zero__less__numeral,axiom,
    ! [N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),numeral_numeral_rat(N)) ).

% zero_less_numeral
tff(fact_4209_not__numeral__less__zero,axiom,
    ! [N: num] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_num_Code_natural(numera5444537566228673987atural,N)),zero_z2226904508553997617atural) ).

% not_numeral_less_zero
tff(fact_4210_not__numeral__less__zero,axiom,
    ! [N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_4211_not__numeral__less__zero,axiom,
    ! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_4212_not__numeral__less__zero,axiom,
    ! [N: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),zero_z3403309356797280102nteger) ).

% not_numeral_less_zero
tff(fact_4213_not__numeral__less__zero,axiom,
    ! [N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(N)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_4214_one__le__numeral,axiom,
    ! [N: num] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),aa_num_Code_natural(numera5444537566228673987atural,N)) ).

% one_le_numeral
tff(fact_4215_one__le__numeral,axiom,
    ! [N: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% one_le_numeral
tff(fact_4216_one__le__numeral,axiom,
    ! [N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),numeral_numeral_rat(N)) ).

% one_le_numeral
tff(fact_4217_one__le__numeral,axiom,
    ! [N: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) ).

% one_le_numeral
tff(fact_4218_one__le__numeral,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N)) ).

% one_le_numeral
tff(fact_4219_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q3: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_4220_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_num_Code_natural(numera5444537566228673987atural,N)),one_one_Code_natural) ).

% not_numeral_less_one
tff(fact_4221_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N)),one_one_int) ).

% not_numeral_less_one
tff(fact_4222_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat) ).

% not_numeral_less_one
tff(fact_4223_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),one_one_Code_integer) ).

% not_numeral_less_one
tff(fact_4224_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(N)),one_one_rat) ).

% not_numeral_less_one
tff(fact_4225_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% zero_neq_neg_numeral
tff(fact_4226_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% zero_neq_neg_numeral
tff(fact_4227_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)) ).

% zero_neq_neg_numeral
tff(fact_4228_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_num_Code_natural(numera5444537566228673987atural,Xa)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_num_Code_natural(numera5444537566228673987atural,Xa)),one_one_Code_natural) ).

% one_plus_numeral_commute
tff(fact_4229_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Xa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xa)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_4230_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xa)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xa)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_4231_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,Xa)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa)),one_one_Code_integer) ).

% one_plus_numeral_commute
tff(fact_4232_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),numeral_numeral_rat(Xa)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(Xa)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_4233_nat__mod__eq__iff,axiom,
    ! [Xa: nat,N: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y3),N) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_4234_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,N) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_4235_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_Code_integer(numera6620942414471956472nteger,N) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_4236_numeral__neq__neg__one,axiom,
    ! [N: num] : numeral_numeral_rat(N) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_4237_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% one_neq_neg_numeral
tff(fact_4238_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% one_neq_neg_numeral
tff(fact_4239_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)) ).

% one_neq_neg_numeral
tff(fact_4240_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = R ) ) ).

% mod_int_unique
tff(fact_4241_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4242_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4243_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4244_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4245_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4246_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4247_unit__imp__mod__eq__0,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( modulo8411746178871703098atural(A,B) = zero_z2226904508553997617atural ) ) ).

% unit_imp_mod_eq_0
tff(fact_4248_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_4249_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_4250_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_4251_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),divide5121882707175180666atural(A,B))),modulo8411746178871703098atural(A,B))),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_4252_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_4253_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_4254_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_4255_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(divide5121882707175180666atural(A,B)),B)),modulo8411746178871703098atural(A,B))),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_4256_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_4257_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_4258_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_4259_mod__div__decomp,axiom,
    ! [A: code_natural,B: code_natural] : A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(divide5121882707175180666atural(A,B)),B)),modulo8411746178871703098atural(A,B)) ).

% mod_div_decomp
tff(fact_4260_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_div_decomp
tff(fact_4261_mod__div__decomp,axiom,
    ! [A: nat,B: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ).

% mod_div_decomp
tff(fact_4262_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)) ).

% mod_div_decomp
tff(fact_4263_div__mult__mod__eq,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(divide5121882707175180666atural(A,B)),B)),modulo8411746178871703098atural(A,B)) = A ).

% div_mult_mod_eq
tff(fact_4264_div__mult__mod__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% div_mult_mod_eq
tff(fact_4265_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% div_mult_mod_eq
tff(fact_4266_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)) = A ).

% div_mult_mod_eq
tff(fact_4267_mod__div__mult__eq,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,B)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(divide5121882707175180666atural(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4268_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4269_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4270_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4271_mod__mult__div__eq,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,B)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),divide5121882707175180666atural(A,B))) = A ).

% mod_mult_div_eq
tff(fact_4272_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).

% mod_mult_div_eq
tff(fact_4273_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = A ).

% mod_mult_div_eq
tff(fact_4274_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B))) = A ).

% mod_mult_div_eq
tff(fact_4275_mult__div__mod__eq,axiom,
    ! [B: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),divide5121882707175180666atural(A,B))),modulo8411746178871703098atural(A,B)) = A ).

% mult_div_mod_eq
tff(fact_4276_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% mult_div_mod_eq
tff(fact_4277_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% mult_div_mod_eq
tff(fact_4278_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)) = A ).

% mult_div_mod_eq
tff(fact_4279_div__mult1__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),divide5121882707175180666atural(B,C))),divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),modulo8411746178871703098atural(B,C)),C)) ).

% div_mult1_eq
tff(fact_4280_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_mult1_eq
tff(fact_4281_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_4282_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),C))),C)) ).

% div_mult1_eq
tff(fact_4283_minus__div__mult__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,aa_Cod4272041906116450446atural(times_2397367101498566445atural(divide5121882707175180666atural(A,B)),B)) = modulo8411746178871703098atural(A,B) ).

% minus_div_mult_eq_mod
tff(fact_4284_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_div_mult_eq_mod
tff(fact_4285_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_div_mult_eq_mod
tff(fact_4286_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).

% minus_div_mult_eq_mod
tff(fact_4287_minus__mod__eq__div__mult,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,modulo8411746178871703098atural(A,B)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(divide5121882707175180666atural(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4288_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4289_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4290_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4291_minus__mod__eq__mult__div,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,modulo8411746178871703098atural(A,B)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),divide5121882707175180666atural(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_4292_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4293_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4294_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4295_minus__mult__div__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),divide5121882707175180666atural(A,B))) = modulo8411746178871703098atural(A,B) ).

% minus_mult_div_eq_mod
tff(fact_4296_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_mult_div_eq_mod
tff(fact_4297_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_mult_div_eq_mod
tff(fact_4298_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).

% minus_mult_div_eq_mod
tff(fact_4299_le__floor__add,axiom,
    ! [Xa: rat,Y3: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xa)),archim3151403230148437115or_rat(Y3))),archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y3))) ).

% le_floor_add
tff(fact_4300_floor__add__int,axiom,
    ! [Xa: rat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xa)),Z) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z))) ).

% floor_add_int
tff(fact_4301_int__add__floor,axiom,
    ! [Z: int,Xa: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),archim3151403230148437115or_rat(Xa)) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xa)) ).

% int_add_floor
tff(fact_4302_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N) ) ).

% mod_le_divisor
tff(fact_4303_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( modulo8411746178871703098atural(aa_nat_Code_natural(semiri2447717529341329178atural,N),aa_nat_Code_natural(semiri2447717529341329178atural,M)) = zero_z2226904508553997617atural ) ) ).

% fact_mod
tff(fact_4304_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1406184849735516958ct_int,M)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_4305_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,N)),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_4306_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_4307_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% not_zero_le_neg_numeral
tff(fact_4308_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) ).

% not_zero_le_neg_numeral
tff(fact_4309_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) ).

% not_zero_le_neg_numeral
tff(fact_4310_neg__numeral__le__zero,axiom,
    ! [N: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_4311_neg__numeral__le__zero,axiom,
    ! [N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_4312_neg__numeral__le__zero,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_4313_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) ).

% not_zero_less_neg_numeral
tff(fact_4314_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% not_zero_less_neg_numeral
tff(fact_4315_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) ).

% not_zero_less_neg_numeral
tff(fact_4316_neg__numeral__less__zero,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_4317_neg__numeral__less__zero,axiom,
    ! [N: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_4318_neg__numeral__less__zero,axiom,
    ! [N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_4319_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_4320_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% not_one_le_neg_numeral
tff(fact_4321_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))) ).

% not_one_le_neg_numeral
tff(fact_4322_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_one_le_neg_numeral
tff(fact_4323_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_4324_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_4325_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_4326_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_4327_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_4328_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_4329_neg__one__le__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M)) ).

% neg_one_le_numeral
tff(fact_4330_neg__one__le__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),numeral_numeral_rat(M)) ).

% neg_one_le_numeral
tff(fact_4331_neg__one__le__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M)) ).

% neg_one_le_numeral
tff(fact_4332_neg__numeral__le__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_4333_neg__numeral__le__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_4334_neg__numeral__le__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) ).

% neg_numeral_le_one
tff(fact_4335_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B,C) = numeral_numeral_rat(W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C),numeral_numeral_rat(W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_4336_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( numeral_numeral_rat(W) = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C) = B,numeral_numeral_rat(W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_4337_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_4338_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_4339_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))) ).

% not_neg_one_less_neg_numeral
tff(fact_4340_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_one_less_neg_numeral
tff(fact_4341_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% not_one_less_neg_numeral
tff(fact_4342_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))) ).

% not_one_less_neg_numeral
tff(fact_4343_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_4344_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_4345_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_4346_neg__one__less__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M)) ).

% neg_one_less_numeral
tff(fact_4347_neg__one__less__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M)) ).

% neg_one_less_numeral
tff(fact_4348_neg__one__less__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),numeral_numeral_rat(M)) ).

% neg_one_less_numeral
tff(fact_4349_neg__numeral__less__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) ).

% neg_numeral_less_one
tff(fact_4350_neg__numeral__less__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_4351_neg__numeral__less__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_4352_nat__mod__eq__lemma,axiom,
    ! [Xa: nat,N: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y3),N) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xa)
       => ? [Q3: nat] : Xa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_4353_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => ~ ! [S4: nat] : N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S4)) ) ) ).

% mod_eq_nat2E
tff(fact_4354_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ~ ! [S4: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S4)) ) ) ).

% mod_eq_nat1E
tff(fact_4355_div__less__mono,axiom,
    ! [A3: nat,B3: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),N) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),N)) ) ) ) ) ).

% div_less_mono
tff(fact_4356_zdiv__mono__strict,axiom,
    ! [A3: int,B3: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),N) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),N) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),N)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_4357_div__mod__decomp,axiom,
    ! [A3: nat,N: nat] : A3 = 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,A3),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),N)) ).

% div_mod_decomp
tff(fact_4358_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).

% mod_mult2_eq
tff(fact_4359_div__mod__decomp__int,axiom,
    ! [A3: int,N: int] : A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N)),N)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),N)) ).

% div_mod_decomp_int
tff(fact_4360_eucl__rel__int,axiom,
    ! [K: int,L: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ).

% eucl_rel_int
tff(fact_4361_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_4362_of__nat__floor,axiom,
    ! [R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,archim3151403230148437115or_rat(R)))),R) ) ).

% of_nat_floor
tff(fact_4363_mod__mult2__eq_H,axiom,
    ! [A: code_natural,M: nat,N: nat] : modulo8411746178871703098atural(A,aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M)),modulo8411746178871703098atural(divide5121882707175180666atural(A,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N)))),modulo8411746178871703098atural(A,aa_nat_Code_natural(semiri3763490453095760265atural,M))) ).

% mod_mult2_eq'
tff(fact_4364_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_4365_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,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(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_4366_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_4367_one__add__floor,axiom,
    ! [Xa: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xa)),one_one_int) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),one_one_rat)) ).

% one_add_floor
tff(fact_4368_split__mod,axiom,
    ! [Pa: fun_nat_o,M: nat,N: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))
    <=> ( ( ( N = zero_zero_nat )
         => aa_nat_o(Pa,M) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),N)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
               => aa_nat_o(Pa,J3) ) ) ) ) ) ).

% split_mod
tff(fact_4369_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_4370_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_4371_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C) = B,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4372_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B,C) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4373_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      aa_num_Code_natural(numera5444537566228673987atural,aa_nat_num(num_of_nat,N)) = $ite(N = zero_zero_nat,one_one_Code_natural,aa_nat_Code_natural(semiri3763490453095760265atural,N)) ).

% numeral_num_of_nat_unfold
tff(fact_4374_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      aa_num_Code_integer(numera6620942414471956472nteger,aa_nat_num(num_of_nat,N)) = $ite(N = zero_zero_nat,one_one_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% numeral_num_of_nat_unfold
tff(fact_4375_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      numeral_numeral_rat(aa_nat_num(num_of_nat,N)) = $ite(N = zero_zero_nat,one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% numeral_num_of_nat_unfold
tff(fact_4376_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      aa_num_int(numeral_numeral_int,aa_nat_num(num_of_nat,N)) = $ite(N = zero_zero_nat,one_one_int,aa_nat_int(semiri1314217659103216013at_int,N)) ).

% numeral_num_of_nat_unfold
tff(fact_4377_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N)) = $ite(N = zero_zero_nat,one_one_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% numeral_num_of_nat_unfold
tff(fact_4378_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,N: nat] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,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))) = $ite(
        ( ( 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(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)),
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(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_4379_floor__unique,axiom,
    ! [Z: int,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
       => ( archim3151403230148437115or_rat(Xa) = Z ) ) ) ).

% floor_unique
tff(fact_4380_floor__eq__iff,axiom,
    ! [Xa: rat,A: int] :
      ( ( archim3151403230148437115or_rat(Xa) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)) ) ) ).

% floor_eq_iff
tff(fact_4381_floor__split,axiom,
    ! [Pa: fun_int_o,T: rat] :
      ( aa_int_o(Pa,archim3151403230148437115or_rat(T))
    <=> ! [I4: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I4)),T)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,I4)),one_one_rat)) )
         => aa_int_o(Pa,I4) ) ) ).

% floor_split
tff(fact_4382_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),C))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4383_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4384_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4385_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B))),archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% le_mult_floor
tff(fact_4386_less__floor__iff,axiom,
    ! [Z: int,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),archim3151403230148437115or_rat(Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xa) ) ).

% less_floor_iff
tff(fact_4387_floor__le__iff,axiom,
    ! [Xa: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xa)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% floor_le_iff
tff(fact_4388_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_4389_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_4390_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_4391_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_4392_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_4393_floor__divide__lower,axiom,
    ! [Q2: rat,P: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P,Q2)))),Q2)),P) ) ).

% floor_divide_lower
tff(fact_4394_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B)))),aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_4395_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B)))),aa_int_int(ring_1_of_int_int,archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_4396_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_4397_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_4398_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_minus_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_4399_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),N)) ).

% diff_numeral_special(3)
tff(fact_4400_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(one),N)) ).

% diff_numeral_special(3)
tff(fact_4401_diff__numeral__special_I3_J,axiom,
    ! [N: num] : minus_minus_rat(one_one_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = numeral_numeral_rat(aa_num_num(plus_plus_num(one),N)) ).

% diff_numeral_special(3)
tff(fact_4402_diff__numeral__special_I5_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,N))) ).

% diff_numeral_special(5)
tff(fact_4403_diff__numeral__special_I5_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,N))) ).

% diff_numeral_special(5)
tff(fact_4404_diff__numeral__special_I5_J,axiom,
    ! [N: num] : minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat),numeral_numeral_rat(N)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(inc,N))) ).

% diff_numeral_special(5)
tff(fact_4405_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_4406_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_4407_diff__numeral__special_I6_J,axiom,
    ! [M: num] : minus_minus_rat(numeral_numeral_rat(M),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = numeral_numeral_rat(aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_4408_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,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_4409_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_4410_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_4411_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,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,N))) ).

% add_neg_numeral_special(5)
tff(fact_4412_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,N))) ).

% add_neg_numeral_special(5)
tff(fact_4413_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(inc,N))) ).

% add_neg_numeral_special(5)
tff(fact_4414_prod__decode__aux_Osimps,axiom,
    ! [K: nat,M: nat] :
      aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),K),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,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_4415_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_Code_natural(numera5444537566228673987atural,N) = one_one_Code_natural )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_4416_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_int(numeral_numeral_int,N) = one_one_int )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_4417_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,N) = one_one_nat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_4418_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_Code_integer(numera6620942414471956472nteger,N) = one_one_Code_integer )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_4419_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat(N) = one_one_rat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_4420_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_natural = aa_num_Code_natural(numera5444537566228673987atural,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_4421_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_4422_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_4423_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_integer = aa_num_Code_integer(numera6620942414471956472nteger,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_4424_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat = numeral_numeral_rat(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_4425_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_num_Code_natural(numera5444537566228673987atural,N)),one_one_Code_natural)
    <=> aa_num_o(ord_less_eq_num(N),one) ) ).

% numeral_le_one_iff
tff(fact_4426_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),one_one_Code_integer)
    <=> aa_num_o(ord_less_eq_num(N),one) ) ).

% numeral_le_one_iff
tff(fact_4427_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(N)),one_one_rat)
    <=> aa_num_o(ord_less_eq_num(N),one) ) ).

% numeral_le_one_iff
tff(fact_4428_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat)
    <=> aa_num_o(ord_less_eq_num(N),one) ) ).

% numeral_le_one_iff
tff(fact_4429_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),one_one_int)
    <=> aa_num_o(ord_less_eq_num(N),one) ) ).

% numeral_le_one_iff
tff(fact_4430_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_num_Code_natural(numera5444537566228673987atural,N))
    <=> aa_num_o(ord_less_num(one),N) ) ).

% one_less_numeral_iff
tff(fact_4431_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,N))
    <=> aa_num_o(ord_less_num(one),N) ) ).

% one_less_numeral_iff
tff(fact_4432_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N))
    <=> aa_num_o(ord_less_num(one),N) ) ).

% one_less_numeral_iff
tff(fact_4433_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N))
    <=> aa_num_o(ord_less_num(one),N) ) ).

% one_less_numeral_iff
tff(fact_4434_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),numeral_numeral_rat(N))
    <=> aa_num_o(ord_less_num(one),N) ) ).

% one_less_numeral_iff
tff(fact_4435_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4436_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4437_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4438_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4439_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4440_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4441_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4442_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4443_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4444_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4445_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4446_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4447_one__plus__numeral,axiom,
    ! [N: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_num_Code_natural(numera5444537566228673987atural,N)) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_4448_one__plus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_4449_one__plus__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_4450_one__plus__numeral,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_4451_one__plus__numeral,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_4452_numeral__plus__one,axiom,
    ! [N: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_num_Code_natural(numera5444537566228673987atural,N)),one_one_Code_natural) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_4453_numeral__plus__one,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_4454_numeral__plus__one,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_4455_numeral__plus__one,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_4456_numeral__plus__one,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(N)),one_one_rat) = numeral_numeral_rat(aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_4457_prod__decode__aux_Ocases,axiom,
    ! [Xa: product_prod_nat_nat] :
      ~ ! [K2: nat,M4: nat] : Xa != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M4) ).

% prod_decode_aux.cases
tff(fact_4458_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_4459_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_4460_numeral__One,axiom,
    aa_num_Code_natural(numera5444537566228673987atural,one) = one_one_Code_natural ).

% numeral_One
tff(fact_4461_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_4462_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_4463_numeral__One,axiom,
    aa_num_Code_integer(numera6620942414471956472nteger,one) = one_one_Code_integer ).

% numeral_One
tff(fact_4464_numeral__One,axiom,
    numeral_numeral_rat(one) = one_one_rat ).

% numeral_One
tff(fact_4465_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_4466_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N),aa_num_num(inc,aa_nat_num(num_of_nat,N)),one) ).

% num_of_nat.simps(2)
tff(fact_4467_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_4468_uminus__numeral__One,axiom,
    aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% uminus_numeral_One
tff(fact_4469_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_4470_uminus__numeral__One,axiom,
    aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(one)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% uminus_numeral_One
tff(fact_4471_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_4472_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_4473_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_num_Code_integer(numera6620942414471956472nteger,one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_4474_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),one))),N) ).

% Suc_nat_number_of_add
tff(fact_4475_num__of__nat__One,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),one_one_nat)
     => ( aa_nat_num(num_of_nat,N) = one ) ) ).

% num_of_nat_One
tff(fact_4476_numeral__inc,axiom,
    ! [Xa: num] : aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(inc,Xa)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_num_Code_natural(numera5444537566228673987atural,Xa)),one_one_Code_natural) ).

% numeral_inc
tff(fact_4477_numeral__inc,axiom,
    ! [Xa: num] : aa_num_int(numeral_numeral_int,aa_num_num(inc,Xa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xa)),one_one_int) ).

% numeral_inc
tff(fact_4478_numeral__inc,axiom,
    ! [Xa: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(inc,Xa)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xa)),one_one_nat) ).

% numeral_inc
tff(fact_4479_numeral__inc,axiom,
    ! [Xa: num] : aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,Xa)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa)),one_one_Code_integer) ).

% numeral_inc
tff(fact_4480_numeral__inc,axiom,
    ! [Xa: num] : numeral_numeral_rat(aa_num_num(inc,Xa)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(Xa)),one_one_rat) ).

% numeral_inc
tff(fact_4481_prod__decode__aux_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y3: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(Xa),Xaa) = Y3 )
     => ( Y3 = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xa),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Xaa)),aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xa)))) ) ) ).

% prod_decode_aux.elims
tff(fact_4482_pochhammer__double,axiom,
    ! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_4483_even__succ__mod__exp,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),A),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),modulo8411746178871703098atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_4484_even__succ__mod__exp,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_4485_even__succ__mod__exp,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_4486_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_4487_even__succ__div__exp,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_4488_even__succ__div__exp,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_4489_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_4490_even__succ__div__exp,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),A),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)) = divide5121882707175180666atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_4491_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_4492_fact__double,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_rat(comm_s4028243227959126397er_rat(divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))),N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% fact_double
tff(fact_4493_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4494_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4495_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4496_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_natural,M: nat,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M)),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N) = zero_z2226904508553997617atural )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),divide5121882707175180666atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4497_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)))
         => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4498_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4499_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4500_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( aa_num_num(bit0,X2) = aa_num_num(bit0,Y2) )
    <=> ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_4501_one__add__one,axiom,
    aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),one_one_Code_natural) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_4502_one__add__one,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_4503_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_4504_one__add__one,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_4505_one__add__one,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_4506_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_4507_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_4508_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))
    <=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_4509_even__add,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_4510_even__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_4511_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_4512_one__mod__two__eq__one,axiom,
    modulo8411746178871703098atural(one_one_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = one_one_Code_natural ).

% one_mod_two_eq_one
tff(fact_4513_one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_4514_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_4515_one__mod__two__eq__one,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_4516_bits__one__mod__two__eq__one,axiom,
    modulo8411746178871703098atural(one_one_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = one_one_Code_natural ).

% bits_one_mod_two_eq_one
tff(fact_4517_bits__one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_4518_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_4519_bits__one__mod__two__eq__one,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_4520_zero__eq__power2,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% zero_eq_power2
tff(fact_4521_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_4522_zero__eq__power2,axiom,
    ! [A: code_natural] :
      ( ( aa_nat_Code_natural(power_7079662738309270450atural(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z2226904508553997617atural )
    <=> ( A = zero_z2226904508553997617atural ) ) ).

% zero_eq_power2
tff(fact_4523_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_4524_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_4525_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc'
tff(fact_4526_add__2__eq__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc
tff(fact_4527_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% Suc_1
tff(fact_4528_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = M ).

% add_self_div_2
tff(fact_4529_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_4530_dbl__simps_I3_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu8804712462038260780nteger,one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_4531_dbl__simps_I3_J,axiom,
    neg_numeral_dbl_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_4532_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_4533_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_4534_bits__1__div__2,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% bits_1_div_2
tff(fact_4535_bits__1__div__2,axiom,
    divide5121882707175180666atural(one_one_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = zero_z2226904508553997617atural ).

% bits_1_div_2
tff(fact_4536_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_4537_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_4538_one__div__two__eq__zero,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% one_div_two_eq_zero
tff(fact_4539_one__div__two__eq__zero,axiom,
    divide5121882707175180666atural(one_one_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = zero_z2226904508553997617atural ).

% one_div_two_eq_zero
tff(fact_4540_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4541_add__neg__numeral__special_I9_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4542_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4543_diff__numeral__special_I11_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4544_diff__numeral__special_I11_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4545_diff__numeral__special_I11_J,axiom,
    minus_minus_rat(one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4546_diff__numeral__special_I10_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4547_diff__numeral__special_I10_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4548_diff__numeral__special_I10_J,axiom,
    minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat),one_one_rat) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4549_even__plus__one__iff,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),one_one_Code_natural))
    <=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_4550_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_4551_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_4552_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_4553_even__diff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ).

% even_diff
tff(fact_4554_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ).

% even_diff
tff(fact_4555_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) != one_one_Code_natural )
    <=> ( modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = zero_z2226904508553997617atural ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4556_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4557_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4558_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4559_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) != zero_z2226904508553997617atural )
    <=> ( modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = one_one_Code_natural ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4560_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4561_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4562_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4563_minus__1__div__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_4564_minus__1__div__2__eq,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_4565_power2__less__eq__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
tff(fact_4566_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_4567_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_4568_power2__eq__iff__nonneg,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y3)
       => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xa = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_4569_power2__eq__iff__nonneg,axiom,
    ! [Xa: code_natural,Y3: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),Xa)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),Y3)
       => ( ( aa_nat_Code_natural(power_7079662738309270450atural(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_natural(power_7079662738309270450atural(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xa = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_4570_power2__eq__iff__nonneg,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y3)
       => ( ( aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xa = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_4571_power2__eq__iff__nonneg,axiom,
    ! [Xa: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y3)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xa = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_4572_power2__eq__iff__nonneg,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
       => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xa = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_4573_zero__less__power2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
tff(fact_4574_zero__less__power2,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_4575_zero__less__power2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_4576_minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_4577_minus__1__mod__2__eq,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_4578_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_4579_bits__minus__1__mod__2__eq,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_4580_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,zero_n356916108424825756nteger((B))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_4581_of__bool__half__eq__0,axiom,
    ! [B: $o] : divide5121882707175180666atural(zero_n8403883297036319079atural((B)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = zero_z2226904508553997617atural ).

% of_bool_half_eq_0
tff(fact_4582_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_o_int(zero_n2684676970156552555ol_int,(B))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_4583_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_4584_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_4585_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_4586_dbl__simps_I4_J,axiom,
    aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4587_dbl__simps_I4_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu8804712462038260780nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4588_dbl__simps_I4_J,axiom,
    neg_numeral_dbl_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4589_even__succ__div__2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_4590_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_4591_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_4592_even__succ__div__2,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),A),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_4593_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_4594_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_4595_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_4596_odd__succ__div__two,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),one_one_Code_natural),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),one_one_Code_natural) ) ) ).

% odd_succ_div_two
tff(fact_4597_even__succ__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_4598_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_4599_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_4600_even__succ__div__two,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),one_one_Code_natural),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_4601_even__power,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% even_power
tff(fact_4602_even__power,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% even_power
tff(fact_4603_even__power,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% even_power
tff(fact_4604_zero__le__power__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_4605_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_4606_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_4607_power__less__zero__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W))),zero_z3403309356797280102nteger)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq_numeral
tff(fact_4608_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_4609_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_4610_power__less__zero__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),zero_z3403309356797280102nteger)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq
tff(fact_4611_power__less__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_4612_power__less__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_4613_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_int ).

% power_minus1_even
tff(fact_4614_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_4615_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_rat ).

% power_minus1_even
tff(fact_4616_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_4617_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_4618_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_4619_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_4620_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_4621_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_4622_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( 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_4623_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ) ) ).

% even_diff_nat
tff(fact_4624_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_4625_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_4626_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_4627_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_4628_odd__two__times__div__two__succ,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))))),one_one_Code_natural) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_4629_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N),one_one_Code_natural))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4630_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4631_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4632_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)),one_one_Code_integer))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4633_zero__less__power__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_4634_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_4635_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_4636_bits__1__div__exp,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(N = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4637_bits__1__div__exp,axiom,
    ! [N: nat] : divide5121882707175180666atural(one_one_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)) = zero_n8403883297036319079atural(N = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4638_bits__1__div__exp,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_o_int(zero_n2684676970156552555ol_int,N = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4639_bits__1__div__exp,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_o_nat(zero_n2687167440665602831ol_nat,N = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4640_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(N = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4641_one__div__2__pow__eq,axiom,
    ! [N: nat] : divide5121882707175180666atural(one_one_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)) = zero_n8403883297036319079atural(N = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4642_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_o_int(zero_n2684676970156552555ol_int,N = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4643_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_o_nat(zero_n2687167440665602831ol_nat,N = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4644_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_4645_power__le__zero__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W))),zero_z3403309356797280102nteger)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_4646_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_4647_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_4648_one__mod__2__pow__eq,axiom,
    ! [N: nat] : modulo8411746178871703098atural(one_one_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)) = zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4649_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4650_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4651_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4652_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one != aa_num_num(bit0,X2) ).

% verit_eq_simplify(10)
tff(fact_4653_numeral__Bit0,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_Bit0
tff(fact_4654_numeral__Bit0,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_Bit0
tff(fact_4655_numeral__Bit0,axiom,
    ! [N: num] : aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% numeral_Bit0
tff(fact_4656_numeral__Bit0,axiom,
    ! [N: num] : numeral_numeral_rat(aa_num_num(bit0,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(N)),numeral_numeral_rat(N)) ).

% numeral_Bit0
tff(fact_4657_numeral__code_I2_J,axiom,
    ! [N: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,N),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m) ) ).

% numeral_code(2)
tff(fact_4658_numeral__code_I2_J,axiom,
    ! [N: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,N),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m) ) ).

% numeral_code(2)
tff(fact_4659_numeral__code_I2_J,axiom,
    ! [N: num] :
      aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N)) = $let(
        m: code_integer,
        m:= aa_num_Code_integer(numera6620942414471956472nteger,N),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,m),m) ) ).

% numeral_code(2)
tff(fact_4660_numeral__code_I2_J,axiom,
    ! [N: num] :
      numeral_numeral_rat(aa_num_num(bit0,N)) = $let(
        m: rat,
        m:= numeral_numeral_rat(N),
        aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m),m) ) ).

% numeral_code(2)
tff(fact_4661_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4662_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4663_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = zero_z3403309356797280102nteger )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4664_num_Osize_I5_J,axiom,
    ! [X2: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_4665_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ).

% power_numeral_even
tff(fact_4666_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: int,
        w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ).

% power_numeral_even
tff(fact_4667_power__numeral__even,axiom,
    ! [Z: code_integer,W: num] :
      aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: code_integer,
        w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,w),w) ) ).

% power_numeral_even
tff(fact_4668_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,w),w) ) ).

% power_numeral_even
tff(fact_4669_left__add__twice,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_4670_left__add__twice,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_4671_left__add__twice,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_4672_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_4673_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2_right
tff(fact_4674_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_4675_mult__2__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),Z) ).

% mult_2_right
tff(fact_4676_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),numeral_numeral_rat(aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2_right
tff(fact_4677_mult__2,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2
tff(fact_4678_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_4679_mult__2,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Z) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),Z) ).

% mult_2
tff(fact_4680_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Z) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2
tff(fact_4681_even__zero,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),zero_z2226904508553997617atural) ).

% even_zero
tff(fact_4682_even__zero,axiom,
    aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int) ).

% even_zero
tff(fact_4683_even__zero,axiom,
    aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat) ).

% even_zero
tff(fact_4684_even__zero,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).

% even_zero
tff(fact_4685_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% odd_even_add
tff(fact_4686_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% odd_even_add
tff(fact_4687_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% odd_even_add
tff(fact_4688_odd__one,axiom,
    ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),one_one_Code_natural) ).

% odd_one
tff(fact_4689_odd__one,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int) ).

% odd_one
tff(fact_4690_odd__one,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat) ).

% odd_one
tff(fact_4691_odd__one,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),one_one_Code_integer) ).

% odd_one
tff(fact_4692_zero__power2,axiom,
    aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% zero_power2
tff(fact_4693_zero__power2,axiom,
    aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat ).

% zero_power2
tff(fact_4694_zero__power2,axiom,
    aa_nat_Code_natural(power_7079662738309270450atural(zero_z2226904508553997617atural),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z2226904508553997617atural ).

% zero_power2
tff(fact_4695_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% zero_power2
tff(fact_4696_zero__power2,axiom,
    aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int ).

% zero_power2
tff(fact_4697_numeral__2__eq__2,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_4698_one__power2,axiom,
    aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_power2
tff(fact_4699_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat ).

% one_power2
tff(fact_4700_one__power2,axiom,
    aa_nat_Code_natural(power_7079662738309270450atural(one_one_Code_natural),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_natural ).

% one_power2
tff(fact_4701_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_power2
tff(fact_4702_one__power2,axiom,
    aa_nat_int(aa_int_fun_nat_int(power_power_int,one_one_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int ).

% one_power2
tff(fact_4703_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% nat_1_add_1
tff(fact_4704_num__of__nat__double,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N)) ) ) ).

% num_of_nat_double
tff(fact_4705_half__gt__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A,numeral_numeral_rat(aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_4706_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A,numeral_numeral_rat(aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% half_gt_zero_iff
tff(fact_4707_zero__le__power2,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_4708_zero__le__power2,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_4709_zero__le__power2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_4710_power2__eq__imp__eq,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y3)
         => ( Xa = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_4711_power2__eq__imp__eq,axiom,
    ! [Xa: code_natural,Y3: code_natural] :
      ( ( aa_nat_Code_natural(power_7079662738309270450atural(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_natural(power_7079662738309270450atural(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),Xa)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),Y3)
         => ( Xa = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_4712_power2__eq__imp__eq,axiom,
    ! [Xa: rat,Y3: rat] :
      ( ( aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y3)
         => ( Xa = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_4713_power2__eq__imp__eq,axiom,
    ! [Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y3)
         => ( Xa = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_4714_power2__eq__imp__eq,axiom,
    ! [Xa: int,Y3: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
         => ( Xa = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_4715_power2__le__imp__le,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y3)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y3) ) ) ).

% power2_le_imp_le
tff(fact_4716_power2__le__imp__le,axiom,
    ! [Xa: code_natural,Y3: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_natural(power_7079662738309270450atural(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),Y3)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,Xa),Y3) ) ) ).

% power2_le_imp_le
tff(fact_4717_power2__le__imp__le,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y3) ) ) ).

% power2_le_imp_le
tff(fact_4718_power2__le__imp__le,axiom,
    ! [Xa: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y3) ) ) ).

% power2_le_imp_le
tff(fact_4719_power2__le__imp__le,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y3) ) ) ).

% power2_le_imp_le
tff(fact_4720_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
    <=> ( modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = zero_z2226904508553997617atural ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_4721_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_4722_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_4723_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_4724_power2__less__0,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_z3403309356797280102nteger) ).

% power2_less_0
tff(fact_4725_power2__less__0,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_4726_power2__less__0,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int) ).

% power2_less_0
tff(fact_4727_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z2226904508553997617atural )
     => ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M) != zero_z2226904508553997617atural ) ) ).

% exp_add_not_zero_imp_left
tff(fact_4728_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_4729_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_4730_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
tff(fact_4731_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z2226904508553997617atural )
     => ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N) != zero_z2226904508553997617atural ) ) ).

% exp_add_not_zero_imp_right
tff(fact_4732_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_4733_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_4734_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
tff(fact_4735_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
    <=> ( modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = one_one_Code_natural ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_4736_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_4737_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_4738_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_4739_sum__power2__eq__zero__iff,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger )
    <=> ( ( Xa = zero_z3403309356797280102nteger )
        & ( Y3 = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_4740_sum__power2__eq__zero__iff,axiom,
    ! [Xa: rat,Y3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( Xa = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_4741_sum__power2__eq__zero__iff,axiom,
    ! [Xa: int,Y3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( Xa = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_4742_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N) != zero_z2226904508553997617atural )
     => ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_z2226904508553997617atural ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4743_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4744_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4745_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4746_div__exp__eq,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_4747_div__exp__eq,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_4748_div__exp__eq,axiom,
    ! [A: code_integer,M: nat,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_4749_div__exp__eq,axiom,
    ! [A: code_natural,M: nat,N: nat] : divide5121882707175180666atural(divide5121882707175180666atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M)),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)) = divide5121882707175180666atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_4750_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( ( A = one_one_int )
        | ( A = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_4751_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( A = one_one_Code_integer )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_4752_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_4753_less__2__cases,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_4754_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_4755_abs__square__eq__1,axiom,
    ! [Xa: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xa) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_4756_abs__square__eq__1,axiom,
    ! [Xa: rat] :
      ( ( aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( abs_abs_rat(Xa) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_4757_abs__square__eq__1,axiom,
    ! [Xa: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,Xa) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_4758_odd__pos,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% odd_pos
tff(fact_4759_nat__2,axiom,
    aa_int_nat(nat2,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_4760_nat__induct2,axiom,
    ! [Pa: fun_nat_o,N: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( aa_nat_o(Pa,one_one_nat)
       => ( ! [N2: nat] :
              ( aa_nat_o(Pa,N2)
             => aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(Pa,N) ) ) ) ).

% nat_induct2
tff(fact_4761_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4762_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4763_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4764_oddE,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ~ ! [B2: code_natural] : A != aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),B2)),one_one_Code_natural) ) ).

% oddE
tff(fact_4765_oddE,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B2: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),one_one_int) ) ).

% oddE
tff(fact_4766_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)),one_one_nat) ) ).

% oddE
tff(fact_4767_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ~ ! [B2: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B2)),one_one_Code_integer) ) ).

% oddE
tff(fact_4768_power2__less__imp__less,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y3)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y3) ) ) ).

% power2_less_imp_less
tff(fact_4769_power2__less__imp__less,axiom,
    ! [Xa: code_natural,Y3: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_natural(power_7079662738309270450atural(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),Y3)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Xa),Y3) ) ) ).

% power2_less_imp_less
tff(fact_4770_power2__less__imp__less,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y3) ) ) ).

% power2_less_imp_less
tff(fact_4771_power2__less__imp__less,axiom,
    ! [Xa: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y3) ) ) ).

% power2_less_imp_less
tff(fact_4772_power2__less__imp__less,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y3) ) ) ).

% power2_less_imp_less
tff(fact_4773_sum__power2__ge__zero,axiom,
    ! [Xa: code_integer,Y3: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_4774_sum__power2__ge__zero,axiom,
    ! [Xa: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_4775_sum__power2__ge__zero,axiom,
    ! [Xa: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_4776_sum__power2__le__zero__iff,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger)
    <=> ( ( Xa = zero_z3403309356797280102nteger )
        & ( Y3 = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_4777_sum__power2__le__zero__iff,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( Xa = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_4778_sum__power2__le__zero__iff,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( Xa = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_4779_parity__cases,axiom,
    ! [A: code_natural] :
      ( ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
       => ( modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) != zero_z2226904508553997617atural ) )
     => ~ ( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
         => ( modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) != one_one_Code_natural ) ) ) ).

% parity_cases
tff(fact_4780_parity__cases,axiom,
    ! [A: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_4781_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_4782_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
         => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_4783_mod2__eq__if,axiom,
    ! [A: code_natural] :
      modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = $ite(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A),zero_z2226904508553997617atural,one_one_Code_natural) ).

% mod2_eq_if
tff(fact_4784_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_4785_mod2__eq__if,axiom,
    ! [A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_4786_mod2__eq__if,axiom,
    ! [A: code_integer] :
      aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_4787_bits__stable__imp__add__self,axiom,
    ! [A: code_natural] :
      ( ( divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = A )
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))) = zero_z2226904508553997617atural ) ) ).

% bits_stable_imp_add_self
tff(fact_4788_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_4789_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_4790_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_4791_not__sum__power2__lt__zero,axiom,
    ! [Xa: code_integer,Y3: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger) ).

% not_sum_power2_lt_zero
tff(fact_4792_not__sum__power2__lt__zero,axiom,
    ! [Xa: rat,Y3: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_4793_not__sum__power2__lt__zero,axiom,
    ! [Xa: int,Y3: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_4794_sum__power2__gt__zero__iff,axiom,
    ! [Xa: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xa != zero_z3403309356797280102nteger )
        | ( Y3 != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_4795_sum__power2__gt__zero__iff,axiom,
    ! [Xa: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xa != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_4796_sum__power2__gt__zero__iff,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xa != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_4797_power2__sum,axiom,
    ! [Xa: int,Y3: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa)),Y3)) ).

% power2_sum
tff(fact_4798_power2__sum,axiom,
    ! [Xa: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xa)),Y3)) ).

% power2_sum
tff(fact_4799_power2__sum,axiom,
    ! [Xa: code_integer,Y3: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Xa),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Xa)),Y3)) ).

% power2_sum
tff(fact_4800_power2__sum,axiom,
    ! [Xa: rat,Y3: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Xa)),Y3)) ).

% power2_sum
tff(fact_4801_square__le__1,axiom,
    ! [Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Xa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_4802_square__le__1,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_4803_square__le__1,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_4804_zero__le__even__power_H,axiom,
    ! [A: code_integer,N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% zero_le_even_power'
tff(fact_4805_zero__le__even__power_H,axiom,
    ! [A: rat,N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% zero_le_even_power'
tff(fact_4806_zero__le__even__power_H,axiom,
    ! [A: int,N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% zero_le_even_power'
tff(fact_4807_power2__le__iff__abs__le,axiom,
    ! [Y3: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xa)),Y3) ) ) ).

% power2_le_iff_abs_le
tff(fact_4808_power2__le__iff__abs__le,axiom,
    ! [Y3: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Xa)),Y3) ) ) ).

% power2_le_iff_abs_le
tff(fact_4809_power2__le__iff__abs__le,axiom,
    ! [Y3: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xa)),Y3) ) ) ).

% power2_le_iff_abs_le
tff(fact_4810_zero__le__even__power,axiom,
    ! [N: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ).

% zero_le_even_power
tff(fact_4811_zero__le__even__power,axiom,
    ! [N: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)) ) ).

% zero_le_even_power
tff(fact_4812_zero__le__even__power,axiom,
    ! [N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)) ) ).

% zero_le_even_power
tff(fact_4813_zero__le__odd__power,axiom,
    ! [N: nat,A: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ).

% zero_le_odd_power
tff(fact_4814_zero__le__odd__power,axiom,
    ! [N: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_4815_zero__le__odd__power,axiom,
    ! [N: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_4816_zero__le__power__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_le_power_eq
tff(fact_4817_zero__le__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_4818_zero__le__power__eq,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_4819_abs__square__le__1,axiom,
    ! [Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xa)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_4820_abs__square__le__1,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Xa)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_4821_abs__square__le__1,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xa)),one_one_int) ) ).

% abs_square_le_1
tff(fact_4822_abs__square__less__1,axiom,
    ! [Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xa)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_4823_abs__square__less__1,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(Xa)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_4824_abs__square__less__1,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xa)),one_one_int) ) ).

% abs_square_less_1
tff(fact_4825_bits__induct,axiom,
    ! [Pa: fun_Code_integer_o,A: code_integer] :
      ( ! [A2: code_integer] :
          ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A2),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A2 )
         => aa_Code_integer_o(Pa,A2) )
     => ( ! [A2: code_integer,B2: $o] :
            ( aa_Code_integer_o(Pa,A2)
           => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_n356916108424825756nteger((B2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A2))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A2 )
             => aa_Code_integer_o(Pa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_n356916108424825756nteger((B2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A2))) ) )
       => aa_Code_integer_o(Pa,A) ) ) ).

% bits_induct
tff(fact_4826_bits__induct,axiom,
    ! [Pa: fun_Code_natural_o,A: code_natural] :
      ( ! [A2: code_natural] :
          ( ( divide5121882707175180666atural(A2,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = A2 )
         => aa_Code_natural_o(Pa,A2) )
     => ( ! [A2: code_natural,B2: $o] :
            ( aa_Code_natural_o(Pa,A2)
           => ( ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,zero_n8403883297036319079atural((B2))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A2)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = A2 )
             => aa_Code_natural_o(Pa,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,zero_n8403883297036319079atural((B2))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A2))) ) )
       => aa_Code_natural_o(Pa,A) ) ) ).

% bits_induct
tff(fact_4827_bits__induct,axiom,
    ! [Pa: fun_int_o,A: int] :
      ( ! [A2: int] :
          ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A2 )
         => aa_int_o(Pa,A2) )
     => ( ! [A2: int,B2: $o] :
            ( aa_int_o(Pa,A2)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A2 )
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))) ) )
       => aa_int_o(Pa,A) ) ) ).

% bits_induct
tff(fact_4828_bits__induct,axiom,
    ! [Pa: fun_nat_o,A: nat] :
      ( ! [A2: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A2 )
         => aa_nat_o(Pa,A2) )
     => ( ! [A2: nat,B2: $o] :
            ( aa_nat_o(Pa,A2)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A2 )
             => aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))) ) )
       => aa_nat_o(Pa,A) ) ) ).

% bits_induct
tff(fact_4829_div__exp__mod__exp__eq,axiom,
    ! [A: code_natural,N: nat,M: nat] : modulo8411746178871703098atural(divide5121882707175180666atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M)) = divide5121882707175180666atural(modulo8411746178871703098atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_4830_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_4831_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_4832_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_4833_nat__bit__induct,axiom,
    ! [Pa: fun_nat_o,N: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(Pa,N2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ) )
       => ( ! [N2: nat] :
              ( aa_nat_o(Pa,N2)
             => aa_nat_o(Pa,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) )
         => aa_nat_o(Pa,N) ) ) ) ).

% nat_bit_induct
tff(fact_4834_minus__one__power__iff,axiom,
    ! [N: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_4835_minus__one__power__iff,axiom,
    ! [N: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_4836_minus__one__power__iff,axiom,
    ! [N: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% minus_one_power_iff
tff(fact_4837_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_4838_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_4839_fact__code,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_nat)) ).

% fact_code
tff(fact_4840_fact__code,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_nat)) ).

% fact_code
tff(fact_4841_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4842_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4843_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4844_power2__diff,axiom,
    ! [Xa: int,Y3: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa)),Y3)) ).

% power2_diff
tff(fact_4845_power2__diff,axiom,
    ! [Xa: code_integer,Y3: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xa),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Xa)),Y3)) ).

% power2_diff
tff(fact_4846_power2__diff,axiom,
    ! [Xa: rat,Y3: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(Xa,Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Xa)),Y3)) ).

% power2_diff
tff(fact_4847_odd__0__le__power__imp__0__le,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_4848_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_4849_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_4850_odd__power__less__zero,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_z3403309356797280102nteger) ) ).

% odd_power_less_zero
tff(fact_4851_odd__power__less__zero,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_4852_odd__power__less__zero,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_4853_zero__less__power__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))
    <=> ( ( N = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_less_power_eq
tff(fact_4854_zero__less__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))
    <=> ( ( N = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_4855_zero__less__power__eq,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N))
    <=> ( ( N = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_4856_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_4857_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_4858_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_4859_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_4860_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_4861_mod__double__modulus,axiom,
    ! [M: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Xa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Xa),M) )
          | ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Xa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Xa),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_4862_mod__double__modulus,axiom,
    ! [M: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_4863_mod__double__modulus,axiom,
    ! [M: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),M) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_4864_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)))
         => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4865_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4866_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4867_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% even_mask_div_iff'
tff(fact_4868_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% even_mask_div_iff'
tff(fact_4869_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),one_one_Code_integer)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% even_mask_div_iff'
tff(fact_4870_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),divide5121882707175180666atural(minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M),one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% even_mask_div_iff'
tff(fact_4871_power__le__zero__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),zero_z3403309356797280102nteger)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_4872_power__le__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_4873_power__le__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_4874_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% even_mod_4_div_2
tff(fact_4875_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
    <=> ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% even_mask_div_iff
tff(fact_4876_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% even_mask_div_iff
tff(fact_4877_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),one_one_Code_integer)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% even_mask_div_iff
tff(fact_4878_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),divide5121882707175180666atural(minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M),one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)))
    <=> ( ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N) = zero_z2226904508553997617atural )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% even_mask_div_iff
tff(fact_4879_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          zero_n356916108424825756nteger(( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_4880_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      divide5121882707175180666atural(aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(zero_n8403883297036319079atural(( ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M) != zero_z2226904508553997617atural )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ))),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_4881_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_4882_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_4883_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_4884_divmod__step__eq,axiom,
    ! [L: num,Q2: code_integer,R: code_integer] :
      aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(L),produc1086072967326762835nteger(Q2,R)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,L)),R),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q2)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,R),aa_num_Code_integer(numera6620942414471956472nteger,L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q2),R)) ).

% divmod_step_eq
tff(fact_4885_divmod__step__eq,axiom,
    ! [L: num,Q2: nat,R: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R),aa_num_nat(numeral_numeral_nat,L))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q2)),R)) ).

% divmod_step_eq
tff(fact_4886_divmod__step__eq,axiom,
    ! [L: num,Q2: int,R: int] :
      aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R),aa_num_int(numeral_numeral_int,L))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),R)) ).

% divmod_step_eq
tff(fact_4887_flip__bit__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4888_flip__bit__0,axiom,
    ! [A: code_natural] : bit_se168947363167071951atural(zero_zero_nat,A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,zero_n8403883297036319079atural(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4889_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4890_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4891_set__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4892_set__bit__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4893_set__bit__0,axiom,
    ! [A: code_natural] : bit_se1617098188084679374atural(zero_zero_nat,A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4894_set__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4895_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_4896_unset__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_4897_unset__bit__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_4898_unset__bit__0,axiom,
    ! [A: code_natural] : bit_se7083795435491715335atural(zero_zero_nat,A) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_4899_set__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se1617098188084679374atural(aa_nat_nat(suc,N),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),bit_se1617098188084679374atural(N,divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4900_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4901_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4902_set__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4903_flip__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se168947363167071951atural(aa_nat_nat(suc,N),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),bit_se168947363167071951atural(N,divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4904_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4905_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4906_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4907_unset__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se7083795435491715335atural(aa_nat_nat(suc,N),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),bit_se7083795435491715335atural(N,divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4908_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4909_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4910_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4911_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_4912_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(M),A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_4913_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_4914_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_4915_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(M),A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_4916_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_4917_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_4918_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_4919_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(M),A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_4920_signed__take__bit__rec,axiom,
    ! [N: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),A) = $ite(N = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_4921_signed__take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),A) = $ite(N = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_4922_round__unique,axiom,
    ! [Xa: rat,Y3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(Xa,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y3))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))))
       => ( archim7778729529865785530nd_rat(Xa) = Y3 ) ) ) ).

% round_unique
tff(fact_4923_round__altdef,axiom,
    ! [Xa: rat] :
      archim7778729529865785530nd_rat(Xa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))),archimedean_frac_rat(Xa)),archim2889992004027027881ng_rat(Xa),archim3151403230148437115or_rat(Xa)) ).

% round_altdef
tff(fact_4924_binomial__code,axiom,
    ! [N: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,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_4925_round__unique_H,axiom,
    ! [Xa: rat,N: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(minus_minus_rat(Xa,aa_int_rat(ring_1_of_int_rat,N)))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))
     => ( archim7778729529865785530nd_rat(Xa) = N ) ) ).

% round_unique'
tff(fact_4926_of__int__round__abs__le,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xa)),Xa))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_4927_of__int__round__gt,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(Xa,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xa))) ).

% of_int_round_gt
tff(fact_4928_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_4929_signed__take__bit__of__0,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% signed_take_bit_of_0
tff(fact_4930_binomial__n__n,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),N) = one_one_nat ).

% binomial_n_n
tff(fact_4931_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% signed_take_bit_of_minus_1
tff(fact_4932_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_4933_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_4934_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_4935_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_4936_signed__take__bit__Suc__1,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(suc,N)),one_one_Code_integer) = one_one_Code_integer ).

% signed_take_bit_Suc_1
tff(fact_4937_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K) ) ).

% binomial_eq_0_iff
tff(fact_4938_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_4939_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,K)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_4940_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_num_nat(numeral_numeral_nat,K)),one_one_Code_integer) = one_one_Code_integer ).

% signed_take_bit_numeral_of_1
tff(fact_4941_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_4942_round__0,axiom,
    archim7778729529865785530nd_rat(zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_4943_round__1,axiom,
    archim7778729529865785530nd_rat(one_one_rat) = one_one_int ).

% round_1
tff(fact_4944_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) ) ).

% zero_less_binomial_iff
tff(fact_4945_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_4946_signed__take__bit__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_4947_choose__one,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),one_one_nat) = N ).

% choose_one
tff(fact_4948_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_4949_choose__mult__lemma,axiom,
    ! [M: nat,R: 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),R)),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),R)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),M)) ).

% choose_mult_lemma
tff(fact_4950_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) ) ).

% zero_less_binomial
tff(fact_4951_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),A)) ).

% Suc_times_binomial_add
tff(fact_4952_binomial__absorb__comp,axiom,
    ! [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,N),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)),K)) ).

% binomial_absorb_comp
tff(fact_4953_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_4954_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_4955_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,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_4956_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_4957_choose__two,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_4958_round__def,axiom,
    ! [Xa: rat] : archim7778729529865785530nd_rat(Xa) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).

% round_def
tff(fact_4959_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_4960_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_4961_of__int__round__le,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xa))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_4962_of__int__round__ge,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(Xa,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xa))) ).

% of_int_round_ge
tff(fact_4963_set__decode__0,axiom,
    ! [Xa: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),aa_nat_set_nat(nat_set_decode,Xa))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xa) ) ).

% set_decode_0
tff(fact_4964_set__decode__def,axiom,
    ! [Xa: nat] : aa_nat_set_nat(nat_set_decode,Xa) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cz,Xa)) ).

% set_decode_def
tff(fact_4965_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(L),Qr) = aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_da(L)),Qr) ).

% divmod_step_def
tff(fact_4966_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_db(L)),Qr) ).

% divmod_step_def
tff(fact_4967_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_dc(L)),Qr) ).

% divmod_step_def
tff(fact_4968_take__bit__rec,axiom,
    ! [N: nat,A: code_natural] :
      bit_se569199155075624693atural(N,A) = $ite(N = zero_zero_nat,zero_z2226904508553997617atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(bit_se569199155075624693atural(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))))),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_4969_take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),A) = $ite(N = zero_zero_nat,zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_4970_take__bit__rec,axiom,
    ! [N: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = $ite(N = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_4971_take__bit__rec,axiom,
    ! [N: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = $ite(N = zero_zero_nat,zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(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(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_4972_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : aa_num_nat(size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_4973_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      unique3479559517661332726nteger(M,N) = $ite(aa_num_o(ord_less_num(M),N),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(N),unique3479559517661332726nteger(M,aa_num_num(bit0,N)))) ).

% divmod_divmod_step
tff(fact_4974_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      unique5055182867167087721od_nat(M,N) = $ite(aa_num_o(ord_less_num(M),N),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)),unique5026877609467782581ep_nat(N,unique5055182867167087721od_nat(M,aa_num_num(bit0,N)))) ).

% divmod_divmod_step
tff(fact_4975_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      unique5052692396658037445od_int(M,N) = $ite(aa_num_o(ord_less_num(M),N),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,M)),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(N),unique5052692396658037445od_int(M,aa_num_num(bit0,N)))) ).

% divmod_divmod_step
tff(fact_4976_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% Suc_0_xor_eq
tff(fact_4977_take__bit__of__0,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% take_bit_of_0
tff(fact_4978_take__bit__of__0,axiom,
    ! [N: nat] : bit_se569199155075624693atural(N,zero_z2226904508553997617atural) = zero_z2226904508553997617atural ).

% take_bit_of_0
tff(fact_4979_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_4980_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_4981_bit_Oxor__self,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xa),Xa) = zero_zero_int ).

% bit.xor_self
tff(fact_4982_bit_Oxor__self,axiom,
    ! [Xa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,Xa),Xa) = zero_z3403309356797280102nteger ).

% bit.xor_self
tff(fact_4983_xor__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),A) = zero_zero_int ).

% xor_self_eq
tff(fact_4984_xor__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,A),A) = zero_z3403309356797280102nteger ).

% xor_self_eq
tff(fact_4985_xor__self__eq,axiom,
    ! [A: code_natural] : bit_se2046307713759805098atural(A,A) = zero_z2226904508553997617atural ).

% xor_self_eq
tff(fact_4986_xor__self__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),A) = zero_zero_nat ).

% xor_self_eq
tff(fact_4987_xor_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,zero_zero_int),A) = A ).

% xor.left_neutral
tff(fact_4988_xor_Oleft__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,zero_z3403309356797280102nteger),A) = A ).

% xor.left_neutral
tff(fact_4989_xor_Oleft__neutral,axiom,
    ! [A: code_natural] : bit_se2046307713759805098atural(zero_z2226904508553997617atural,A) = A ).

% xor.left_neutral
tff(fact_4990_xor_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,zero_zero_nat),A) = A ).

% xor.left_neutral
tff(fact_4991_xor_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),zero_zero_int) = A ).

% xor.right_neutral
tff(fact_4992_xor_Oright__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,A),zero_z3403309356797280102nteger) = A ).

% xor.right_neutral
tff(fact_4993_xor_Oright__neutral,axiom,
    ! [A: code_natural] : bit_se2046307713759805098atural(A,zero_z2226904508553997617atural) = A ).

% xor.right_neutral
tff(fact_4994_xor_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),zero_zero_nat) = A ).

% xor.right_neutral
tff(fact_4995_case__prod__conv,axiom,
    ! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,A),B) ).

% case_prod_conv
tff(fact_4996_case__prod__conv,axiom,
    ! [F: fun_na5914473681949881652_nat_o,A: nat,B: nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,A),B) ).

% case_prod_conv
tff(fact_4997_case__prod__conv,axiom,
    ! [F: fun_in1696471567231710611nt_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,A),B) ).

% case_prod_conv
tff(fact_4998_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prod_conv
tff(fact_4999_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_int,A: int,B: int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int_int(aa_int_fun_int_int(F,A),B) ).

% case_prod_conv
tff(fact_5000_case__prod__curry,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,produc4701859392678344816at_nat(F)) = F ).

% case_prod_curry
tff(fact_5001_case__prod__curry,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,produc2062876521443793799_nat_o(F)) = F ).

% case_prod_curry
tff(fact_5002_case__prod__curry,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,produc8249235968001453780nt_int(F)) = F ).

% case_prod_curry
tff(fact_5003_case__prod__curry,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,produc175634133007206835_int_o(F)) = F ).

% case_prod_curry
tff(fact_5004_case__prod__curry,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,produc1016772743285680337nt_int(F)) = F ).

% case_prod_curry
tff(fact_5005_curry__case__prod,axiom,
    ! [F: fun_na1668100129766719005at_nat] : produc4701859392678344816at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F)) = F ).

% curry_case_prod
tff(fact_5006_curry__case__prod,axiom,
    ! [F: fun_na5914473681949881652_nat_o] : produc2062876521443793799_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F)) = F ).

% curry_case_prod
tff(fact_5007_curry__case__prod,axiom,
    ! [F: fun_in1696471567231710611nt_int] : produc8249235968001453780nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F)) = F ).

% curry_case_prod
tff(fact_5008_curry__case__prod,axiom,
    ! [F: fun_int_fun_int_o] : produc175634133007206835_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F)) = F ).

% curry_case_prod
tff(fact_5009_curry__case__prod,axiom,
    ! [F: fun_int_fun_int_int] : produc1016772743285680337nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F)) = F ).

% curry_case_prod
tff(fact_5010_take__bit__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(zero_zero_nat),A) = zero_z3403309356797280102nteger ).

% take_bit_0
tff(fact_5011_take__bit__0,axiom,
    ! [A: code_natural] : bit_se569199155075624693atural(zero_zero_nat,A) = zero_z2226904508553997617atural ).

% take_bit_0
tff(fact_5012_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_5013_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_5014_take__bit__Suc__1,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),one_one_Code_integer) = one_one_Code_integer ).

% take_bit_Suc_1
tff(fact_5015_take__bit__Suc__1,axiom,
    ! [N: nat] : bit_se569199155075624693atural(aa_nat_nat(suc,N),one_one_Code_natural) = one_one_Code_natural ).

% take_bit_Suc_1
tff(fact_5016_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_5017_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_5018_take__bit__numeral__1,axiom,
    ! [L: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,L)),one_one_Code_integer) = one_one_Code_integer ).

% take_bit_numeral_1
tff(fact_5019_take__bit__numeral__1,axiom,
    ! [L: num] : bit_se569199155075624693atural(aa_num_nat(numeral_numeral_nat,L),one_one_Code_natural) = one_one_Code_natural ).

% take_bit_numeral_1
tff(fact_5020_take__bit__numeral__1,axiom,
    ! [L: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),one_one_nat) = one_one_nat ).

% take_bit_numeral_1
tff(fact_5021_take__bit__numeral__1,axiom,
    ! [L: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,L)),one_one_int) = one_one_int ).

% take_bit_numeral_1
tff(fact_5022_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),one_one_Code_integer) = zero_z3403309356797280102nteger )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_5023_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( bit_se569199155075624693atural(N,one_one_Code_natural) = zero_z2226904508553997617atural )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_5024_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_5025_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_5026_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_Suc_0
tff(fact_5027_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5052692396658037445od_int(M,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,M)),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_5028_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_5029_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique3479559517661332726nteger(M,one) = produc1086072967326762835nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_5030_take__bit__of__1,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),one_one_Code_integer) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_5031_take__bit__of__1,axiom,
    ! [N: nat] : bit_se569199155075624693atural(N,one_one_Code_natural) = zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_5032_take__bit__of__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_5033_take__bit__of__1,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_5034_even__take__bit__eq,axiom,
    ! [N: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),A))
    <=> ( ( N = zero_zero_nat )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_5035_even__take__bit__eq,axiom,
    ! [N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A))
    <=> ( ( N = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_5036_even__take__bit__eq,axiom,
    ! [N: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A))
    <=> ( ( N = zero_zero_nat )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_5037_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_5038_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_5039_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(3)
tff(fact_5040_take__bit__Suc__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,zero_zero_nat)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_5041_take__bit__Suc__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,zero_zero_nat)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_5042_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_5043_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_dd),unique5052692396658037445od_int(M,N)) ).

% divmod_algorithm_code(5)
tff(fact_5044_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_de),unique5055182867167087721od_nat(M,N)) ).

% divmod_algorithm_code(5)
tff(fact_5045_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_df),unique3479559517661332726nteger(M,N)) ).

% divmod_algorithm_code(5)
tff(fact_5046_case__prod__app,axiom,
    ! [F: fun_na1668100129766719005at_nat,Xa: product_prod_nat_nat,Y3: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),Xa),Y3) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aa_Pro6895555690143131534at_nat(aTP_Lamm_dg(F),Y3)),Xa) ).

% case_prod_app
tff(fact_5047_case__prod__app,axiom,
    ! [F: fun_na5914473681949881652_nat_o,Xa: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),Xa),Y3)
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_Pro1957341097241355813_nat_o(aTP_Lamm_dh(F),Y3)),Xa) ) ).

% case_prod_app
tff(fact_5048_prod_Ocase__distrib,axiom,
    ! [Ha: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(Ha,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun6133342665515662012_int_o(aTP_Lamm_di(Ha),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5049_prod_Ocase__distrib,axiom,
    ! [Ha: fun_o_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_int(Ha,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun7323606162024495668nt_int(aTP_Lamm_dj(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_5050_prod_Ocase__distrib,axiom,
    ! [Ha: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(Ha,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun82482788810740534_int_o(aTP_Lamm_dk(Ha),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5051_prod_Ocase__distrib,axiom,
    ! [Ha: fun_int_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int_int(Ha,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun6119123525526402106nt_int(aTP_Lamm_dl(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_5052_prod_Ocase__distrib,axiom,
    ! [Ha: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Ha,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun4545072666309863515_int_o(aTP_Lamm_dm(Ha),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5053_prod_Ocase__distrib,axiom,
    ! [Ha: fun_Pr982841511892225912nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(Ha,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun7461304428735181397nt_int(aTP_Lamm_dn(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_5054_prod_Ocase__distrib,axiom,
    ! [Ha: fun_o_8988054265482332260nt_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_P3134438092330336605nt_int(Ha,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun4023131440832735197nt_int(aTP_Lamm_do(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_5055_prod_Ocase__distrib,axiom,
    ! [Ha: fun_in4211310926729683942nt_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(Ha,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun1034830265760468567nt_int(aTP_Lamm_dp(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_5056_prod_Ocase__distrib,axiom,
    ! [Ha: fun_Pr2478283820680522243nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(Ha,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun4152872853588681340nt_int(aTP_Lamm_dq(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_5057_prod_Ocase__distrib,axiom,
    ! [Ha: fun_fu3275390434952296369_nat_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_fun359341089595923442_nat_o(Ha,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),Prod)) = aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aa_fun824319262034445606_nat_o(aTP_Lamm_dr(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_5058_take__bit__add,axiom,
    ! [N: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% take_bit_add
tff(fact_5059_take__bit__add,axiom,
    ! [N: nat,A: int,B: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),B))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ).

% take_bit_add
tff(fact_5060_old_Oprod_Ocase,axiom,
    ! [F: fun_na1668100129766719005at_nat,X1: nat,X2: nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X2) ).

% old.prod.case
tff(fact_5061_old_Oprod_Ocase,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X1: nat,X2: nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X1),X2) ).

% old.prod.case
tff(fact_5062_old_Oprod_Ocase,axiom,
    ! [F: fun_in1696471567231710611nt_int,X1: int,X2: int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X2) ).

% old.prod.case
tff(fact_5063_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_o,X1: int,X2: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2))
    <=> aa_int_o(aa_int_fun_int_o(F,X1),X2) ) ).

% old.prod.case
tff(fact_5064_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_int,X1: int,X2: int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = aa_int_int(aa_int_fun_int_int(F,X1),X2) ).

% old.prod.case
tff(fact_5065_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_na1668100129766719005at_nat,G: fun_na1668100129766719005at_nat,P: product_prod_nat_nat] :
      ( ! [X4: nat,Y4: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4) = Q2 )
         => ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X4),Y4) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(G,X4),Y4) ) )
     => ( ( P = Q2 )
       => ( aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),P) = aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,G),Q2) ) ) ) ).

% split_cong
tff(fact_5066_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_na5914473681949881652_nat_o,G: fun_na5914473681949881652_nat_o,P: product_prod_nat_nat] :
      ( ! [X4: nat,Y4: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4) = Q2 )
         => ( aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X4),Y4) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(G,X4),Y4) ) )
     => ( ( P = Q2 )
       => ( aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),P) = aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,G),Q2) ) ) ) ).

% split_cong
tff(fact_5067_split__cong,axiom,
    ! [Q2: product_prod_int_int,F: fun_in1696471567231710611nt_int,G: fun_in1696471567231710611nt_int,P: product_prod_int_int] :
      ( ! [X4: int,Y4: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4) = Q2 )
         => ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X4),Y4) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(G,X4),Y4) ) )
     => ( ( P = Q2 )
       => ( aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),P) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,G),Q2) ) ) ) ).

% split_cong
tff(fact_5068_split__cong,axiom,
    ! [Q2: product_prod_int_int,F: fun_int_fun_int_o,G: fun_int_fun_int_o,P: product_prod_int_int] :
      ( ! [X4: int,Y4: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4) = Q2 )
         => ( aa_int_o(aa_int_fun_int_o(F,X4),Y4)
          <=> aa_int_o(aa_int_fun_int_o(G,X4),Y4) ) )
     => ( ( P = Q2 )
       => ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),P)
        <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,G),Q2) ) ) ) ).

% split_cong
tff(fact_5069_split__cong,axiom,
    ! [Q2: product_prod_int_int,F: fun_int_fun_int_int,G: fun_int_fun_int_int,P: product_prod_int_int] :
      ( ! [X4: int,Y4: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4) = Q2 )
         => ( aa_int_int(aa_int_fun_int_int(F,X4),Y4) = aa_int_int(aa_int_fun_int_int(G,X4),Y4) ) )
     => ( ( P = Q2 )
       => ( aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),P) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,G),Q2) ) ) ) ).

% split_cong
tff(fact_5070_case__prod__Pair__iden,axiom,
    ! [P: produc7509114972854644512nteger] : produc6766542753387345183nteger(produc7264323995540650962nteger,P) = P ).

% case_prod_Pair_iden
tff(fact_5071_case__prod__Pair__iden,axiom,
    ! [P: produc9140211490168418807atural] : produc237422109222252493atural(produc3399137497101226409atural,P) = P ).

% case_prod_Pair_iden
tff(fact_5072_case__prod__Pair__iden,axiom,
    ! [P: produc2453742332425399446nteger] : produc697581972998607883nteger(produc565448451787809096nteger,P) = P ).

% case_prod_Pair_iden
tff(fact_5073_case__prod__Pair__iden,axiom,
    ! [P: produc3043004577601930755nt_int] : produc6823905543215200229nt_int(produc7320969046608631221nt_int,P) = P ).

% case_prod_Pair_iden
tff(fact_5074_case__prod__Pair__iden,axiom,
    ! [P: produc4788131091073775919nt_int] : produc7102243260366339645nt_int(produc4729913460834754785nt_int,P) = P ).

% case_prod_Pair_iden
tff(fact_5075_case__prod__Pair__iden,axiom,
    ! [P: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,product_Pair_int_int),P) = P ).

% case_prod_Pair_iden
tff(fact_5076_case__prodE2,axiom,
    ! [Q: fun_fu8391177546919232046_nat_o,Pa: fun_na1668100129766719005at_nat,Z: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(Q,aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,Pa),Z))
     => ~ ! [X4: nat,Y4: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4) )
           => ~ aa_fun3538214411680374423_nat_o(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Pa,X4),Y4)) ) ) ).

% case_prodE2
tff(fact_5077_case__prodE2,axiom,
    ! [Q: fun_fu6891447142836604759at_o_o,Pa: fun_na5914473681949881652_nat_o,Z: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(Q,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Pa),Z))
     => ~ ! [X4: nat,Y4: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4) )
           => ~ aa_fun8408151826566693824at_o_o(Q,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Pa,X4),Y4)) ) ) ).

% case_prodE2
tff(fact_5078_case__prodE2,axiom,
    ! [Q: fun_Pr3238374608701176162_int_o,Pa: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Q,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Pa),Z))
     => ~ ! [X4: int,Y4: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4) )
           => ~ aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Pa,X4),Y4)) ) ) ).

% case_prodE2
tff(fact_5079_case__prodE2,axiom,
    ! [Q: fun_o_o,Pa: fun_int_fun_int_o,Z: product_prod_int_int] :
      ( aa_o_o(Q,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Pa),Z))
     => ~ ! [X4: int,Y4: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4) )
           => ~ aa_o_o(Q,aa_int_o(aa_int_fun_int_o(Pa,X4),Y4)) ) ) ).

% case_prodE2
tff(fact_5080_case__prodE2,axiom,
    ! [Q: fun_int_o,Pa: fun_int_fun_int_int,Z: product_prod_int_int] :
      ( aa_int_o(Q,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Pa),Z))
     => ~ ! [X4: int,Y4: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4) )
           => ~ aa_int_o(Q,aa_int_int(aa_int_fun_int_int(Pa,X4),Y4)) ) ) ).

% case_prodE2
tff(fact_5081_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_ds(F)) = F ).

% case_prod_eta
tff(fact_5082_case__prod__eta,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_dt(F)) = F ).

% case_prod_eta
tff(fact_5083_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_du(F)) = F ).

% case_prod_eta
tff(fact_5084_case__prod__eta,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_dv(F)) = F ).

% case_prod_eta
tff(fact_5085_case__prod__eta,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_dw(F)) = F ).

% case_prod_eta
tff(fact_5086_cond__case__prod__eta,axiom,
    ! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
      ( ! [X4: nat,Y4: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X4),Y4) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4))
     => ( aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5087_cond__case__prod__eta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,G: fun_Pr2028404973800837934_nat_o] :
      ( ! [X4: nat,Y4: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X4),Y4) = aa_Pro989595756709598047_nat_o(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4))
     => ( aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5088_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X4: int,Y4: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X4),Y4) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4))
     => ( aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5089_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X4: int,Y4: int] :
          ( aa_int_o(aa_int_fun_int_o(F,X4),Y4)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4)) )
     => ( aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5090_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_int,G: fun_Pr982841511892225912nt_int] :
      ( ! [X4: int,Y4: int] : aa_int_int(aa_int_fun_int_int(F,X4),Y4) = aa_Pro6799258434465214121nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4))
     => ( aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5091_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(L),Qr) = aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_da(L)),Qr) ).

% divmod_step_integer_def
tff(fact_5092_internal__case__prod__def,axiom,
    produc8104577556401574439at_nat = produc4528391901023682385at_nat ).

% internal_case_prod_def
tff(fact_5093_internal__case__prod__def,axiom,
    produc9189008374811844670_nat_o = produc6395135316500328296_nat_o ).

% internal_case_prod_def
tff(fact_5094_internal__case__prod__def,axiom,
    produc297006045350968285nt_int = produc4245557441103728435nt_int ).

% internal_case_prod_def
tff(fact_5095_internal__case__prod__def,axiom,
    produc8005341501107743676_int_o = produc4947309494688390418_int_o ).

% internal_case_prod_def
tff(fact_5096_internal__case__prod__def,axiom,
    produc7926200574084438792nt_int = produc8211389475949308722nt_int ).

% internal_case_prod_def
tff(fact_5097_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_5098_divmod__int__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_int_def
tff(fact_5099_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se569199155075624693atural(N,A) = zero_z2226904508553997617atural )
    <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)),A) ) ).

% take_bit_eq_0_iff
tff(fact_5100_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),A) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)),A) ) ).

% take_bit_eq_0_iff
tff(fact_5101_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 )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),A) ) ).

% take_bit_eq_0_iff
tff(fact_5102_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 )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),A) ) ).

% take_bit_eq_0_iff
tff(fact_5103_divmod__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_def
tff(fact_5104_divmod__def,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ).

% divmod_def
tff(fact_5105_divmod__def,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% divmod_def
tff(fact_5106_divmod_H__nat__def,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ).

% divmod'_nat_def
tff(fact_5107_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),one_one_Code_integer) ).

% take_bit_Suc_minus_1_eq
tff(fact_5108_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(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_5109_take__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se569199155075624693atural(aa_nat_nat(suc,N),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(bit_se569199155075624693atural(N,divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))))),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5110_take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5111_take__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5112_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(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5113_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,K)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K))),one_one_Code_integer) ).

% take_bit_numeral_minus_1_eq
tff(fact_5114_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,K)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K))),one_one_int) ).

% take_bit_numeral_minus_1_eq
tff(fact_5115_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_int_o(aa_int_fun_int_o(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(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% take_bit_int_less_eq
tff(fact_5116_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = $ite(
        M = zero_zero_nat,
        N,
        $ite(N = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_5117_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,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_5118_xor__one__eq,axiom,
    ! [A: code_natural] : bit_se2046307713759805098atural(A,one_one_Code_natural) = minus_7197305767214868737atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_n8403883297036319079atural(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))),zero_n8403883297036319079atural(~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_5119_xor__one__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,A),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)))),zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_5120_xor__one__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_5121_xor__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_5122_one__xor__eq,axiom,
    ! [A: code_natural] : bit_se2046307713759805098atural(one_one_Code_natural,A) = minus_7197305767214868737atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_n8403883297036319079atural(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))),zero_n8403883297036319079atural(~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_5123_one__xor__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,one_one_Code_integer),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)))),zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_5124_one__xor__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_5125_one__xor__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_5126_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A )
     => ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)),one_one_Code_integer)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5127_stable__imp__take__bit__eq,axiom,
    ! [A: code_natural,N: nat] :
      ( ( divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = A )
     => ( bit_se569199155075624693atural(N,A) = $ite(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A),zero_z2226904508553997617atural,minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N),one_one_Code_natural)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5128_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5129_stable__imp__take__bit__eq,axiom,
    ! [A: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5130_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_db(L)),Qr) ).

% divmod_step_nat_def
tff(fact_5131_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_dc(L)),Qr) ).

% divmod_step_int_def
tff(fact_5132_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% xor_Suc_0_eq
tff(fact_5133_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_option_num_int(case_option_int_num(zero_zero_int,aTP_Lamm_dx(M)),aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_5134_divmod__nat__if,axiom,
    ! [M: nat,N: nat] :
      divmod_nat(M,N) = $ite(
        ( ( N = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_dy),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ).

% divmod_nat_if
tff(fact_5135_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_dz),unique5052692396658037445od_int(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_5136_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ea),unique5055182867167087721od_nat(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_5137_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_eb),unique3479559517661332726nteger(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_5138_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(aa_num_num(bitM,M),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_5139_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = $ite(aa_num_o(ord_less_eq_num(M),N),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M))),aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(aa_num_num(bit1,N)),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ).

% divmod_algorithm_code(7)
tff(fact_5140_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N)) = $ite(aa_num_o(ord_less_eq_num(M),N),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ).

% divmod_algorithm_code(7)
tff(fact_5141_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = $ite(aa_num_o(ord_less_eq_num(M),N),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,N)),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ).

% divmod_algorithm_code(7)
tff(fact_5142_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = $ite(aa_num_o(ord_less_num(M),N),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M))),aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(aa_num_num(bit1,N)),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ).

% divmod_algorithm_code(8)
tff(fact_5143_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = $ite(aa_num_o(ord_less_num(M),N),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ).

% divmod_algorithm_code(8)
tff(fact_5144_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = $ite(aa_num_o(ord_less_num(M),N),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,N)),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ).

% divmod_algorithm_code(8)
tff(fact_5145_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% odd_mod_4_div_2
tff(fact_5146_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa_num_num(bit1,X32) = aa_num_num(bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_5147_case__prodI2,axiom,
    ! [P: produc7509114972854644512nteger,C: fun_fu3422731660242565741eger_o] :
      ( ! [A2: fun_Co3460408610320954864e_term,B2: produc8923325533196201883nteger] :
          ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A2),B2) )
         => aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(C,A2),B2) )
     => aa_Pro4652312591602923194eger_o(produc7733317944277263833eger_o(C),P) ) ).

% case_prodI2
tff(fact_5148_case__prodI2,axiom,
    ! [P: produc9140211490168418807atural,C: fun_fu3694549533497745878ural_o] :
      ( ! [A2: fun_Pr285225112257423215e_term,B2: produc7822875418678951345atural] :
          ( ( P = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A2),B2) )
         => aa_Pro6073791466322389475ural_o(aa_fun9089857049899810327ural_o(C,A2),B2) )
     => aa_Pro866224779355485219ural_o(produc3453235027721926722ural_o(C),P) ) ).

% case_prodI2
tff(fact_5149_case__prodI2,axiom,
    ! [P: produc2453742332425399446nteger,C: fun_fu7695273699059480439eger_o] :
      ( ! [A2: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] :
          ( ( P = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A2),B2) )
         => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(C,A2),B2) )
     => aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(C),P) ) ).

% case_prodI2
tff(fact_5150_case__prodI2,axiom,
    ! [P: produc3043004577601930755nt_int,C: fun_fu6865642474282033866_int_o] :
      ( ! [A2: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] :
          ( ( P = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A2),B2) )
         => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(C,A2),B2) )
     => aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(C),P) ) ).

% case_prodI2
tff(fact_5151_case__prodI2,axiom,
    ! [P: produc4788131091073775919nt_int,C: fun_fu2978539155105838238_int_o] :
      ( ! [A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A2),B2) )
         => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(C,A2),B2) )
     => aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(C),P) ) ).

% case_prodI2
tff(fact_5152_case__prodI2,axiom,
    ! [P: product_prod_int_int,C: fun_int_fun_int_o] :
      ( ! [A2: int,B2: int] :
          ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
         => aa_int_o(aa_int_fun_int_o(C,A2),B2) )
     => aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,C),P) ) ).

% case_prodI2
tff(fact_5153_case__prodI,axiom,
    ! [F: fun_fu3422731660242565741eger_o,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(F,A),B)
     => aa_Pro4652312591602923194eger_o(produc7733317944277263833eger_o(F),aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B)) ) ).

% case_prodI
tff(fact_5154_case__prodI,axiom,
    ! [F: fun_fu3694549533497745878ural_o,A: fun_Pr285225112257423215e_term,B: produc7822875418678951345atural] :
      ( aa_Pro6073791466322389475ural_o(aa_fun9089857049899810327ural_o(F,A),B)
     => aa_Pro866224779355485219ural_o(produc3453235027721926722ural_o(F),aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A),B)) ) ).

% case_prodI
tff(fact_5155_case__prodI,axiom,
    ! [F: fun_fu7695273699059480439eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(F,A),B)
     => aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(F),aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B)) ) ).

% case_prodI
tff(fact_5156_case__prodI,axiom,
    ! [F: fun_fu6865642474282033866_int_o,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(F,A),B)
     => aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(F),aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B)) ) ).

% case_prodI
tff(fact_5157_case__prodI,axiom,
    ! [F: fun_fu2978539155105838238_int_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(F,A),B)
     => aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(F),aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B)) ) ).

% case_prodI
tff(fact_5158_case__prodI,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(F,A),B)
     => aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% case_prodI
tff(fact_5159_mem__case__prodI2,axiom,
    ! [P: produc7509114972854644512nteger,Z: nat,C: fun_fu6383124537394581533et_nat] :
      ( ! [A2: fun_Co3460408610320954864e_term,B2: produc8923325533196201883nteger] :
          ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A2),B2) )
         => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro381016180347496421et_nat(aa_fun3554285010146031238et_nat(C,A2),B2)) )
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc7961109426542857093et_nat(C,P)) ) ).

% mem_case_prodI2
tff(fact_5160_mem__case__prodI2,axiom,
    ! [P: produc7509114972854644512nteger,Z: int,C: fun_fu1569576118606662777et_int] :
      ( ! [A2: fun_Co3460408610320954864e_term,B2: produc8923325533196201883nteger] :
          ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A2),B2) )
         => member_int(Z,aa_Pro5426537197693075521et_int(aa_fun1064201865214304482et_int(C,A2),B2)) )
     => member_int(Z,produc3783258407033660385et_int(C,P)) ) ).

% mem_case_prodI2
tff(fact_5161_mem__case__prodI2,axiom,
    ! [P: produc4788131091073775919nt_int,Z: nat,C: fun_fu9079770784915425324et_nat] :
      ( ! [A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A2),B2) )
         => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(C,A2),B2)) )
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc5011043943518292116et_nat(C,P)) ) ).

% mem_case_prodI2
tff(fact_5162_mem__case__prodI2,axiom,
    ! [P: produc4788131091073775919nt_int,Z: int,C: fun_fu4266222366127506568et_int] :
      ( ! [A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A2),B2) )
         => member_int(Z,aa_Pro8644714012989290591et_int(aa_fun5919120857301092337et_int(C,A2),B2)) )
     => member_int(Z,produc833192924009095408et_int(C,P)) ) ).

% mem_case_prodI2
tff(fact_5163_mem__case__prodI2,axiom,
    ! [P: produc7509114972854644512nteger,Z: set_nat,C: fun_fu963894927305104083et_nat] :
      ( ! [A2: fun_Co3460408610320954864e_term,B2: produc8923325533196201883nteger] :
          ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A2),B2) )
         => member_set_nat(Z,aa_Pro3787439844975407259et_nat(aa_fun966729468685823036et_nat(C,A2),B2)) )
     => member_set_nat(Z,produc3793265978605622331et_nat(C,P)) ) ).

% mem_case_prodI2
tff(fact_5164_mem__case__prodI2,axiom,
    ! [P: produc4788131091073775919nt_int,Z: set_nat,C: fun_fu5458086908000294626et_nat] :
      ( ! [A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A2),B2) )
         => member_set_nat(Z,aa_Pro1963545105414287545et_nat(aa_fun8341152516354955083et_nat(C,A2),B2)) )
     => member_set_nat(Z,produc4945675248400274762et_nat(C,P)) ) ).

% mem_case_prodI2
tff(fact_5165_mem__case__prodI2,axiom,
    ! [P: produc7509114972854644512nteger,Z: set_Pr1261947904930325089at_nat,C: fun_fu5924095598813543062at_nat] :
      ( ! [A2: fun_Co3460408610320954864e_term,B2: produc8923325533196201883nteger] :
          ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A2),B2) )
         => member2643936169264416010at_nat(Z,aa_Pro3625874441308352674at_nat(aa_fun7691711108118702679at_nat(C,A2),B2)) )
     => member2643936169264416010at_nat(Z,produc2447002572354077954at_nat(C,P)) ) ).

% mem_case_prodI2
tff(fact_5166_mem__case__prodI2,axiom,
    ! [P: produc7509114972854644512nteger,Z: set_Pr958786334691620121nt_int,C: fun_fu6439581347498178894nt_int] :
      ( ! [A2: fun_Co3460408610320954864e_term,B2: produc8923325533196201883nteger] :
          ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A2),B2) )
         => member2340774599025711042nt_int(Z,aa_Pro2031029616684803930nt_int(aa_fun5905053582222755087nt_int(C,A2),B2)) )
     => member2340774599025711042nt_int(Z,produc852157747730529210nt_int(C,P)) ) ).

% mem_case_prodI2
tff(fact_5167_mem__case__prodI2,axiom,
    ! [P: produc4788131091073775919nt_int,Z: set_Pr1261947904930325089at_nat,C: fun_fu204026027145793479at_nat] :
      ( ! [A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A2),B2) )
         => member2643936169264416010at_nat(Z,aa_Pro8634818098949444356at_nat(aa_fun6421688493056135304at_nat(C,A2),B2)) )
     => member2643936169264416010at_nat(Z,produc8248746225639445811at_nat(C,P)) ) ).

% mem_case_prodI2
tff(fact_5168_mem__case__prodI2,axiom,
    ! [P: produc4788131091073775919nt_int,Z: set_Pr958786334691620121nt_int,C: fun_fu719511775830429311nt_int] :
      ( ! [A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
          ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A2),B2) )
         => member2340774599025711042nt_int(Z,aa_Pro7039973274325895612nt_int(aa_fun4635030967160187712nt_int(C,A2),B2)) )
     => member2340774599025711042nt_int(Z,produc6653901401015897067nt_int(C,P)) ) ).

% mem_case_prodI2
tff(fact_5169_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_fu6383124537394581533et_nat,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro381016180347496421et_nat(aa_fun3554285010146031238et_nat(C,A),B))
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc7961109426542857093et_nat(C,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B))) ) ).

% mem_case_prodI
tff(fact_5170_mem__case__prodI,axiom,
    ! [Z: int,C: fun_fu1569576118606662777et_int,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( member_int(Z,aa_Pro5426537197693075521et_int(aa_fun1064201865214304482et_int(C,A),B))
     => member_int(Z,produc3783258407033660385et_int(C,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B))) ) ).

% mem_case_prodI
tff(fact_5171_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_fu9079770784915425324et_nat,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(C,A),B))
     => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc5011043943518292116et_nat(C,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5172_mem__case__prodI,axiom,
    ! [Z: int,C: fun_fu4266222366127506568et_int,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( member_int(Z,aa_Pro8644714012989290591et_int(aa_fun5919120857301092337et_int(C,A),B))
     => member_int(Z,produc833192924009095408et_int(C,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5173_mem__case__prodI,axiom,
    ! [Z: set_nat,C: fun_fu963894927305104083et_nat,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( member_set_nat(Z,aa_Pro3787439844975407259et_nat(aa_fun966729468685823036et_nat(C,A),B))
     => member_set_nat(Z,produc3793265978605622331et_nat(C,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B))) ) ).

% mem_case_prodI
tff(fact_5174_mem__case__prodI,axiom,
    ! [Z: set_nat,C: fun_fu5458086908000294626et_nat,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( member_set_nat(Z,aa_Pro1963545105414287545et_nat(aa_fun8341152516354955083et_nat(C,A),B))
     => member_set_nat(Z,produc4945675248400274762et_nat(C,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5175_mem__case__prodI,axiom,
    ! [Z: set_Pr1261947904930325089at_nat,C: fun_fu5924095598813543062at_nat,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( member2643936169264416010at_nat(Z,aa_Pro3625874441308352674at_nat(aa_fun7691711108118702679at_nat(C,A),B))
     => member2643936169264416010at_nat(Z,produc2447002572354077954at_nat(C,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B))) ) ).

% mem_case_prodI
tff(fact_5176_mem__case__prodI,axiom,
    ! [Z: set_Pr958786334691620121nt_int,C: fun_fu6439581347498178894nt_int,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( member2340774599025711042nt_int(Z,aa_Pro2031029616684803930nt_int(aa_fun5905053582222755087nt_int(C,A),B))
     => member2340774599025711042nt_int(Z,produc852157747730529210nt_int(C,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B))) ) ).

% mem_case_prodI
tff(fact_5177_mem__case__prodI,axiom,
    ! [Z: set_Pr1261947904930325089at_nat,C: fun_fu204026027145793479at_nat,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( member2643936169264416010at_nat(Z,aa_Pro8634818098949444356at_nat(aa_fun6421688493056135304at_nat(C,A),B))
     => member2643936169264416010at_nat(Z,produc8248746225639445811at_nat(C,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5178_mem__case__prodI,axiom,
    ! [Z: set_Pr958786334691620121nt_int,C: fun_fu719511775830429311nt_int,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( member2340774599025711042nt_int(Z,aa_Pro7039973274325895612nt_int(aa_fun4635030967160187712nt_int(C,A),B))
     => member2340774599025711042nt_int(Z,produc6653901401015897067nt_int(C,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_5179_case__prodI2_H,axiom,
    ! [P: product_prod_nat_nat,C: fun_na5914473681949881652_nat_o,Xa: product_prod_nat_nat] :
      ( ! [A2: nat,B2: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) = P )
         => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,A2),B2),Xa) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,C),P),Xa) ) ).

% case_prodI2'
tff(fact_5180_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : aa_num_option_num(bit_take_bit_num(zero_zero_nat),M) = none_num ).

% take_bit_num_simps(1)
tff(fact_5181_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : aa_num_option_num(bit_take_bit_num(aa_nat_nat(suc,N)),aa_num_num(bit0,M)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_ec),aa_num_option_num(bit_take_bit_num(N),M)) ).

% take_bit_num_simps(3)
tff(fact_5182_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : aa_num_option_num(bit_take_bit_num(aa_nat_nat(suc,N)),aa_num_num(bit1,M)) = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_take_bit_num(N),M))) ).

% take_bit_num_simps(4)
tff(fact_5183_xor__numerals_I8_J,axiom,
    ! [Xa: num] : bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Xa)),one_one_Code_natural) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Xa)) ).

% xor_numerals(8)
tff(fact_5184_xor__numerals_I8_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xa)) ).

% xor_numerals(8)
tff(fact_5185_xor__numerals_I8_J,axiom,
    ! [Xa: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Xa))),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Xa)) ).

% xor_numerals(8)
tff(fact_5186_xor__numerals_I8_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa)) ).

% xor_numerals(8)
tff(fact_5187_xor__numerals_I5_J,axiom,
    ! [Xa: num] : bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Xa)),one_one_Code_natural) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Xa)) ).

% xor_numerals(5)
tff(fact_5188_xor__numerals_I5_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xa))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa)) ).

% xor_numerals(5)
tff(fact_5189_xor__numerals_I5_J,axiom,
    ! [Xa: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Xa))),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Xa)) ).

% xor_numerals(5)
tff(fact_5190_xor__numerals_I5_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% xor_numerals(5)
tff(fact_5191_xor__numerals_I2_J,axiom,
    ! [Y3: num] : bit_se2046307713759805098atural(one_one_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y3))) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Y3)) ).

% xor_numerals(2)
tff(fact_5192_xor__numerals_I2_J,axiom,
    ! [Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y3)) ).

% xor_numerals(2)
tff(fact_5193_xor__numerals_I2_J,axiom,
    ! [Y3: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y3))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y3)) ).

% xor_numerals(2)
tff(fact_5194_xor__numerals_I2_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3)) ).

% xor_numerals(2)
tff(fact_5195_xor__numerals_I1_J,axiom,
    ! [Y3: num] : bit_se2046307713759805098atural(one_one_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Y3))) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y3)) ).

% xor_numerals(1)
tff(fact_5196_xor__numerals_I1_J,axiom,
    ! [Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y3))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3)) ).

% xor_numerals(1)
tff(fact_5197_xor__numerals_I1_J,axiom,
    ! [Y3: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y3))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y3)) ).

% xor_numerals(1)
tff(fact_5198_xor__numerals_I1_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% xor_numerals(1)
tff(fact_5199_dbl__inc__simps_I3_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5200_dbl__inc__simps_I3_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5201_dbl__inc__simps_I3_J,axiom,
    neg_nu5219082963157363817nc_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5202_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_5203_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_5204_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(4)
tff(fact_5205_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_5206_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_5207_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_5208_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_5209_xor__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa)) ).

% xor_nat_numerals(4)
tff(fact_5210_xor__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% xor_nat_numerals(3)
tff(fact_5211_xor__nat__numerals_I2_J,axiom,
    ! [Y3: 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,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3)) ).

% xor_nat_numerals(2)
tff(fact_5212_xor__nat__numerals_I1_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% xor_nat_numerals(1)
tff(fact_5213_dbl__dec__simps_I4_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5214_dbl__dec__simps_I4_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5215_dbl__dec__simps_I4_J,axiom,
    neg_nu3179335615603231917ec_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5216_take__bit__numeral__numeral,axiom,
    ! [M: num,N: num] : bit_se569199155075624693atural(aa_num_nat(numeral_numeral_nat,M),aa_num_Code_natural(numera5444537566228673987atural,N)) = case_o5621594795226839503al_num(zero_z2226904508553997617atural,numera5444537566228673987atural,aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N)) ).

% take_bit_numeral_numeral
tff(fact_5217_take__bit__numeral__numeral,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = case_o356765784539232260er_num(zero_z3403309356797280102nteger,numera6620942414471956472nteger,aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N)) ).

% take_bit_numeral_numeral
tff(fact_5218_take__bit__numeral__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_option_num_nat(case_option_nat_num(zero_zero_nat,numeral_numeral_nat),aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N)) ).

% take_bit_numeral_numeral
tff(fact_5219_take__bit__numeral__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,M)),aa_num_int(numeral_numeral_int,N)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N)) ).

% take_bit_numeral_numeral
tff(fact_5220_xor__numerals_I4_J,axiom,
    ! [Xa: num,Y3: num] : bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Xa)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y3))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,Xa),aa_num_Code_natural(numera5444537566228673987atural,Y3)))) ).

% xor_numerals(4)
tff(fact_5221_xor__numerals_I4_J,axiom,
    ! [Xa: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y3)))) ).

% xor_numerals(4)
tff(fact_5222_xor__numerals_I4_J,axiom,
    ! [Xa: num,Y3: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Xa))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y3))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa)),aa_num_Code_integer(numera6620942414471956472nteger,Y3)))) ).

% xor_numerals(4)
tff(fact_5223_xor__numerals_I4_J,axiom,
    ! [Xa: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,Xa)),aa_num_nat(numeral_numeral_nat,Y3)))) ).

% xor_numerals(4)
tff(fact_5224_xor__numerals_I6_J,axiom,
    ! [Xa: num,Y3: num] : bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Xa)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Y3))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,Xa),aa_num_Code_natural(numera5444537566228673987atural,Y3)))) ).

% xor_numerals(6)
tff(fact_5225_xor__numerals_I6_J,axiom,
    ! [Xa: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y3)))) ).

% xor_numerals(6)
tff(fact_5226_xor__numerals_I6_J,axiom,
    ! [Xa: num,Y3: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Xa))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y3))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa)),aa_num_Code_integer(numera6620942414471956472nteger,Y3)))) ).

% xor_numerals(6)
tff(fact_5227_xor__numerals_I6_J,axiom,
    ! [Xa: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,Xa)),aa_num_nat(numeral_numeral_nat,Y3)))) ).

% xor_numerals(6)
tff(fact_5228_divmod__integer_H__def,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% divmod_integer'_def
tff(fact_5229_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] : aa_num_num(bit0,X2) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_5230_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_5231_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_fu6383124537394581533et_nat,P: produc7509114972854644512nteger] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc7961109426542857093et_nat(C,P))
     => ~ ! [X4: fun_Co3460408610320954864e_term,Y4: produc8923325533196201883nteger] :
            ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X4),Y4) )
           => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro381016180347496421et_nat(aa_fun3554285010146031238et_nat(C,X4),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5232_mem__case__prodE,axiom,
    ! [Z: int,C: fun_fu1569576118606662777et_int,P: produc7509114972854644512nteger] :
      ( member_int(Z,produc3783258407033660385et_int(C,P))
     => ~ ! [X4: fun_Co3460408610320954864e_term,Y4: produc8923325533196201883nteger] :
            ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X4),Y4) )
           => ~ member_int(Z,aa_Pro5426537197693075521et_int(aa_fun1064201865214304482et_int(C,X4),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5233_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_fu9079770784915425324et_nat,P: produc4788131091073775919nt_int] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),produc5011043943518292116et_nat(C,P))
     => ~ ! [X4: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
            ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y4) )
           => ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(C,X4),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5234_mem__case__prodE,axiom,
    ! [Z: int,C: fun_fu4266222366127506568et_int,P: produc4788131091073775919nt_int] :
      ( member_int(Z,produc833192924009095408et_int(C,P))
     => ~ ! [X4: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
            ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y4) )
           => ~ member_int(Z,aa_Pro8644714012989290591et_int(aa_fun5919120857301092337et_int(C,X4),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5235_mem__case__prodE,axiom,
    ! [Z: set_nat,C: fun_fu963894927305104083et_nat,P: produc7509114972854644512nteger] :
      ( member_set_nat(Z,produc3793265978605622331et_nat(C,P))
     => ~ ! [X4: fun_Co3460408610320954864e_term,Y4: produc8923325533196201883nteger] :
            ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X4),Y4) )
           => ~ member_set_nat(Z,aa_Pro3787439844975407259et_nat(aa_fun966729468685823036et_nat(C,X4),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5236_mem__case__prodE,axiom,
    ! [Z: set_nat,C: fun_fu5458086908000294626et_nat,P: produc4788131091073775919nt_int] :
      ( member_set_nat(Z,produc4945675248400274762et_nat(C,P))
     => ~ ! [X4: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
            ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y4) )
           => ~ member_set_nat(Z,aa_Pro1963545105414287545et_nat(aa_fun8341152516354955083et_nat(C,X4),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5237_mem__case__prodE,axiom,
    ! [Z: set_Pr1261947904930325089at_nat,C: fun_fu5924095598813543062at_nat,P: produc7509114972854644512nteger] :
      ( member2643936169264416010at_nat(Z,produc2447002572354077954at_nat(C,P))
     => ~ ! [X4: fun_Co3460408610320954864e_term,Y4: produc8923325533196201883nteger] :
            ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X4),Y4) )
           => ~ member2643936169264416010at_nat(Z,aa_Pro3625874441308352674at_nat(aa_fun7691711108118702679at_nat(C,X4),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5238_mem__case__prodE,axiom,
    ! [Z: set_Pr958786334691620121nt_int,C: fun_fu6439581347498178894nt_int,P: produc7509114972854644512nteger] :
      ( member2340774599025711042nt_int(Z,produc852157747730529210nt_int(C,P))
     => ~ ! [X4: fun_Co3460408610320954864e_term,Y4: produc8923325533196201883nteger] :
            ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X4),Y4) )
           => ~ member2340774599025711042nt_int(Z,aa_Pro2031029616684803930nt_int(aa_fun5905053582222755087nt_int(C,X4),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5239_mem__case__prodE,axiom,
    ! [Z: set_Pr1261947904930325089at_nat,C: fun_fu204026027145793479at_nat,P: produc4788131091073775919nt_int] :
      ( member2643936169264416010at_nat(Z,produc8248746225639445811at_nat(C,P))
     => ~ ! [X4: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
            ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y4) )
           => ~ member2643936169264416010at_nat(Z,aa_Pro8634818098949444356at_nat(aa_fun6421688493056135304at_nat(C,X4),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5240_mem__case__prodE,axiom,
    ! [Z: set_Pr958786334691620121nt_int,C: fun_fu719511775830429311nt_int,P: produc4788131091073775919nt_int] :
      ( member2340774599025711042nt_int(Z,produc6653901401015897067nt_int(C,P))
     => ~ ! [X4: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
            ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y4) )
           => ~ member2340774599025711042nt_int(Z,aa_Pro7039973274325895612nt_int(aa_fun4635030967160187712nt_int(C,X4),Y4)) ) ) ).

% mem_case_prodE
tff(fact_5241_case__prodE,axiom,
    ! [C: fun_fu3422731660242565741eger_o,P: produc7509114972854644512nteger] :
      ( aa_Pro4652312591602923194eger_o(produc7733317944277263833eger_o(C),P)
     => ~ ! [X4: fun_Co3460408610320954864e_term,Y4: produc8923325533196201883nteger] :
            ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X4),Y4) )
           => ~ aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(C,X4),Y4) ) ) ).

% case_prodE
tff(fact_5242_case__prodE,axiom,
    ! [C: fun_fu3694549533497745878ural_o,P: produc9140211490168418807atural] :
      ( aa_Pro866224779355485219ural_o(produc3453235027721926722ural_o(C),P)
     => ~ ! [X4: fun_Pr285225112257423215e_term,Y4: produc7822875418678951345atural] :
            ( ( P = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,X4),Y4) )
           => ~ aa_Pro6073791466322389475ural_o(aa_fun9089857049899810327ural_o(C,X4),Y4) ) ) ).

% case_prodE
tff(fact_5243_case__prodE,axiom,
    ! [C: fun_fu7695273699059480439eger_o,P: produc2453742332425399446nteger] :
      ( aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(C),P)
     => ~ ! [X4: fun_Pr5098669352780829690e_term,Y4: produc8923325533196201883nteger] :
            ( ( P = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X4),Y4) )
           => ~ aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(C,X4),Y4) ) ) ).

% case_prodE
tff(fact_5244_case__prodE,axiom,
    ! [C: fun_fu6865642474282033866_int_o,P: produc3043004577601930755nt_int] :
      ( aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(C),P)
     => ~ ! [X4: fun_Pr7763680713840462443e_term,Y4: product_prod_int_int] :
            ( ( P = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Y4) )
           => ~ aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(C,X4),Y4) ) ) ).

% case_prodE
tff(fact_5245_case__prodE,axiom,
    ! [C: fun_fu2978539155105838238_int_o,P: produc4788131091073775919nt_int] :
      ( aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(C),P)
     => ~ ! [X4: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
            ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y4) )
           => ~ aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(C,X4),Y4) ) ) ).

% case_prodE
tff(fact_5246_case__prodE,axiom,
    ! [C: fun_int_fun_int_o,P: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,C),P)
     => ~ ! [X4: int,Y4: int] :
            ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4) )
           => ~ aa_int_o(aa_int_fun_int_o(C,X4),Y4) ) ) ).

% case_prodE
tff(fact_5247_case__prodD,axiom,
    ! [F: fun_fu3422731660242565741eger_o,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4652312591602923194eger_o(produc7733317944277263833eger_o(F),aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B))
     => aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(F,A),B) ) ).

% case_prodD
tff(fact_5248_case__prodD,axiom,
    ! [F: fun_fu3694549533497745878ural_o,A: fun_Pr285225112257423215e_term,B: produc7822875418678951345atural] :
      ( aa_Pro866224779355485219ural_o(produc3453235027721926722ural_o(F),aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,A),B))
     => aa_Pro6073791466322389475ural_o(aa_fun9089857049899810327ural_o(F,A),B) ) ).

% case_prodD
tff(fact_5249_case__prodD,axiom,
    ! [F: fun_fu7695273699059480439eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(F),aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B))
     => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(F,A),B) ) ).

% case_prodD
tff(fact_5250_case__prodD,axiom,
    ! [F: fun_fu6865642474282033866_int_o,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(F),aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(F,A),B) ) ).

% case_prodD
tff(fact_5251_case__prodD,axiom,
    ! [F: fun_fu2978539155105838238_int_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(F),aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(F,A),B) ) ).

% case_prodD
tff(fact_5252_case__prodD,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prodD
tff(fact_5253_case__prodE_H,axiom,
    ! [C: fun_na5914473681949881652_nat_o,P: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,C),P),Z)
     => ~ ! [X4: nat,Y4: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,X4),Y4),Z) ) ) ).

% case_prodE'
tff(fact_5254_case__prodD_H,axiom,
    ! [R4: fun_na5914473681949881652_nat_o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,R4),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),C)
     => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(R4,A),B),C) ) ).

% case_prodD'
tff(fact_5255_xor__num_Ocases,axiom,
    ! [Xa: product_prod_num_num] :
      ( ( Xa != product_Pair_num_num(one,one) )
     => ( ! [N2: num] : Xa != product_Pair_num_num(one,aa_num_num(bit0,N2))
       => ( ! [N2: num] : Xa != product_Pair_num_num(one,aa_num_num(bit1,N2))
         => ( ! [M4: num] : Xa != product_Pair_num_num(aa_num_num(bit0,M4),one)
           => ( ! [M4: num,N2: num] : Xa != product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit0,N2))
             => ( ! [M4: num,N2: num] : Xa != product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit1,N2))
               => ( ! [M4: num] : Xa != product_Pair_num_num(aa_num_num(bit1,M4),one)
                 => ( ! [M4: num,N2: num] : Xa != product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit0,N2))
                   => ~ ! [M4: num,N2: num] : Xa != product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_5256_Collect__case__prod__mono,axiom,
    ! [A3: fun_int_fun_int_o,B3: fun_int_fun_int_o] :
      ( ord_le77980682512674322_int_o(A3,B3)
     => aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,A3))),collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,B3))) ) ).

% Collect_case_prod_mono
tff(fact_5257_execute__bind__case,axiom,
    ! [F: heap_Time_Heap_a,G: fun_a_6093291656547882009Heap_a,Ha: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_a_a(F,G)),Ha) = case_o6061097939050036047it_nat(none_P2651198173097904984it_nat,produc7850007024774191849it_nat(aTP_Lamm_ee(G)),aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha)) ).

% execute_bind_case
tff(fact_5258_numeral__Bit1,axiom,
    ! [N: num] : aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_num_Code_natural(numera5444537566228673987atural,N)),aa_num_Code_natural(numera5444537566228673987atural,N))),one_one_Code_natural) ).

% numeral_Bit1
tff(fact_5259_numeral__Bit1,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_Bit1
tff(fact_5260_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_5261_numeral__Bit1,axiom,
    ! [N: num] : aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_num_Code_integer(numera6620942414471956472nteger,N))),one_one_Code_integer) ).

% numeral_Bit1
tff(fact_5262_numeral__Bit1,axiom,
    ! [N: num] : numeral_numeral_rat(aa_num_num(bit1,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(N)),numeral_numeral_rat(N))),one_one_rat) ).

% numeral_Bit1
tff(fact_5263_numeral__code_I3_J,axiom,
    ! [N: num] :
      aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,N)) = $let(
        m: code_natural,
        m:= aa_num_Code_natural(numera5444537566228673987atural,N),
        aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,m),m)),one_one_Code_natural) ) ).

% numeral_code(3)
tff(fact_5264_numeral__code_I3_J,axiom,
    ! [N: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,N),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m)),one_one_int) ) ).

% numeral_code(3)
tff(fact_5265_numeral__code_I3_J,axiom,
    ! [N: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,N),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_5266_numeral__code_I3_J,axiom,
    ! [N: num] :
      aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N)) = $let(
        m: code_integer,
        m:= aa_num_Code_integer(numera6620942414471956472nteger,N),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,m),m)),one_one_Code_integer) ) ).

% numeral_code(3)
tff(fact_5267_numeral__code_I3_J,axiom,
    ! [N: num] :
      numeral_numeral_rat(aa_num_num(bit1,N)) = $let(
        m: rat,
        m:= numeral_numeral_rat(N),
        aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m),m)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_5268_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_5269_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: int,
        w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_5270_power__numeral__odd,axiom,
    ! [Z: code_integer,W: num] :
      aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: code_integer,
        w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_5271_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_5272_Heap__Time__Monad_Obind__def,axiom,
    ! [F: heap_Time_Heap_a,G: fun_a_6093291656547882009Heap_a] : heap_Time_bind_a_a(F,G) = heap_Time_Heap_a2(aa_fun6119220767042692876it_nat(aTP_Lamm_ef(F),G)) ).

% Heap_Time_Monad.bind_def
tff(fact_5273_Heap__Time__Monad_Obind__def,axiom,
    ! [F: heap_Time_Heap_a,G: fun_a_2990225006182978344t_unit] : heap_T8685611227969916822t_unit(F,G) = heap_T6183433275982383450t_unit(aa_fun372765568861188684it_nat(aTP_Lamm_ei(F),G)) ).

% Heap_Time_Monad.bind_def
tff(fact_5274_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_5275_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_5276_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_5277_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_5278_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_5279_take__bit__num__eq__None__imp,axiom,
    ! [M: nat,N: num] :
      ( ( aa_num_option_num(bit_take_bit_num(M),N) = none_num )
     => ( bit_se569199155075624693atural(M,aa_num_Code_natural(numera5444537566228673987atural,N)) = zero_z2226904508553997617atural ) ) ).

% take_bit_num_eq_None_imp
tff(fact_5280_take__bit__num__eq__None__imp,axiom,
    ! [M: nat,N: num] :
      ( ( aa_num_option_num(bit_take_bit_num(M),N) = none_num )
     => ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(M),aa_num_Code_integer(numera6620942414471956472nteger,N)) = zero_z3403309356797280102nteger ) ) ).

% take_bit_num_eq_None_imp
tff(fact_5281_take__bit__num__eq__None__imp,axiom,
    ! [M: nat,N: num] :
      ( ( aa_num_option_num(bit_take_bit_num(M),N) = none_num )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_num_nat(numeral_numeral_nat,N)) = zero_zero_nat ) ) ).

% take_bit_num_eq_None_imp
tff(fact_5282_take__bit__num__eq__None__imp,axiom,
    ! [M: nat,N: num] :
      ( ( aa_num_option_num(bit_take_bit_num(M),N) = none_num )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_num_int(numeral_numeral_int,N)) = zero_zero_int ) ) ).

% take_bit_num_eq_None_imp
tff(fact_5283_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_5284_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_5285_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5286_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5287_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5288_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5289_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5290_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5291_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      aa_num_option_num(bit_take_bit_num(N),M) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat,none_num,aa_num_option_num(some_num,aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M))))) ).

% take_bit_num_def
tff(fact_5292_numeral__BitM,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bitM,N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),one_one_int) ).

% numeral_BitM
tff(fact_5293_numeral__BitM,axiom,
    ! [N: num] : aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bitM,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N))),one_one_Code_integer) ).

% numeral_BitM
tff(fact_5294_numeral__BitM,axiom,
    ! [N: num] : numeral_numeral_rat(aa_num_num(bitM,N)) = minus_minus_rat(numeral_numeral_rat(aa_num_num(bit0,N)),one_one_rat) ).

% numeral_BitM
tff(fact_5295_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_5296_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_5297_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : aa_Pro6799258434465214121nt_int(adjust_div,Qr) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_ej),Qr) ).

% Divides.adjust_div_def
tff(fact_5298_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : bit_se569199155075624693atural(aa_nat_nat(suc,N),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,K))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(bit_se569199155075624693atural(N,aa_num_Code_natural(numera5444537566228673987atural,K))),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),one_one_Code_natural) ).

% take_bit_Suc_bit1
tff(fact_5299_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),aa_num_Code_integer(numera6620942414471956472nteger,K))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) ).

% take_bit_Suc_bit1
tff(fact_5300_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_5301_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_5302_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).

% divmod_nat_def
tff(fact_5303_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = produc6677183202524767010eger_o(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,K),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),K)) ).

% bit_cut_integer_def
tff(fact_5304_integer__of__int__code,axiom,
    ! [K: int] :
      aa_int_Code_integer(code_integer_of_int,K) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))),
        $ite(
          K = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,l),one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_5305_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_ek(D))) ).

% int_ge_less_than_def
tff(fact_5306_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_el(D))) ).

% int_ge_less_than2_def
tff(fact_5307_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : bit_se569199155075624693atural(aa_num_nat(numeral_numeral_nat,L),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,K))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(bit_se569199155075624693atural(aa_num_nat(pred_numeral,L),aa_num_Code_natural(numera5444537566228673987atural,K))),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),one_one_Code_natural) ).

% take_bit_numeral_bit1
tff(fact_5308_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,L)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_num_nat(pred_numeral,L)),aa_num_Code_integer(numera6620942414471956472nteger,K))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) ).

% take_bit_numeral_bit1
tff(fact_5309_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_5310_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_5311_Code__Numeral_Onegative__def,axiom,
    code_negative = comp_C3531382070062128313er_num(uminus1351360451143612070nteger,numera6620942414471956472nteger) ).

% Code_Numeral.negative_def
tff(fact_5312_push__bit__numeral__minus__1,axiom,
    ! [N: num] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,aa_num_nat(numeral_numeral_nat,N)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,N))) ).

% push_bit_numeral_minus_1
tff(fact_5313_push__bit__numeral__minus__1,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_num_nat(numeral_numeral_nat,N)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,N))) ).

% push_bit_numeral_minus_1
tff(fact_5314_split__part,axiom,
    ! [Pa: $o,Q: fun_int_fun_int_o,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun6133342665515662012_int_o(aTP_Lamm_em((Pa)),Q)),X)
    <=> ( (Pa)
        & aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Q),X) ) ) ).

% split_part
tff(fact_5315_push__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),zero_zero_int) = zero_zero_int ).

% push_bit_of_0
tff(fact_5316_push__bit__of__0,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% push_bit_of_0
tff(fact_5317_push__bit__of__0,axiom,
    ! [N: nat] : bit_se6611745700429515170atural(N,zero_z2226904508553997617atural) = zero_z2226904508553997617atural ).

% push_bit_of_0
tff(fact_5318_push__bit__of__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),zero_zero_nat) = zero_zero_nat ).

% push_bit_of_0
tff(fact_5319_push__bit__eq__0__iff,axiom,
    ! [N: nat,A: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% push_bit_eq_0_iff
tff(fact_5320_push__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% push_bit_eq_0_iff
tff(fact_5321_push__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se6611745700429515170atural(N,A) = zero_z2226904508553997617atural )
    <=> ( A = zero_z2226904508553997617atural ) ) ).

% push_bit_eq_0_iff
tff(fact_5322_push__bit__eq__0__iff,axiom,
    ! [N: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),A) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% push_bit_eq_0_iff
tff(fact_5323_push__bit__push__bit,axiom,
    ! [M: nat,N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),A) ).

% push_bit_push_bit
tff(fact_5324_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_5325_push__bit__of__1,axiom,
    ! [N: nat] : bit_se6611745700429515170atural(N,one_one_Code_natural) = aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N) ).

% push_bit_of_1
tff(fact_5326_push__bit__of__1,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ).

% push_bit_of_1
tff(fact_5327_push__bit__of__1,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N) ).

% push_bit_of_1
tff(fact_5328_push__bit__of__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% push_bit_of_1
tff(fact_5329_even__push__bit__iff,axiom,
    ! [N: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),A))
    <=> ( ( N != zero_zero_nat )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% even_push_bit_iff
tff(fact_5330_even__push__bit__iff,axiom,
    ! [N: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),A))
    <=> ( ( N != zero_zero_nat )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ) ).

% even_push_bit_iff
tff(fact_5331_even__push__bit__iff,axiom,
    ! [N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),A))
    <=> ( ( N != zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% even_push_bit_iff
tff(fact_5332_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% push_bit_of_Suc_0
tff(fact_5333_take__bit__num__simps_I7_J,axiom,
    ! [R: num,M: num] : aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R)),aa_num_num(bit1,M)) = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_take_bit_num(aa_num_nat(pred_numeral,R)),M))) ).

% take_bit_num_simps(7)
tff(fact_5334_take__bit__num__simps_I6_J,axiom,
    ! [R: num,M: num] : aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R)),aa_num_num(bit0,M)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_ec),aa_num_option_num(bit_take_bit_num(aa_num_nat(pred_numeral,R)),M)) ).

% take_bit_num_simps(6)
tff(fact_5335_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_en),Prod) ).

% prod.disc_eq_case
tff(fact_5336_push__bit__add,axiom,
    ! [N: nat,A: int,B: int] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),A)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),B)) ).

% push_bit_add
tff(fact_5337_push__bit__add,axiom,
    ! [N: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),B)) ).

% push_bit_add
tff(fact_5338_push__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),A)) ).

% push_bit_take_bit
tff(fact_5339_push__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),A)) ).

% push_bit_take_bit
tff(fact_5340_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_5341_pred__numeral__def,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),one_one_nat) ).

% pred_numeral_def
tff(fact_5342_flip__bit__eq__xor,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),A) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) ).

% flip_bit_eq_xor
tff(fact_5343_flip__bit__eq__xor,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,A),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer)) ).

% flip_bit_eq_xor
tff(fact_5344_flip__bit__eq__xor,axiom,
    ! [N: nat,A: code_natural] : bit_se168947363167071951atural(N,A) = bit_se2046307713759805098atural(A,bit_se6611745700429515170atural(N,one_one_Code_natural)) ).

% flip_bit_eq_xor
tff(fact_5345_flip__bit__eq__xor,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat)) ).

% flip_bit_eq_xor
tff(fact_5346_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_eo(K),code_divmod_abs(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_5347_mask__numeral,axiom,
    ! [N: num] : aa_nat_Code_natural(bit_se943457434206027407atural,aa_num_nat(numeral_numeral_nat,N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_Code_natural(bit_se943457434206027407atural,aa_num_nat(pred_numeral,N)))) ).

% mask_numeral
tff(fact_5348_mask__numeral,axiom,
    ! [N: num] : aa_nat_Code_integer(bit_se2119862282449309892nteger,aa_num_nat(numeral_numeral_nat,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,aa_num_nat(pred_numeral,N)))) ).

% mask_numeral
tff(fact_5349_mask__numeral,axiom,
    ! [N: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(pred_numeral,N)))) ).

% mask_numeral
tff(fact_5350_mask__numeral,axiom,
    ! [N: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(pred_numeral,N)))) ).

% mask_numeral
tff(fact_5351_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_5352_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(bit_and_not_num(aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_ep),aa_num_option_num(bit_and_not_num(M),N)) ).

% and_not_num.simps(8)
tff(fact_5353_take__bit__num__code,axiom,
    ! [N: nat,M: num] : aa_num_option_num(bit_take_bit_num(N),M) = produc478579273971653890on_num(aTP_Lamm_et,product_Pair_nat_num(N,M)) ).

% take_bit_num_code
tff(fact_5354_num__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_num(code_num_of_integer,K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_eu,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_5355_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% mask_nat_positive_iff
tff(fact_5356_mask__0,axiom,
    aa_nat_Code_integer(bit_se2119862282449309892nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% mask_0
tff(fact_5357_mask__0,axiom,
    aa_nat_Code_natural(bit_se943457434206027407atural,zero_zero_nat) = zero_z2226904508553997617atural ).

% mask_0
tff(fact_5358_mask__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,zero_zero_nat) = zero_zero_nat ).

% mask_0
tff(fact_5359_mask__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,zero_zero_nat) = zero_zero_int ).

% mask_0
tff(fact_5360_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_Code_integer(bit_se2119862282449309892nteger,N) = zero_z3403309356797280102nteger )
    <=> ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_5361_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_Code_natural(bit_se943457434206027407atural,N) = zero_z2226904508553997617atural )
    <=> ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_5362_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_5363_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_5364_case__nat__numeral,axiom,
    ! [A: option_num,F: fun_nat_option_num,V: num] : aa_nat_option_num(case_nat_option_num(A,F),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_option_num(F,aa_num_nat(pred_numeral,V)) ).

% case_nat_numeral
tff(fact_5365_case__nat__numeral,axiom,
    ! [A: $o,F: fun_nat_o,V: num] :
      ( aa_nat_o(case_nat_o((A),F),aa_num_nat(numeral_numeral_nat,V))
    <=> aa_nat_o(F,aa_num_nat(pred_numeral,V)) ) ).

% case_nat_numeral
tff(fact_5366_case__nat__numeral,axiom,
    ! [A: nat,F: fun_nat_nat,V: num] : aa_nat_nat(case_nat_nat(A,F),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(F,aa_num_nat(pred_numeral,V)) ).

% case_nat_numeral
tff(fact_5367_mask__Suc__0,axiom,
    aa_nat_Code_integer(bit_se2119862282449309892nteger,aa_nat_nat(suc,zero_zero_nat)) = one_one_Code_integer ).

% mask_Suc_0
tff(fact_5368_mask__Suc__0,axiom,
    aa_nat_Code_natural(bit_se943457434206027407atural,aa_nat_nat(suc,zero_zero_nat)) = one_one_Code_natural ).

% mask_Suc_0
tff(fact_5369_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_5370_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_5371_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_nat_Code_integer(bit_se2119862282449309892nteger,N) ).

% take_bit_minus_one_eq_mask
tff(fact_5372_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_nat_int(bit_se2000444600071755411sk_int,N) ).

% take_bit_minus_one_eq_mask
tff(fact_5373_case__nat__add__eq__if,axiom,
    ! [A: option_num,F: fun_nat_option_num,V: num,N: nat] : aa_nat_option_num(case_nat_option_num(A,F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N)) = aa_nat_option_num(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(pred_numeral,V)),N)) ).

% case_nat_add_eq_if
tff(fact_5374_case__nat__add__eq__if,axiom,
    ! [A: $o,F: fun_nat_o,V: num,N: nat] :
      ( aa_nat_o(case_nat_o((A),F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N))
    <=> aa_nat_o(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(pred_numeral,V)),N)) ) ).

% case_nat_add_eq_if
tff(fact_5375_case__nat__add__eq__if,axiom,
    ! [A: nat,F: fun_nat_nat,V: num,N: nat] : aa_nat_nat(case_nat_nat(A,F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N)) = aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(pred_numeral,V)),N)) ).

% case_nat_add_eq_if
tff(fact_5376_nat_Ocase__distrib,axiom,
    ! [Ha: fun_op8989173174992673105on_num,F1: option_num,F22: fun_nat_option_num,Nat: nat] : aa_opt5369159329878751634on_num(Ha,aa_nat_option_num(case_nat_option_num(F1,F22),Nat)) = aa_nat_option_num(case_nat_option_num(aa_opt5369159329878751634on_num(Ha,F1),aa_fun7716400287237121964on_num(aTP_Lamm_ev(Ha),F22)),Nat) ).

% nat.case_distrib
tff(fact_5377_nat_Ocase__distrib,axiom,
    ! [Ha: fun_option_num_o,F1: option_num,F22: fun_nat_option_num,Nat: nat] :
      ( aa_option_num_o(Ha,aa_nat_option_num(case_nat_option_num(F1,F22),Nat))
    <=> aa_nat_o(case_nat_o(aa_option_num_o(Ha,F1),aa_fun1675964276245824108_nat_o(aTP_Lamm_ew(Ha),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_5378_nat_Ocase__distrib,axiom,
    ! [Ha: fun_option_num_nat,F1: option_num,F22: fun_nat_option_num,Nat: nat] : aa_option_num_nat(Ha,aa_nat_option_num(case_nat_option_num(F1,F22),Nat)) = aa_nat_nat(case_nat_nat(aa_option_num_nat(Ha,F1),aa_fun7387201323300553874at_nat(aTP_Lamm_ex(Ha),F22)),Nat) ).

% nat.case_distrib
tff(fact_5379_nat_Ocase__distrib,axiom,
    ! [Ha: fun_o_option_num,F1: $o,F22: fun_nat_o,Nat: nat] : aa_o_option_num(Ha,aa_nat_o(case_nat_o((F1),F22),Nat)) = aa_nat_option_num(case_nat_option_num(aa_o_option_num(Ha,(F1)),aa_fun6990899178391151900on_num(aTP_Lamm_ey(Ha),F22)),Nat) ).

% nat.case_distrib
tff(fact_5380_nat_Ocase__distrib,axiom,
    ! [Ha: fun_o_o,F1: $o,F22: fun_nat_o,Nat: nat] :
      ( aa_o_o(Ha,aa_nat_o(case_nat_o((F1),F22),Nat))
    <=> aa_nat_o(case_nat_o(aa_o_o(Ha,(F1)),aa_fun5937193402907943164_nat_o(aTP_Lamm_ez(Ha),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_5381_nat_Ocase__distrib,axiom,
    ! [Ha: fun_o_nat,F1: $o,F22: fun_nat_o,Nat: nat] : aa_o_nat(Ha,aa_nat_o(case_nat_o((F1),F22),Nat)) = aa_nat_nat(case_nat_nat(aa_o_nat(Ha,(F1)),aa_fun4204860950123276802at_nat(aTP_Lamm_fa(Ha),F22)),Nat) ).

% nat.case_distrib
tff(fact_5382_nat_Ocase__distrib,axiom,
    ! [Ha: fun_nat_option_num,F1: nat,F22: fun_nat_nat,Nat: nat] : aa_nat_option_num(Ha,aa_nat_nat(case_nat_nat(F1,F22),Nat)) = aa_nat_option_num(case_nat_option_num(aa_nat_option_num(Ha,F1),aa_fun6517009837249254290on_num(aTP_Lamm_fb(Ha),F22)),Nat) ).

% nat.case_distrib
tff(fact_5383_nat_Ocase__distrib,axiom,
    ! [Ha: fun_nat_o,F1: nat,F22: fun_nat_nat,Nat: nat] :
      ( aa_nat_o(Ha,aa_nat_nat(case_nat_nat(F1,F22),Nat))
    <=> aa_nat_o(case_nat_o(aa_nat_o(Ha,F1),aa_fun8673969331427483654_nat_o(aTP_Lamm_fc(Ha),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_5384_nat_Ocase__distrib,axiom,
    ! [Ha: fun_nat_nat,F1: nat,F22: fun_nat_nat,Nat: nat] : aa_nat_nat(Ha,aa_nat_nat(case_nat_nat(F1,F22),Nat)) = aa_nat_nat(case_nat_nat(aa_nat_nat(Ha,F1),aa_fun5791449513213591928at_nat(aTP_Lamm_fd(Ha),F22)),Nat) ).

% nat.case_distrib
tff(fact_5385_num_Ocase__distrib,axiom,
    ! [Ha: fun_op8989173174992673105on_num,F1: option_num,F22: fun_num_option_num,F3: fun_num_option_num,Num: num] : aa_opt5369159329878751634on_num(Ha,aa_num_option_num(case_num_option_num(F1,F22,F3),Num)) = aa_num_option_num(case_num_option_num(aa_opt5369159329878751634on_num(Ha,F1),aa_fun3636977833914307392on_num(aTP_Lamm_bb(Ha),F22),aa_fun3636977833914307392on_num(aTP_Lamm_bb(Ha),F3)),Num) ).

% num.case_distrib
tff(fact_5386_old_Onat_Osimps_I5_J,axiom,
    ! [F1: option_num,F22: fun_nat_option_num,X2: nat] : aa_nat_option_num(case_nat_option_num(F1,F22),aa_nat_nat(suc,X2)) = aa_nat_option_num(F22,X2) ).

% old.nat.simps(5)
tff(fact_5387_old_Onat_Osimps_I5_J,axiom,
    ! [F1: $o,F22: fun_nat_o,X2: nat] :
      ( aa_nat_o(case_nat_o((F1),F22),aa_nat_nat(suc,X2))
    <=> aa_nat_o(F22,X2) ) ).

% old.nat.simps(5)
tff(fact_5388_old_Onat_Osimps_I5_J,axiom,
    ! [F1: nat,F22: fun_nat_nat,X2: nat] : aa_nat_nat(case_nat_nat(F1,F22),aa_nat_nat(suc,X2)) = aa_nat_nat(F22,X2) ).

% old.nat.simps(5)
tff(fact_5389_old_Onat_Osimps_I4_J,axiom,
    ! [F1: option_num,F22: fun_nat_option_num] : aa_nat_option_num(case_nat_option_num(F1,F22),zero_zero_nat) = F1 ).

% old.nat.simps(4)
tff(fact_5390_old_Onat_Osimps_I4_J,axiom,
    ! [F1: $o,F22: fun_nat_o] :
      ( aa_nat_o(case_nat_o((F1),F22),zero_zero_nat)
    <=> (F1) ) ).

% old.nat.simps(4)
tff(fact_5391_old_Onat_Osimps_I4_J,axiom,
    ! [F1: nat,F22: fun_nat_nat] : aa_nat_nat(case_nat_nat(F1,F22),zero_zero_nat) = F1 ).

% old.nat.simps(4)
tff(fact_5392_verit__eq__simplify_I17_J,axiom,
    ! [F1: option_num,F22: fun_num_option_num,F3: fun_num_option_num,X2: num] : aa_num_option_num(case_num_option_num(F1,F22,F3),aa_num_num(bit0,X2)) = aa_num_option_num(F22,X2) ).

% verit_eq_simplify(17)
tff(fact_5393_verit__eq__simplify_I16_J,axiom,
    ! [F1: option_num,F22: fun_num_option_num,F3: fun_num_option_num] : aa_num_option_num(case_num_option_num(F1,F22,F3),one) = F1 ).

% verit_eq_simplify(16)
tff(fact_5394_verit__eq__simplify_I18_J,axiom,
    ! [F1: option_num,F22: fun_num_option_num,F3: fun_num_option_num,X32: num] : aa_num_option_num(case_num_option_num(F1,F22,F3),aa_num_num(bit1,X32)) = aa_num_option_num(F3,X32) ).

% verit_eq_simplify(18)
tff(fact_5395_less__mask,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N)) ) ).

% less_mask
tff(fact_5396_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,J)) ).

% divmod_abs_code(5)
tff(fact_5397_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_z3403309356797280102nteger,J) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ).

% divmod_abs_code(6)
tff(fact_5398_Nitpick_Ocase__nat__unfold,axiom,
    ! [Xa: option_num,F: fun_nat_option_num,N: nat] :
      aa_nat_option_num(case_nat_option_num(Xa,F),N) = $ite(N = zero_zero_nat,Xa,aa_nat_option_num(F,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% Nitpick.case_nat_unfold
tff(fact_5399_Nitpick_Ocase__nat__unfold,axiom,
    ! [Xa: $o,F: fun_nat_o,N: nat] :
      ( aa_nat_o(case_nat_o((Xa),F),N)
    <=> $ite(N = zero_zero_nat,(Xa),aa_nat_o(F,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% Nitpick.case_nat_unfold
tff(fact_5400_Nitpick_Ocase__nat__unfold,axiom,
    ! [Xa: nat,F: fun_nat_nat,N: nat] :
      aa_nat_nat(case_nat_nat(Xa,F),N) = $ite(N = zero_zero_nat,Xa,aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% Nitpick.case_nat_unfold
tff(fact_5401_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,N))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_5402_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,N))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_5403_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,N))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_5404_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L))) ).

% divmod_abs_def
tff(fact_5405_mask__half__int,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ).

% mask_half_int
tff(fact_5406_mask__nat__def,axiom,
    ! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) ).

% mask_nat_def
tff(fact_5407_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : aa_nat_Code_natural(bit_se943457434206027407atural,N) = minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N),one_one_Code_natural) ).

% mask_eq_exp_minus_1
tff(fact_5408_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : aa_nat_Code_integer(bit_se2119862282449309892nteger,N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)),one_one_Code_integer) ).

% mask_eq_exp_minus_1
tff(fact_5409_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) ).

% mask_eq_exp_minus_1
tff(fact_5410_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ).

% mask_eq_exp_minus_1
tff(fact_5411_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,K),L),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,K),L)) ).

% divmod_integer_def
tff(fact_5412_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_nat(code_nat_of_integer,K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_fe,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_5413_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_ff(L)),code_divmod_abs(K,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_fg(L)),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_5414_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
          aa_Pro4693618923521437820nteger(aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer),L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_fh(L)),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_5415_int__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_int(code_int_of_integer,K) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
        $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_fi,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_5416_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),aa_num_num(bitM,N))) ).

% and_minus_numerals(7)
tff(fact_5417_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),aa_num_num(bitM,N))) ).

% and_minus_numerals(3)
tff(fact_5418_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),aa_num_num(bit0,N))) ).

% and_minus_numerals(8)
tff(fact_5419_and__zero__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% and_zero_eq
tff(fact_5420_and__zero__eq,axiom,
    ! [A: code_natural] : bit_se2773287842338716102atural(A,zero_z2226904508553997617atural) = zero_z2226904508553997617atural ).

% and_zero_eq
tff(fact_5421_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_5422_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_5423_zero__and__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).

% zero_and_eq
tff(fact_5424_zero__and__eq,axiom,
    ! [A: code_natural] : bit_se2773287842338716102atural(zero_z2226904508553997617atural,A) = zero_z2226904508553997617atural ).

% zero_and_eq
tff(fact_5425_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_5426_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_5427_bit_Oconj__zero__left,axiom,
    ! [Xa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,zero_z3403309356797280102nteger),Xa) = zero_z3403309356797280102nteger ).

% bit.conj_zero_left
tff(fact_5428_bit_Oconj__zero__left,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),Xa) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_5429_bit_Oconj__zero__right,axiom,
    ! [Xa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,Xa),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% bit.conj_zero_right
tff(fact_5430_bit_Oconj__zero__right,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_5431_apsnd__conv,axiom,
    ! [F: fun_Pr5876161306168176259nteger,Xa: fun_Co3460408610320954864e_term,Y3: produc8923325533196201883nteger] : produc4532778850088682670e_term(F,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Xa),Y3)) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Xa),aa_Pro4693618923521437820nteger(F,Y3)) ).

% apsnd_conv
tff(fact_5432_apsnd__conv,axiom,
    ! [F: fun_Pr74676805955934211atural,Xa: fun_Pr285225112257423215e_term,Y3: produc7822875418678951345atural] : produc1166422551560059565e_term(F,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,Xa),Y3)) = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,Xa),aa_Pro4262726347734369532atural(F,Y3)) ).

% apsnd_conv
tff(fact_5433_apsnd__conv,axiom,
    ! [F: fun_Pr5876161306168176259nteger,Xa: fun_Pr5098669352780829690e_term,Y3: produc8923325533196201883nteger] : produc2265439103302406840e_term(F,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Xa),Y3)) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Xa),aa_Pro4693618923521437820nteger(F,Y3)) ).

% apsnd_conv
tff(fact_5434_apsnd__conv,axiom,
    ! [F: fun_Pr2478283820680522243nt_int,Xa: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int] : produc8061333830752525225e_term(F,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Xa),Y3)) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Xa),aa_Pro6970504127699319548nt_int(F,Y3)) ).

% apsnd_conv
tff(fact_5435_apsnd__conv,axiom,
    ! [F: fun_Pr2478283820680522243nt_int,Xa: fun_in5730379987311621695e_term,Y3: product_prod_int_int] : produc2254893824690362237e_term(F,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Xa),Y3)) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Xa),aa_Pro6970504127699319548nt_int(F,Y3)) ).

% apsnd_conv
tff(fact_5436_apsnd__conv,axiom,
    ! [F: fun_Co8499238673658658103nteger,Xa: code_integer,Y3: code_integer] : aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),produc1086072967326762835nteger(Xa,Y3)) = produc1086072967326762835nteger(Xa,aa_Cod1783974652533396728nteger(F,Y3)) ).

% apsnd_conv
tff(fact_5437_bit_Oconj__one__right,axiom,
    ! [Xa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,Xa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = Xa ).

% bit.conj_one_right
tff(fact_5438_bit_Oconj__one__right,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),aa_int_int(uminus_uminus_int,one_one_int)) = Xa ).

% bit.conj_one_right
tff(fact_5439_and_Oright__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = A ).

% and.right_neutral
tff(fact_5440_and_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = A ).

% and.right_neutral
tff(fact_5441_and_Oleft__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),A) = A ).

% and.left_neutral
tff(fact_5442_and_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,one_one_int)),A) = A ).

% and.left_neutral
tff(fact_5443_and__numerals_I2_J,axiom,
    ! [Y3: num] : bit_se2773287842338716102atural(one_one_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y3))) = one_one_Code_natural ).

% and_numerals(2)
tff(fact_5444_and__numerals_I2_J,axiom,
    ! [Y3: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y3))) = one_one_Code_integer ).

% and_numerals(2)
tff(fact_5445_and__numerals_I2_J,axiom,
    ! [Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3))) = one_one_int ).

% and_numerals(2)
tff(fact_5446_and__numerals_I2_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = one_one_nat ).

% and_numerals(2)
tff(fact_5447_and__numerals_I8_J,axiom,
    ! [Xa: num] : bit_se2773287842338716102atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Xa)),one_one_Code_natural) = one_one_Code_natural ).

% and_numerals(8)
tff(fact_5448_and__numerals_I8_J,axiom,
    ! [Xa: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Xa))),one_one_Code_integer) = one_one_Code_integer ).

% and_numerals(8)
tff(fact_5449_and__numerals_I8_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa))),one_one_int) = one_one_int ).

% and_numerals(8)
tff(fact_5450_and__numerals_I8_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),one_one_nat) = one_one_nat ).

% and_numerals(8)
tff(fact_5451_and__numerals_I5_J,axiom,
    ! [Xa: num] : bit_se2773287842338716102atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Xa)),one_one_Code_natural) = zero_z2226904508553997617atural ).

% and_numerals(5)
tff(fact_5452_and__numerals_I5_J,axiom,
    ! [Xa: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Xa))),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% and_numerals(5)
tff(fact_5453_and__numerals_I5_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xa))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_5454_and__numerals_I5_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_5455_and__numerals_I1_J,axiom,
    ! [Y3: num] : bit_se2773287842338716102atural(one_one_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Y3))) = zero_z2226904508553997617atural ).

% and_numerals(1)
tff(fact_5456_and__numerals_I1_J,axiom,
    ! [Y3: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y3))) = zero_z3403309356797280102nteger ).

% and_numerals(1)
tff(fact_5457_and__numerals_I1_J,axiom,
    ! [Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y3))) = zero_zero_int ).

% and_numerals(1)
tff(fact_5458_and__numerals_I1_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_5459_and__numerals_I7_J,axiom,
    ! [Xa: num,Y3: num] : bit_se2773287842338716102atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Xa)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y3))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),bit_se2773287842338716102atural(aa_num_Code_natural(numera5444537566228673987atural,Xa),aa_num_Code_natural(numera5444537566228673987atural,Y3)))) ).

% and_numerals(7)
tff(fact_5460_and__numerals_I7_J,axiom,
    ! [Xa: num,Y3: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Xa))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y3))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa)),aa_num_Code_integer(numera6620942414471956472nteger,Y3)))) ).

% and_numerals(7)
tff(fact_5461_and__numerals_I7_J,axiom,
    ! [Xa: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y3)))) ).

% and_numerals(7)
tff(fact_5462_and__numerals_I7_J,axiom,
    ! [Xa: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,Xa)),aa_num_nat(numeral_numeral_nat,Y3)))) ).

% and_numerals(7)
tff(fact_5463_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),aa_num_num(bit0,N))) ).

% and_minus_numerals(4)
tff(fact_5464_apsnd__compose,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger,Xa: produc8923325533196201883nteger] : aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,G),Xa)) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,aa_fun4435975922079263404nteger(comp_C7449957260575251196nteger(F),G)),Xa) ).

% apsnd_compose
tff(fact_5465_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
        & ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_5466_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( ( A = aa_int_int(uminus_uminus_int,one_one_int) )
        & ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_5467_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_fj),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_5468_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_fk),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_5469_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,M)),N) ) ).

% less_eq_nat.simps(2)
tff(fact_5470_one__and__eq,axiom,
    ! [A: code_natural] : bit_se2773287842338716102atural(one_one_Code_natural,A) = modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_5471_one__and__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,one_one_Code_integer),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_5472_one__and__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_5473_one__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_5474_and__one__eq,axiom,
    ! [A: code_natural] : bit_se2773287842338716102atural(A,one_one_Code_natural) = modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_5475_and__one__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_5476_and__one__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_5477_and__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_5478_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_av),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ).

% diff_Suc
tff(fact_5479_integer__of__num_I3_J,axiom,
    ! [N: num] :
      code_integer_of_num(aa_num_num(bit1,N)) = $let(
        k: code_integer,
        k:= code_integer_of_num(N),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,k),k)),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_5480_nat_Osplit__sels_I1_J,axiom,
    ! [Pa: fun_option_num_o,F1: option_num,F22: fun_nat_option_num,Nat: nat] :
      ( aa_option_num_o(Pa,aa_nat_option_num(case_nat_option_num(F1,F22),Nat))
    <=> ( ( ( Nat = zero_zero_nat )
         => aa_option_num_o(Pa,F1) )
        & ( ( Nat = aa_nat_nat(suc,aa_nat_nat(pred,Nat)) )
         => aa_option_num_o(Pa,aa_nat_option_num(F22,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_5481_nat_Osplit__sels_I1_J,axiom,
    ! [Pa: fun_o_o,F1: $o,F22: fun_nat_o,Nat: nat] :
      ( aa_o_o(Pa,aa_nat_o(case_nat_o((F1),F22),Nat))
    <=> ( ( ( Nat = zero_zero_nat )
         => aa_o_o(Pa,(F1)) )
        & ( ( Nat = aa_nat_nat(suc,aa_nat_nat(pred,Nat)) )
         => aa_o_o(Pa,aa_nat_o(F22,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_5482_nat_Osplit__sels_I1_J,axiom,
    ! [Pa: fun_nat_o,F1: nat,F22: fun_nat_nat,Nat: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(case_nat_nat(F1,F22),Nat))
    <=> ( ( ( Nat = zero_zero_nat )
         => aa_nat_o(Pa,F1) )
        & ( ( Nat = aa_nat_nat(suc,aa_nat_nat(pred,Nat)) )
         => aa_nat_o(Pa,aa_nat_nat(F22,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_5483_nat_Osplit__sels_I2_J,axiom,
    ! [Pa: fun_option_num_o,F1: option_num,F22: fun_nat_option_num,Nat: nat] :
      ( aa_option_num_o(Pa,aa_nat_option_num(case_nat_option_num(F1,F22),Nat))
    <=> ~ ( ( ( Nat = zero_zero_nat )
            & ~ aa_option_num_o(Pa,F1) )
          | ( ( Nat = aa_nat_nat(suc,aa_nat_nat(pred,Nat)) )
            & ~ aa_option_num_o(Pa,aa_nat_option_num(F22,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_5484_nat_Osplit__sels_I2_J,axiom,
    ! [Pa: fun_o_o,F1: $o,F22: fun_nat_o,Nat: nat] :
      ( aa_o_o(Pa,aa_nat_o(case_nat_o((F1),F22),Nat))
    <=> ~ ( ( ( Nat = zero_zero_nat )
            & ~ aa_o_o(Pa,(F1)) )
          | ( ( Nat = aa_nat_nat(suc,aa_nat_nat(pred,Nat)) )
            & ~ aa_o_o(Pa,aa_nat_o(F22,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_5485_nat_Osplit__sels_I2_J,axiom,
    ! [Pa: fun_nat_o,F1: nat,F22: fun_nat_nat,Nat: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(case_nat_nat(F1,F22),Nat))
    <=> ~ ( ( ( Nat = zero_zero_nat )
            & ~ aa_nat_o(Pa,F1) )
          | ( ( Nat = aa_nat_nat(suc,aa_nat_nat(pred,Nat)) )
            & ~ aa_nat_o(Pa,aa_nat_nat(F22,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_5486_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fl,N))) ).

% mask_eq_sum_exp_nat
tff(fact_5487_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_5488_Ints__sum,axiom,
    ! [A3: set_se7855581050983116737at_nat,F: fun_se9142667792015812128at_int] :
      ( ! [X4: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(X4,A3)
         => member_int(aa_set4746584241299694929at_int(F,X4),ring_1_Ints_int) )
     => member_int(groups178575644746855891at_int(F,A3),ring_1_Ints_int) ) ).

% Ints_sum
tff(fact_5489_Ints__sum,axiom,
    ! [A3: set_se6260736226359567993nt_int,F: fun_se9146809000320981208nt_int] :
      ( ! [X4: set_Pr958786334691620121nt_int] :
          ( member2340774599025711042nt_int(X4,A3)
         => member_int(aa_set4750725449604864009nt_int(F,X4),ring_1_Ints_int) )
     => member_int(groups182716853052024971nt_int(F,A3),ring_1_Ints_int) ) ).

% Ints_sum
tff(fact_5490_Ints__sum,axiom,
    ! [A3: set_set_nat,F: fun_set_nat_int] :
      ( ! [X4: set_nat] :
          ( member_set_nat(X4,A3)
         => member_int(aa_set_nat_int(F,X4),ring_1_Ints_int) )
     => member_int(groups8292507037921071086at_int(F,A3),ring_1_Ints_int) ) ).

% Ints_sum
tff(fact_5491_Ints__sum,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => member_int(aa_nat_int(F,X4),ring_1_Ints_int) )
     => member_int(aa_set_nat_int(groups3539618377306564664at_int(F),A3),ring_1_Ints_int) ) ).

% Ints_sum
tff(fact_5492_Ints__sum,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => member_int(aa_int_int(F,X4),ring_1_Ints_int) )
     => member_int(groups4538972089207619220nt_int(F,A3),ring_1_Ints_int) ) ).

% Ints_sum
tff(fact_5493_fst__comp__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger] : comp_P3428181540658402820nteger(produc8508995932063986495nteger,aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F)) = produc8508995932063986495nteger ).

% fst_comp_apsnd
tff(fact_5494_fst__comp__apsnd,axiom,
    ! [F: fun_nat_nat] : comp_P1567445206330693457at_nat(product_fst_nat_nat,produc3094765310956027504at_nat(F)) = product_fst_nat_nat ).

% fst_comp_apsnd
tff(fact_5495_fst__comp__apsnd,axiom,
    ! [F: fun_int_int] : comp_P5356232565778667509nt_int(product_fst_int_int,produc4463282112584876420nt_int(F)) = product_fst_int_int ).

% fst_comp_apsnd
tff(fact_5496_fst__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger,Xa: produc8923325533196201883nteger] : aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),Xa)) = aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,Xa) ).

% fst_apsnd
tff(fact_5497_fst__apsnd,axiom,
    ! [F: fun_nat_nat,Xa: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_Pro262675715343573244at_nat(produc3094765310956027504at_nat(F),Xa)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa) ).

% fst_apsnd
tff(fact_5498_fst__apsnd,axiom,
    ! [F: fun_int_int,Xa: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_Pro6970504127699319548nt_int(produc4463282112584876420nt_int(F),Xa)) = aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa) ).

% fst_apsnd
tff(fact_5499_of__int__sum,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A3)) = groups3906332499630173760nt_rat(aTP_Lamm_fm(F),A3) ).

% of_int_sum
tff(fact_5500_of__int__sum,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A3)) = groups4538972089207619220nt_int(aTP_Lamm_fn(F),A3) ).

% of_int_sum
tff(fact_5501_of__nat__sum,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A3)) = groups4538972089207619220nt_int(aTP_Lamm_fo(F),A3) ).

% of_nat_sum
tff(fact_5502_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_fp(F)),A3) ).

% of_nat_sum
tff(fact_5503_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fq(F)),A3) ).

% of_nat_sum
tff(fact_5504_and__nat__numerals_I1_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_5505_and__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_5506_and__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_5507_and__nat__numerals_I2_J,axiom,
    ! [Y3: 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,Y3))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_5508_and__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_5509_Suc__0__and__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_5510_one__div__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_Pro6799258434465214121nt_int(product_fst_int_int,unique5052692396658037445od_int(one,N)) ).

% one_div_numeral
tff(fact_5511_one__div__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,N)) ).

% one_div_numeral
tff(fact_5512_one__div__numeral,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,unique3479559517661332726nteger(one,N)) ).

% one_div_numeral
tff(fact_5513_fstI,axiom,
    ! [Xa: produc7509114972854644512nteger,Y3: fun_Co3460408610320954864e_term,Z: produc8923325533196201883nteger] :
      ( ( Xa = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Y3),Z) )
     => ( aa_Pro1361292240318994510e_term(produc3547674943397867238nteger,Xa) = Y3 ) ) ).

% fstI
tff(fact_5514_fstI,axiom,
    ! [Xa: produc9140211490168418807atural,Y3: fun_Pr285225112257423215e_term,Z: produc7822875418678951345atural] :
      ( ( Xa = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,Y3),Z) )
     => ( aa_Pro1606062155721680164e_term(produc5314839617791632573atural,Xa) = Y3 ) ) ).

% fstI
tff(fact_5515_fstI,axiom,
    ! [Xa: produc2453742332425399446nteger,Y3: fun_Pr5098669352780829690e_term,Z: produc8923325533196201883nteger] :
      ( ( Xa = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Y3),Z) )
     => ( aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,Xa) = Y3 ) ) ).

% fstI
tff(fact_5516_fstI,axiom,
    ! [Xa: produc3043004577601930755nt_int,Y3: fun_Pr7763680713840462443e_term,Z: product_prod_int_int] :
      ( ( Xa = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Y3),Z) )
     => ( aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,Xa) = Y3 ) ) ).

% fstI
tff(fact_5517_fstI,axiom,
    ! [Xa: produc4788131091073775919nt_int,Y3: fun_in5730379987311621695e_term,Z: product_prod_int_int] :
      ( ( Xa = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Y3),Z) )
     => ( aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,Xa) = Y3 ) ) ).

% fstI
tff(fact_5518_fstI,axiom,
    ! [Xa: product_prod_nat_nat,Y3: nat,Z: nat] :
      ( ( Xa = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),Z) )
     => ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa) = Y3 ) ) ).

% fstI
tff(fact_5519_fstI,axiom,
    ! [Xa: product_prod_int_int,Y3: int,Z: int] :
      ( ( Xa = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y3),Z) )
     => ( aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa) = Y3 ) ) ).

% fstI
tff(fact_5520_fst__eqD,axiom,
    ! [Xa: fun_Co3460408610320954864e_term,Y3: produc8923325533196201883nteger,A: fun_Co3460408610320954864e_term] :
      ( ( aa_Pro1361292240318994510e_term(produc3547674943397867238nteger,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Xa),Y3)) = A )
     => ( Xa = A ) ) ).

% fst_eqD
tff(fact_5521_fst__eqD,axiom,
    ! [Xa: fun_Pr285225112257423215e_term,Y3: produc7822875418678951345atural,A: fun_Pr285225112257423215e_term] :
      ( ( aa_Pro1606062155721680164e_term(produc5314839617791632573atural,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,Xa),Y3)) = A )
     => ( Xa = A ) ) ).

% fst_eqD
tff(fact_5522_fst__eqD,axiom,
    ! [Xa: fun_Pr5098669352780829690e_term,Y3: produc8923325533196201883nteger,A: fun_Pr5098669352780829690e_term] :
      ( ( aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Xa),Y3)) = A )
     => ( Xa = A ) ) ).

% fst_eqD
tff(fact_5523_fst__eqD,axiom,
    ! [Xa: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int,A: fun_Pr7763680713840462443e_term] :
      ( ( aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Xa),Y3)) = A )
     => ( Xa = A ) ) ).

% fst_eqD
tff(fact_5524_fst__eqD,axiom,
    ! [Xa: fun_in5730379987311621695e_term,Y3: product_prod_int_int,A: fun_in5730379987311621695e_term] :
      ( ( aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Xa),Y3)) = A )
     => ( Xa = A ) ) ).

% fst_eqD
tff(fact_5525_fst__eqD,axiom,
    ! [Xa: nat,Y3: nat,A: nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y3)) = A )
     => ( Xa = A ) ) ).

% fst_eqD
tff(fact_5526_fst__eqD,axiom,
    ! [Xa: int,Y3: int,A: int] :
      ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y3)) = A )
     => ( Xa = A ) ) ).

% fst_eqD
tff(fact_5527_fst__conv,axiom,
    ! [X1: fun_Co3460408610320954864e_term,X2: produc8923325533196201883nteger] : aa_Pro1361292240318994510e_term(produc3547674943397867238nteger,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X1),X2)) = X1 ).

% fst_conv
tff(fact_5528_fst__conv,axiom,
    ! [X1: fun_Pr285225112257423215e_term,X2: produc7822875418678951345atural] : aa_Pro1606062155721680164e_term(produc5314839617791632573atural,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,X1),X2)) = X1 ).

% fst_conv
tff(fact_5529_fst__conv,axiom,
    ! [X1: fun_Pr5098669352780829690e_term,X2: produc8923325533196201883nteger] : aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X1),X2)) = X1 ).

% fst_conv
tff(fact_5530_fst__conv,axiom,
    ! [X1: fun_Pr7763680713840462443e_term,X2: product_prod_int_int] : aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X1),X2)) = X1 ).

% fst_conv
tff(fact_5531_fst__conv,axiom,
    ! [X1: fun_in5730379987311621695e_term,X2: product_prod_int_int] : aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X1),X2)) = X1 ).

% fst_conv
tff(fact_5532_fst__conv,axiom,
    ! [X1: nat,X2: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = X1 ).

% fst_conv
tff(fact_5533_fst__conv,axiom,
    ! [X1: int,X2: int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = X1 ).

% fst_conv
tff(fact_5534_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fr(F),A)),A3)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),A) ).

% mod_sum_eq
tff(fact_5535_mod__sum__eq,axiom,
    ! [F: fun_int_int,A: int,A3: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_fs(F),A),A3)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(F,A3)),A) ).

% mod_sum_eq
tff(fact_5536_fst__def,axiom,
    ! [Prod: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_ar),Prod) ).

% fst_def
tff(fact_5537_fst__def,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_ft),Prod) ).

% fst_def
tff(fact_5538_sum__power__add,axiom,
    ! [Xa: int,M: nat,I5: set_nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_fu(Xa),M)),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),M)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xa)),I5)) ).

% sum_power_add
tff(fact_5539_sum__power__add,axiom,
    ! [Xa: code_integer,M: nat,I5: set_nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aTP_Lamm_fv(Xa),M)),I5) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),M)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(Xa)),I5)) ).

% sum_power_add
tff(fact_5540_sum__power__add,axiom,
    ! [Xa: rat,M: nat,I5: set_nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_fw(Xa),M)),I5) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xa),M)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(Xa)),I5)) ).

% sum_power_add
tff(fact_5541_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_av),Nat) ).

% pred_def
tff(fact_5542_integer__of__num_I2_J,axiom,
    ! [N: num] :
      code_integer_of_num(aa_num_num(bit0,N)) = $let(
        k: code_integer,
        k:= code_integer_of_num(N),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,k),k) ) ).

% integer_of_num(2)
tff(fact_5543_mask__eq__sum__exp,axiom,
    ! [N: nat] : minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N),one_one_Code_natural) = groups6325495683096345652atural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fl,N))) ).

% mask_eq_sum_exp
tff(fact_5544_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fl,N))) ).

% mask_eq_sum_exp
tff(fact_5545_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)),one_one_Code_integer) = aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fl,N))) ).

% mask_eq_sum_exp
tff(fact_5546_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fl,N))) ).

% mask_eq_sum_exp
tff(fact_5547_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = $ite(
        ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_5548_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,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_5549_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_fx(F),A3)) ).

% sum_abs_ge_zero
tff(fact_5550_sum__abs,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A3))),groups4538972089207619220nt_int(aTP_Lamm_fx(F),A3)) ).

% sum_abs
tff(fact_5551_convex__sum__bound__le,axiom,
    ! [I5: set_nat,Xa: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Xa,I2)) )
     => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Xa),I5) = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(A,I2)),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_fy(Xa),A)),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5552_convex__sum__bound__le,axiom,
    ! [I5: set_int,Xa: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( member_int(I2,I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Xa,I2)) )
     => ( ( groups7873554091576472773nteger(Xa,I5) = one_one_Code_integer )
       => ( ! [I2: int] :
              ( member_int(I2,I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_int_Code_integer(A,I2)),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_fz(Xa),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5553_convex__sum__bound__le,axiom,
    ! [I5: set_nat,Xa: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Xa,I2)) )
     => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(Xa),I5) = one_one_rat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_nat_rat(A,I2),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ga(Xa),A)),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5554_convex__sum__bound__le,axiom,
    ! [I5: set_int,Xa: fun_int_rat,A: fun_int_rat,B: rat,Delta: rat] :
      ( ! [I2: int] :
          ( member_int(I2,I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Xa,I2)) )
     => ( ( groups3906332499630173760nt_rat(Xa,I5) = one_one_rat )
       => ( ! [I2: int] :
              ( member_int(I2,I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_int_rat(A,I2),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_gb(Xa),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5555_convex__sum__bound__le,axiom,
    ! [I5: set_nat,Xa: fun_nat_int,A: fun_nat_int,B: int,Delta: int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(Xa,I2)) )
     => ( ( aa_set_nat_int(groups3539618377306564664at_int(Xa),I5) = one_one_int )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(A,I2)),B))),Delta) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_gc(Xa),A)),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5556_convex__sum__bound__le,axiom,
    ! [I5: set_int,Xa: fun_int_int,A: fun_int_int,B: int,Delta: int] :
      ( ! [I2: int] :
          ( member_int(I2,I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(Xa,I2)) )
     => ( ( groups4538972089207619220nt_int(Xa,I5) = one_one_int )
       => ( ! [I2: int] :
              ( member_int(I2,I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(A,I2)),B))),Delta) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gd(Xa),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5557_convex__sum__bound__le,axiom,
    ! [I5: set_set_nat,Xa: fun_se7586849273513198688nteger,A: fun_se7586849273513198688nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(Xa,I2)) )
     => ( ( groups9190459664516455967nteger(Xa,I5) = one_one_Code_integer )
       => ( ! [I2: set_nat] :
              ( member_set_nat(I2,I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_set871585252387937313nteger(A,I2)),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups9190459664516455967nteger(aa_fun4587287018503165694nteger(aTP_Lamm_ge(Xa),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5558_convex__sum__bound__le,axiom,
    ! [I5: set_set_nat,Xa: fun_set_nat_rat,A: fun_set_nat_rat,B: rat,Delta: rat] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_set_nat_rat(Xa,I2)) )
     => ( ( groups7659867448343625626at_rat(Xa,I5) = one_one_rat )
       => ( ! [I2: set_nat] :
              ( member_set_nat(I2,I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_set_nat_rat(A,I2),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups7659867448343625626at_rat(aa_fun8478617152544252148at_rat(aTP_Lamm_gf(Xa),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5559_convex__sum__bound__le,axiom,
    ! [I5: set_set_nat,Xa: fun_set_nat_int,A: fun_set_nat_int,B: int,Delta: int] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_set_nat_int(Xa,I2)) )
     => ( ( groups8292507037921071086at_int(Xa,I5) = one_one_int )
       => ( ! [I2: set_nat] :
              ( member_set_nat(I2,I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(A,I2)),B))),Delta) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups8292507037921071086at_int(aa_fun3600621996379952540at_int(aTP_Lamm_gg(Xa),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5560_convex__sum__bound__le,axiom,
    ! [I5: set_se7855581050983116737at_nat,Xa: fun_se2775971902884351953nteger,A: fun_se2775971902884351953nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(I2,I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set5601480140167866242nteger(Xa,I2)) )
     => ( ( groups5114918151688220676nteger(Xa,I5) = one_one_Code_integer )
       => ( ! [I2: set_Pr1261947904930325089at_nat] :
              ( member2643936169264416010at_nat(I2,I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_set5601480140167866242nteger(A,I2)),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups5114918151688220676nteger(aa_fun5683560775029648636nteger(aTP_Lamm_gh(Xa),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5561_abs__sum__abs,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_int(abs_abs_int,groups4538972089207619220nt_int(aTP_Lamm_fx(F),A3)) = groups4538972089207619220nt_int(aTP_Lamm_fx(F),A3) ).

% abs_sum_abs
tff(fact_5562_sum_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ci),A3) = zero_zero_nat ).

% sum.neutral_const
tff(fact_5563_sum_Oneutral__const,axiom,
    ! [A3: set_int] : groups4538972089207619220nt_int(aTP_Lamm_cj,A3) = zero_zero_int ).

% sum.neutral_const
tff(fact_5564_sum__SucD,axiom,
    ! [F: fun_nat_nat,A3: set_nat,N: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_nat_nat(suc,N) )
     => ? [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4)) ) ) ).

% sum_SucD
tff(fact_5565_int__sum,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A3)) = groups4538972089207619220nt_int(aTP_Lamm_fo(F),A3) ).

% int_sum
tff(fact_5566_int__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_fp(F)),A3) ).

% int_sum
tff(fact_5567_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B3: set_nat,A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_gi(G),B3)),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_gk(G),A3)),B3) ).

% sum.swap
tff(fact_5568_sum_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B3: set_int,A3: set_int] : groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_gl(G),B3),A3) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_gn(G),A3),B3) ).

% sum.swap
tff(fact_5569_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_nat,A3: set_int] :
      ( ( groups4541462559716669496nt_nat(G,A3) != zero_zero_nat )
     => ~ ! [A2: int] :
            ( member_int(A2,A3)
           => ( aa_int_nat(G,A2) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5570_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_int,A3: set_nat] :
      ( ( aa_set_nat_int(groups3539618377306564664at_int(G),A3) != zero_zero_int )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
           => ( aa_nat_int(G,A2) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5571_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_Code_integer,A3: set_nat] :
      ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),A3) != zero_z3403309356797280102nteger )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
           => ( aa_nat_Code_integer(G,A2) = zero_z3403309356797280102nteger ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5572_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_Code_integer,A3: set_int] :
      ( ( groups7873554091576472773nteger(G,A3) != zero_z3403309356797280102nteger )
     => ~ ! [A2: int] :
            ( member_int(A2,A3)
           => ( aa_int_Code_integer(G,A2) = zero_z3403309356797280102nteger ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5573_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A3: set_nat] :
      ( ( aa_set_nat_rat(groups2906978787729119204at_rat(G),A3) != zero_zero_rat )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
           => ( aa_nat_rat(G,A2) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5574_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A3: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A3) != zero_zero_rat )
     => ~ ! [A2: int] :
            ( member_int(A2,A3)
           => ( aa_int_rat(G,A2) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5575_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_Code_natural,A3: set_nat] :
      ( ( groups6325495683096345652atural(G,A3) != zero_z2226904508553997617atural )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
           => ( aa_nat_Code_natural(G,A2) = zero_z2226904508553997617atural ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5576_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_Code_natural,A3: set_int] :
      ( ( groups6697149243333190288atural(G,A3) != zero_z2226904508553997617atural )
     => ~ ! [A2: int] :
            ( member_int(A2,A3)
           => ( aa_int_Code_natural(G,A2) = zero_z2226904508553997617atural ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5577_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_nat,A3: set_nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) != zero_zero_nat )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
           => ( aa_nat_nat(G,A2) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5578_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_int,A3: set_int] :
      ( ( groups4538972089207619220nt_int(G,A3) != zero_zero_int )
     => ~ ! [A2: int] :
            ( member_int(A2,A3)
           => ( aa_int_int(G,A2) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5579_sum_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_5580_sum_Oneutral,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => ( aa_int_int(G,X4) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A3) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_5581_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),K5)),aa_set_nat_rat(groups2906978787729119204at_rat(G),K5)) ) ).

% sum_mono
tff(fact_5582_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( member_int(I2,K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5)) ) ).

% sum_mono
tff(fact_5583_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( member_int(I2,K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5)) ) ).

% sum_mono
tff(fact_5584_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),K5)),aa_set_nat_int(groups3539618377306564664at_int(G),K5)) ) ).

% sum_mono
tff(fact_5585_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I2)),aa_nat_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),K5)),aa_set_nat_nat(groups3542108847815614940at_nat(G),K5)) ) ).

% sum_mono
tff(fact_5586_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_int,G: fun_int_int] :
      ( ! [I2: int] :
          ( member_int(I2,K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,I2)),aa_int_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4538972089207619220nt_int(F,K5)),groups4538972089207619220nt_int(G,K5)) ) ).

% sum_mono
tff(fact_5587_sum__mono,axiom,
    ! [K5: set_set_nat,F: fun_set_nat_rat,G: fun_set_nat_rat] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(F,I2)),aa_set_nat_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7659867448343625626at_rat(F,K5)),groups7659867448343625626at_rat(G,K5)) ) ).

% sum_mono
tff(fact_5588_sum__mono,axiom,
    ! [K5: set_set_nat,F: fun_set_nat_nat,G: fun_set_nat_nat] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(F,I2)),aa_set_nat_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8294997508430121362at_nat(F,K5)),groups8294997508430121362at_nat(G,K5)) ) ).

% sum_mono
tff(fact_5589_sum__mono,axiom,
    ! [K5: set_set_nat,F: fun_set_nat_int,G: fun_set_nat_int] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(F,I2)),aa_set_nat_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups8292507037921071086at_int(F,K5)),groups8292507037921071086at_int(G,K5)) ) ).

% sum_mono
tff(fact_5590_sum__mono,axiom,
    ! [K5: set_se7855581050983116737at_nat,F: fun_se588918216276836300at_rat,G: fun_se588918216276836300at_rat] :
      ( ! [I2: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(I2,K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set4113944651722249469at_rat(F,I2)),aa_set4113944651722249469at_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups8769308092024186239at_rat(F,K5)),groups8769308092024186239at_rat(G,K5)) ) ).

% sum_mono
tff(fact_5591_sum__distrib__left,axiom,
    ! [R: nat,F: fun_nat_nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_go(R),F)),A3) ).

% sum_distrib_left
tff(fact_5592_sum__distrib__left,axiom,
    ! [R: int,F: fun_int_int,A3: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,R),groups4538972089207619220nt_int(F,A3)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gp(R),F),A3) ).

% sum_distrib_left
tff(fact_5593_sum__distrib__right,axiom,
    ! [F: fun_nat_nat,A3: set_nat,R: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),R) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gq(F),R)),A3) ).

% sum_distrib_right
tff(fact_5594_sum__distrib__right,axiom,
    ! [F: fun_int_int,A3: set_int,R: int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A3)),R) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_gr(F),R),A3) ).

% sum_distrib_right
tff(fact_5595_sum__product,axiom,
    ! [F: fun_nat_nat,A3: set_nat,G: fun_nat_nat,B3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_gt(F),G),B3)),A3) ).

% sum_product
tff(fact_5596_sum__product,axiom,
    ! [F: fun_int_int,A3: set_int,G: fun_int_int,B3: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A3)),groups4538972089207619220nt_int(G,B3)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_gv(F),G),B3),A3) ).

% sum_product
tff(fact_5597_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,Ha: fun_nat_nat,A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gw(G),Ha)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(Ha),A3)) ).

% sum.distrib
tff(fact_5598_sum_Odistrib,axiom,
    ! [G: fun_int_int,Ha: fun_int_int,A3: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gx(G),Ha),A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A3)),groups4538972089207619220nt_int(Ha,A3)) ).

% sum.distrib
tff(fact_5599_sum__subtractf,axiom,
    ! [F: fun_int_int,G: fun_int_int,A3: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gy(F),G),A3) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A3)),groups4538972089207619220nt_int(G,A3)) ).

% sum_subtractf
tff(fact_5600_sum__negf,axiom,
    ! [F: fun_int_int,A3: set_int] : groups4538972089207619220nt_int(aTP_Lamm_gz(F),A3) = aa_int_int(uminus_uminus_int,groups4538972089207619220nt_int(F,A3)) ).

% sum_negf
tff(fact_5601_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X4)),zero_z3403309356797280102nteger) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3)),zero_z3403309356797280102nteger) ) ).

% sum_nonpos
tff(fact_5602_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X4)),zero_z3403309356797280102nteger) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(F,A3)),zero_z3403309356797280102nteger) ) ).

% sum_nonpos
tff(fact_5603_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_Code_natural] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(F,X4)),zero_z2226904508553997617atural) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups6325495683096345652atural(F,A3)),zero_z2226904508553997617atural) ) ).

% sum_nonpos
tff(fact_5604_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_Code_natural] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_int_Code_natural(F,X4)),zero_z2226904508553997617atural) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups6697149243333190288atural(F,A3)),zero_z2226904508553997617atural) ) ).

% sum_nonpos
tff(fact_5605_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),A3)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_5606_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X4)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A3)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_5607_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A3)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_5608_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),A3)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_5609_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_5610_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X4)),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4538972089207619220nt_int(F,A3)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_5611_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X4)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3)) ) ).

% sum_nonneg
tff(fact_5612_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X4)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),groups7873554091576472773nteger(F,A3)) ) ).

% sum_nonneg
tff(fact_5613_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_Code_natural] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,X4)) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),groups6325495683096345652atural(F,A3)) ) ).

% sum_nonneg
tff(fact_5614_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_Code_natural] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,X4)) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),groups6697149243333190288atural(F,A3)) ) ).

% sum_nonneg
tff(fact_5615_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_set_nat_rat(groups2906978787729119204at_rat(F),A3)) ) ).

% sum_nonneg
tff(fact_5616_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A3)) ) ).

% sum_nonneg
tff(fact_5617_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A3)) ) ).

% sum_nonneg
tff(fact_5618_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_set_nat_int(groups3539618377306564664at_int(F),A3)) ) ).

% sum_nonneg
tff(fact_5619_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) ) ).

% sum_nonneg
tff(fact_5620_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(F,A3)) ) ).

% sum_nonneg
tff(fact_5621_sum__cong__Suc,axiom,
    ! [A3: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),A3)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_nat_nat(suc,X4)),A3)
           => ( 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),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) ) ) ) ).

% sum_cong_Suc
tff(fact_5622_sum__subtractf__nat,axiom,
    ! [A3: set_se7855581050983116737at_nat,G: fun_se4097146774670233028at_nat,F: fun_se4097146774670233028at_nat] :
      ( ! [X4: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(X4,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set4749074711808745205at_nat(G,X4)),aa_set4749074711808745205at_nat(F,X4)) )
     => ( groups181066115255906167at_nat(aa_fun4978281634197224252at_nat(aTP_Lamm_ha(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups181066115255906167at_nat(F,A3)),groups181066115255906167at_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_5623_sum__subtractf__nat,axiom,
    ! [A3: set_se6260736226359567993nt_int,G: fun_se4101287982975402108nt_nat,F: fun_se4101287982975402108nt_nat] :
      ( ! [X4: set_Pr958786334691620121nt_int] :
          ( member2340774599025711042nt_int(X4,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set4753215920113914285nt_nat(G,X4)),aa_set4753215920113914285nt_nat(F,X4)) )
     => ( groups185207323561075247nt_nat(aa_fun209864160807355708nt_nat(aTP_Lamm_hb(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups185207323561075247nt_nat(F,A3)),groups185207323561075247nt_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_5624_sum__subtractf__nat,axiom,
    ! [A3: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X4: set_nat] :
          ( member_set_nat(X4,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(G,X4)),aa_set_nat_nat(F,X4)) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_hc(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A3)),groups8294997508430121362at_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_5625_sum__subtractf__nat,axiom,
    ! [A3: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X4)),aa_int_nat(F,X4)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_hd(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_5626_sum__subtractf__nat,axiom,
    ! [A3: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X4)),aa_nat_nat(F,X4)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_he(G),F)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_5627_sum__comp__morphism,axiom,
    ! [Ha: fun_nat_int,G: fun_nat_nat,A3: set_nat] :
      ( ( aa_nat_int(Ha,zero_zero_nat) = zero_zero_int )
     => ( ! [X4: nat,Y4: nat] : aa_nat_int(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),Y4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ha,X4)),aa_nat_int(Ha,Y4))
       => ( aa_set_nat_int(groups3539618377306564664at_int(comp_nat_int_nat(Ha,G)),A3) = aa_nat_int(Ha,aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)) ) ) ) ).

% sum_comp_morphism
tff(fact_5628_sum__comp__morphism,axiom,
    ! [Ha: fun_nat_Code_integer,G: fun_nat_nat,A3: set_nat] :
      ( ( aa_nat_Code_integer(Ha,zero_zero_nat) = zero_z3403309356797280102nteger )
     => ( ! [X4: nat,Y4: nat] : aa_nat_Code_integer(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),Y4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(Ha,X4)),aa_nat_Code_integer(Ha,Y4))
       => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(comp_n3898172953802868194er_nat(Ha,G)),A3) = aa_nat_Code_integer(Ha,aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)) ) ) ) ).

% sum_comp_morphism
tff(fact_5629_sum__comp__morphism,axiom,
    ! [Ha: fun_nat_rat,G: fun_nat_nat,A3: set_nat] :
      ( ( aa_nat_rat(Ha,zero_zero_nat) = zero_zero_rat )
     => ( ! [X4: nat,Y4: nat] : aa_nat_rat(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),Y4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ha,X4)),aa_nat_rat(Ha,Y4))
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(comp_nat_rat_nat(Ha,G)),A3) = aa_nat_rat(Ha,aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)) ) ) ) ).

% sum_comp_morphism
tff(fact_5630_sum__comp__morphism,axiom,
    ! [Ha: fun_nat_Code_natural,G: fun_nat_nat,A3: set_nat] :
      ( ( aa_nat_Code_natural(Ha,zero_zero_nat) = zero_z2226904508553997617atural )
     => ( ! [X4: nat,Y4: nat] : aa_nat_Code_natural(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),Y4)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(Ha,X4)),aa_nat_Code_natural(Ha,Y4))
       => ( groups6325495683096345652atural(comp_n9163001964490475437al_nat(Ha,G),A3) = aa_nat_Code_natural(Ha,aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)) ) ) ) ).

% sum_comp_morphism
tff(fact_5631_sum__comp__morphism,axiom,
    ! [Ha: fun_int_nat,G: fun_int_int,A3: set_int] :
      ( ( aa_int_nat(Ha,zero_zero_int) = zero_zero_nat )
     => ( ! [X4: int,Y4: int] : aa_int_nat(Ha,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),Y4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Ha,X4)),aa_int_nat(Ha,Y4))
       => ( groups4541462559716669496nt_nat(comp_int_nat_int(Ha,G),A3) = aa_int_nat(Ha,groups4538972089207619220nt_int(G,A3)) ) ) ) ).

% sum_comp_morphism
tff(fact_5632_sum__comp__morphism,axiom,
    ! [Ha: fun_int_Code_integer,G: fun_int_int,A3: set_int] :
      ( ( aa_int_Code_integer(Ha,zero_zero_int) = zero_z3403309356797280102nteger )
     => ( ! [X4: int,Y4: int] : aa_int_Code_integer(Ha,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),Y4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(Ha,X4)),aa_int_Code_integer(Ha,Y4))
       => ( groups7873554091576472773nteger(comp_i1585864551200866970er_int(Ha,G),A3) = aa_int_Code_integer(Ha,groups4538972089207619220nt_int(G,A3)) ) ) ) ).

% sum_comp_morphism
tff(fact_5633_sum__comp__morphism,axiom,
    ! [Ha: fun_int_rat,G: fun_int_int,A3: set_int] :
      ( ( aa_int_rat(Ha,zero_zero_int) = zero_zero_rat )
     => ( ! [X4: int,Y4: int] : aa_int_rat(Ha,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),Y4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Ha,X4)),aa_int_rat(Ha,Y4))
       => ( groups3906332499630173760nt_rat(comp_int_rat_int(Ha,G),A3) = aa_int_rat(Ha,groups4538972089207619220nt_int(G,A3)) ) ) ) ).

% sum_comp_morphism
tff(fact_5634_sum__comp__morphism,axiom,
    ! [Ha: fun_int_Code_natural,G: fun_int_int,A3: set_int] :
      ( ( aa_int_Code_natural(Ha,zero_zero_int) = zero_z2226904508553997617atural )
     => ( ! [X4: int,Y4: int] : aa_int_Code_natural(Ha,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),Y4)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_int_Code_natural(Ha,X4)),aa_int_Code_natural(Ha,Y4))
       => ( groups6697149243333190288atural(comp_i6850693561888474213al_int(Ha,G),A3) = aa_int_Code_natural(Ha,groups4538972089207619220nt_int(G,A3)) ) ) ) ).

% sum_comp_morphism
tff(fact_5635_sum__comp__morphism,axiom,
    ! [Ha: fun_int_nat,G: fun_nat_int,A3: set_nat] :
      ( ( aa_int_nat(Ha,zero_zero_int) = zero_zero_nat )
     => ( ! [X4: int,Y4: int] : aa_int_nat(Ha,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),Y4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Ha,X4)),aa_int_nat(Ha,Y4))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(comp_int_nat_nat(Ha,G)),A3) = aa_int_nat(Ha,aa_set_nat_int(groups3539618377306564664at_int(G),A3)) ) ) ) ).

% sum_comp_morphism
tff(fact_5636_sum__comp__morphism,axiom,
    ! [Ha: fun_Code_integer_nat,G: fun_nat_Code_integer,A3: set_nat] :
      ( ( aa_Code_integer_nat(Ha,zero_z3403309356797280102nteger) = zero_zero_nat )
     => ( ! [X4: code_integer,Y4: code_integer] : aa_Code_integer_nat(Ha,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X4),Y4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(Ha,X4)),aa_Code_integer_nat(Ha,Y4))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(comp_C4049556595663050210at_nat(Ha,G)),A3) = aa_Code_integer_nat(Ha,aa_set871585252387937313nteger(groups7501900531339628137nteger(G),A3)) ) ) ) ).

% sum_comp_morphism
tff(fact_5637_fst__diag__fst,axiom,
    comp_P1567445206330693457at_nat(product_fst_nat_nat,comp_n829915685812334579at_nat(aTP_Lamm_hf,product_fst_nat_nat)) = product_fst_nat_nat ).

% fst_diag_fst
tff(fact_5638_fst__diag__fst,axiom,
    comp_P5356232565778667509nt_int(product_fst_int_int,comp_i5941103828406351439nt_int(aTP_Lamm_hg,product_fst_int_int)) = product_fst_int_int ).

% fst_diag_fst
tff(fact_5639_choose__odd__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_hh(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_5640_choose__odd__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_hi(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_5641_choose__odd__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_hj,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_5642_choose__even__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_hk(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_5643_choose__even__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_hl(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_5644_choose__even__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_hm,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_5645_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_hn(A)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_5646_sum__gp,axiom,
    ! [Xa: rat,M: nat,N: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(Xa)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),
        zero_zero_rat,
        $ite(Xa = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)),divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xa),M),aa_nat_rat(power_power_rat(Xa),aa_nat_nat(suc,N))),minus_minus_rat(one_one_rat,Xa))) ) ).

% sum_gp
tff(fact_5647_gbinomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,M))),one_one_rat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_5648_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_5649_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_5650_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_5651_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N)))) ).

% sum.cl_ivl_Suc
tff(fact_5652_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N: nat] :
      aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_Code_integer(G,aa_nat_nat(suc,N)))) ).

% sum.cl_ivl_Suc
tff(fact_5653_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N)))) ).

% sum.cl_ivl_Suc
tff(fact_5654_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_Code_natural,M: nat,N: nat] :
      groups6325495683096345652atural(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),zero_z2226904508553997617atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,groups6325495683096345652atural(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_Code_natural(G,aa_nat_nat(suc,N)))) ).

% sum.cl_ivl_Suc
tff(fact_5655_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N)))) ).

% sum.cl_ivl_Suc
tff(fact_5656_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_5657_atMost__def,axiom,
    ! [U: set_Pr1261947904930325089at_nat] : set_or2762583123196137896at_nat(U) = collec5514110066124741708at_nat(aTP_Lamm_ho(U)) ).

% atMost_def
tff(fact_5658_atMost__def,axiom,
    ! [U: set_Pr958786334691620121nt_int] : set_or2459421552957432928nt_int(U) = collec5210948495886036740nt_int(aTP_Lamm_hp(U)) ).

% atMost_def
tff(fact_5659_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aTP_Lamm_hq(U)) ).

% atMost_def
tff(fact_5660_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_hr,U)) ).

% atMost_def
tff(fact_5661_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aTP_Lamm_hs(U)) ).

% atMost_def
tff(fact_5662_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_ht,U)) ).

% atMost_def
tff(fact_5663_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_hu,U)) ).

% atMost_def
tff(fact_5664_sum_OatLeast__Suc__atMost__Suc__shift,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(comp_nat_nat_nat(G,suc)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_5665_sum_OatLeastAtMost__shift__bounds,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(comp_nat_nat_nat(G,aa_nat_fun_nat_nat(plus_plus_nat,K))),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_5666_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_hv(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_5667_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_hw(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_5668_sum__power__shift,axiom,
    ! [M: nat,N: nat,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xa)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),M)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xa)),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_5669_sum__power__shift,axiom,
    ! [M: nat,N: nat,Xa: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(Xa)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),M)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(Xa)),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_5670_sum__power__shift,axiom,
    ! [M: nat,N: nat,Xa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(Xa)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xa),M)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(Xa)),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_5671_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_hx(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% sum.atLeastAtMost_rev
tff(fact_5672_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hy,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_5673_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_5674_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_Code_integer,K: nat] :
      ( ( aa_nat_Code_integer(F,zero_zero_nat) = zero_z3403309356797280102nteger )
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5675_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5676_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_Code_natural,K: nat] :
      ( ( aa_nat_Code_natural(F,zero_zero_nat) = zero_z2226904508553997617atural )
     => ( groups6325495683096345652atural(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups6325495683096345652atural(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5677_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_5678_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups2906978787729119204at_rat(comp_nat_rat_nat(G,suc)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_5679_sum_OatLeast0__atMost__Suc__shift,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_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(comp_nat_int_nat(G,suc)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_5680_sum_OatLeast0__atMost__Suc__shift,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_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(G,suc)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_5681_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5682_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_5683_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_5684_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,M)),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5685_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_5686_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_5687_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,aa_nat_nat(suc,N))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5688_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_5689_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_5690_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_hz(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_5691_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_ia(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_5692_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_hv(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_5693_sum__telescope,axiom,
    ! [F: fun_nat_int,I: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_ib(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_5694_sum__telescope,axiom,
    ! [F: fun_nat_Code_integer,I: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_ic(F)),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(F,zero_zero_nat)),aa_nat_Code_integer(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_5695_sum__telescope,axiom,
    ! [F: fun_nat_rat,I: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_id(F)),aa_nat_set_nat(set_ord_atMost_nat,I)) = minus_minus_rat(aa_nat_rat(F,zero_zero_nat),aa_nat_rat(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_5696_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,M)),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_hz(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5697_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_ia(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5698_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_hv(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5699_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_ie(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_5700_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_if(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(F,aa_nat_nat(suc,N))),aa_nat_Code_integer(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_5701_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ig(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,N)),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_5702_sum_OatLeast__atMost__pred__shift,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(G,aTP_Lamm_ih)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.atLeast_atMost_pred_shift
tff(fact_5703_sum__choose__lower,axiom,
    ! [R: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ii,R)),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,R),N))),N) ).

% sum_choose_lower
tff(fact_5704_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ij,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_5705_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ij,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_5706_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_ik(F),A),B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_5707_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_Code_integer,A: nat,B: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_il(F),A,B),zero_z3403309356797280102nteger) ).

% sum_atLeastAtMost_code
tff(fact_5708_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo1949268297981939178at_rat(aTP_Lamm_im(F),A,B,zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_5709_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_Code_natural,A: nat,B: nat] : groups6325495683096345652atural(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo9131927060563008058atural(aTP_Lamm_in(F),A,B,zero_z2226904508553997617atural) ).

% sum_atLeastAtMost_code
tff(fact_5710_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_io(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_5711_subset__CollectI,axiom,
    ! [B3: set_se7855581050983116737at_nat,A3: set_se7855581050983116737at_nat,Q: fun_se4782206882984618170_nat_o,Pa: fun_se4782206882984618170_nat_o] :
      ( ord_le2077887516847798113at_nat(B3,A3)
     => ( ! [X4: set_Pr1261947904930325089at_nat] :
            ( member2643936169264416010at_nat(X4,B3)
           => ( aa_set3702741264012733747_nat_o(Q,X4)
             => aa_set3702741264012733747_nat_o(Pa,X4) ) )
       => ord_le2077887516847798113at_nat(collec5514110066124741708at_nat(aa_fun3315923329236548914_nat_o(aTP_Lamm_cc(B3),Q)),collec5514110066124741708at_nat(aa_fun3315923329236548914_nat_o(aTP_Lamm_cc(A3),Pa))) ) ) ).

% subset_CollectI
tff(fact_5712_subset__CollectI,axiom,
    ! [B3: set_se6260736226359567993nt_int,A3: set_se6260736226359567993nt_int,Q: fun_se7249947736802449410_int_o,Pa: fun_se7249947736802449410_int_o] :
      ( ord_le483042692224249369nt_int(B3,A3)
     => ( ! [X4: set_Pr958786334691620121nt_int] :
            ( member2340774599025711042nt_int(X4,B3)
           => ( aa_set6170482117830564987_int_o(Q,X4)
             => aa_set6170482117830564987_int_o(Pa,X4) ) )
       => ord_le483042692224249369nt_int(collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_cd(B3),Q)),collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_cd(A3),Pa))) ) ) ).

% subset_CollectI
tff(fact_5713_subset__CollectI,axiom,
    ! [B3: set_set_nat,A3: set_set_nat,Q: fun_set_nat_o,Pa: fun_set_nat_o] :
      ( ord_le6893508408891458716et_nat(B3,A3)
     => ( ! [X4: set_nat] :
            ( member_set_nat(X4,B3)
           => ( aa_set_nat_o(Q,X4)
             => aa_set_nat_o(Pa,X4) ) )
       => ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ce(B3),Q)),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ce(A3),Pa))) ) ) ).

% subset_CollectI
tff(fact_5714_subset__CollectI,axiom,
    ! [B3: set_int,A3: set_int,Q: fun_int_o,Pa: fun_int_o] :
      ( ord_less_eq_set_int(B3,A3)
     => ( ! [X4: int] :
            ( member_int(X4,B3)
           => ( aa_int_o(Q,X4)
             => aa_int_o(Pa,X4) ) )
       => ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(B3),Q)),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(A3),Pa))) ) ) ).

% subset_CollectI
tff(fact_5715_subset__CollectI,axiom,
    ! [B3: set_nat,A3: set_nat,Q: fun_nat_o,Pa: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B3),A3)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),B3)
           => ( aa_nat_o(Q,X4)
             => aa_nat_o(Pa,X4) ) )
       => aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(B3),Q))),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(A3),Pa))) ) ) ).

% subset_CollectI
tff(fact_5716_subset__Collect__iff,axiom,
    ! [B3: set_se7855581050983116737at_nat,A3: set_se7855581050983116737at_nat,Pa: fun_se4782206882984618170_nat_o] :
      ( ord_le2077887516847798113at_nat(B3,A3)
     => ( ord_le2077887516847798113at_nat(B3,collec5514110066124741708at_nat(aa_fun3315923329236548914_nat_o(aTP_Lamm_cc(A3),Pa)))
      <=> ! [X3: set_Pr1261947904930325089at_nat] :
            ( member2643936169264416010at_nat(X3,B3)
           => aa_set3702741264012733747_nat_o(Pa,X3) ) ) ) ).

% subset_Collect_iff
tff(fact_5717_subset__Collect__iff,axiom,
    ! [B3: set_se6260736226359567993nt_int,A3: set_se6260736226359567993nt_int,Pa: fun_se7249947736802449410_int_o] :
      ( ord_le483042692224249369nt_int(B3,A3)
     => ( ord_le483042692224249369nt_int(B3,collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_cd(A3),Pa)))
      <=> ! [X3: set_Pr958786334691620121nt_int] :
            ( member2340774599025711042nt_int(X3,B3)
           => aa_set6170482117830564987_int_o(Pa,X3) ) ) ) ).

% subset_Collect_iff
tff(fact_5718_subset__Collect__iff,axiom,
    ! [B3: set_set_nat,A3: set_set_nat,Pa: fun_set_nat_o] :
      ( ord_le6893508408891458716et_nat(B3,A3)
     => ( ord_le6893508408891458716et_nat(B3,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ce(A3),Pa)))
      <=> ! [X3: set_nat] :
            ( member_set_nat(X3,B3)
           => aa_set_nat_o(Pa,X3) ) ) ) ).

% subset_Collect_iff
tff(fact_5719_subset__Collect__iff,axiom,
    ! [B3: set_int,A3: set_int,Pa: fun_int_o] :
      ( ord_less_eq_set_int(B3,A3)
     => ( ord_less_eq_set_int(B3,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(A3),Pa)))
      <=> ! [X3: int] :
            ( member_int(X3,B3)
           => aa_int_o(Pa,X3) ) ) ) ).

% subset_Collect_iff
tff(fact_5720_subset__Collect__iff,axiom,
    ! [B3: set_nat,A3: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B3),A3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B3),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(A3),Pa)))
      <=> ! [X3: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),B3)
           => aa_nat_o(Pa,X3) ) ) ) ).

% subset_Collect_iff
tff(fact_5721_sum_OatLeastAtMost__shift__0,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(G,aa_nat_fun_nat_nat(plus_plus_nat,M))),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_5722_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_rat(groups2906978787729119204at_rat(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),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_5723_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),P))) = 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),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_5724_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),P))) = 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),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_5725_gbinomial__parallel__sum,axiom,
    ! [A: rat,N: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ip(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,N))),one_one_rat)),N) ).

% gbinomial_parallel_sum
tff(fact_5726_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : aa_set4749074711808745205at_nat(groups977919841031483927at_nat(produc6842872674320459806at_nat(G)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_iq(N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.triangle_reindex_eq
tff(fact_5727_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_it(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_5728_vandermonde,axiom,
    ! [M: nat,N: nat,R: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_iu(M),N),R)),aa_nat_set_nat(set_ord_atMost_nat,R)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),R) ).

% vandermonde
tff(fact_5729_sum__gp__basic,axiom,
    ! [Xa: 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),Xa)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xa)),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(aa_int_fun_nat_int(power_power_int,Xa),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_5730_sum__gp__basic,axiom,
    ! [Xa: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),Xa)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(Xa)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_5731_sum__gp__basic,axiom,
    ! [Xa: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(one_one_rat,Xa)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(Xa)),aa_nat_set_nat(set_ord_atMost_nat,N))) = minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xa),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_5732_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,M: nat,N: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_iv(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_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))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_5733_sum__natinterval__diff,axiom,
    ! [F: fun_nat_Code_integer,M: nat,N: nat] :
      aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_iw(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(F,M)),aa_nat_Code_integer(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),zero_z3403309356797280102nteger) ).

% sum_natinterval_diff
tff(fact_5734_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,M: nat,N: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ix(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N),minus_minus_rat(aa_nat_rat(F,M),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_5735_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_iy(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_5736_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_iz(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(F,N)),aa_nat_Code_integer(F,M)) ) ) ).

% sum_telescope''
tff(fact_5737_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ja(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = minus_minus_rat(aa_nat_rat(F,N),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_5738_choose__row__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% choose_row_sum
tff(fact_5739_binomial,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jb(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial
tff(fact_5740_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_jc(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_5741_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_jd(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_5742_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_je(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_5743_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_jf(A)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),M)),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),M)) ).

% gbinomial_sum_lower_neg
tff(fact_5744_binomial__ring,axiom,
    ! [A: code_integer,B: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),N) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_jg(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_5745_binomial__ring,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),N) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_jh(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_5746_binomial__ring,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ji(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_5747_binomial__ring,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jb(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_5748_pochhammer__binomial__sum,axiom,
    ! [A: code_integer,B: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),N) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_jj(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_5749_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),N) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_jk(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_5750_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_jl(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_5751_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Xa)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xa)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_5752_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,Xa: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),Xa)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(Xa)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),M)),aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_5753_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,Xa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(one_one_rat,Xa)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(Xa)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = minus_minus_rat(aa_nat_rat(power_power_rat(Xa),M),aa_nat_rat(power_power_rat(Xa),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_5754_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_jc(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_5755_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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_jd(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_5756_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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_je(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_5757_choose__square__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jm,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N) ).

% choose_square_sum
tff(fact_5758_sum_Ozero__middle,axiom,
    ! [P: nat,K: nat,G: fun_nat_int,Ha: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P)
       => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_jn(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_jo(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_5759_sum_Ozero__middle,axiom,
    ! [P: nat,K: nat,G: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P)
       => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_jp(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_jq(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_5760_sum_Ozero__middle,axiom,
    ! [P: nat,K: nat,G: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_jr(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_js(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_5761_sum_Ozero__middle,axiom,
    ! [P: nat,K: nat,G: fun_nat_Code_natural,Ha: fun_nat_Code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P)
       => ( groups6325495683096345652atural(aa_fun56224542555504168atural(aa_fun1651958757531339674atural(aTP_Lamm_jt(K),G),Ha),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups6325495683096345652atural(aa_fun56224542555504168atural(aa_fun1651958757531339674atural(aTP_Lamm_ju(K),G),Ha),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_5762_sum_Ozero__middle,axiom,
    ! [P: nat,K: nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_jv(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_jw(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_5763_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: rat,Xa: rat,Y3: rat] : aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_jx(M),A),Xa),Y3)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_jy(M),A),Xa),Y3)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_5764_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_jz(K)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_5765_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_av),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,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_5766_sum__gp0,axiom,
    ! [Xa: rat,N: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(Xa)),aa_nat_set_nat(set_ord_atMost_nat,N)) = $ite(Xa = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),divide_divide_rat(minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xa),aa_nat_nat(suc,N))),minus_minus_rat(one_one_rat,Xa))) ).

% sum_gp0
tff(fact_5767_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: rat,Xa: rat,Y3: rat] : aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_jx(M),A),Xa),Y3)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ka(M),A),Xa),Y3)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_5768_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_kb(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_5769_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_kc(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_5770_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_kd,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_5771_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ke(M)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_5772_double__arith__series,axiom,
    ! [A: code_natural,D: code_natural,N: nat] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),groups6325495683096345652atural(aa_Cod5320533616963873627atural(aTP_Lamm_kf(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(semiri3763490453095760265atural,N)),one_one_Code_natural)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N)),D))) ).

% double_arith_series
tff(fact_5773_double__arith__series,axiom,
    ! [A: code_integer,D: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_kg(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),D))) ).

% double_arith_series
tff(fact_5774_double__arith__series,axiom,
    ! [A: rat,D: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_kh(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),D))) ).

% double_arith_series
tff(fact_5775_double__arith__series,axiom,
    ! [A: int,D: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ki(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,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D))) ).

% double_arith_series
tff(fact_5776_double__arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kj(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,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D))) ).

% double_arith_series
tff(fact_5777_double__gauss__sum,axiom,
    ! [N: nat] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),groups6325495683096345652atural(semiri3763490453095760265atural,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(semiri3763490453095760265atural,N)),one_one_Code_natural)) ).

% double_gauss_sum
tff(fact_5778_double__gauss__sum,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)) ).

% double_gauss_sum
tff(fact_5779_double__gauss__sum,axiom,
    ! [N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(semiri681578069525770553at_rat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum
tff(fact_5780_double__gauss__sum,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(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_5781_double__gauss__sum,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(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_5782_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% binomial_r_part_sum
tff(fact_5783_choose__linear__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kk,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% choose_linear_sum
tff(fact_5784_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kl(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,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_5785_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_av),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,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_5786_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_km(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_5787_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_kn(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_5788_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ko,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_5789_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),groups6325495683096345652atural(semiri3763490453095760265atural,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(semiri3763490453095760265atural,N)),one_one_Code_natural)) ).

% double_gauss_sum_from_Suc_0
tff(fact_5790_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)) ).

% double_gauss_sum_from_Suc_0
tff(fact_5791_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(semiri681578069525770553at_rat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_5792_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(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_5793_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(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_5794_arith__series,axiom,
    ! [A: code_integer,D: code_integer,N: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_kg(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),D)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_5795_arith__series,axiom,
    ! [A: code_natural,D: code_natural,N: nat] : groups6325495683096345652atural(aa_Cod5320533616963873627atural(aTP_Lamm_kf(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(semiri3763490453095760265atural,N)),one_one_Code_natural)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N)),D))),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_5796_arith__series,axiom,
    ! [A: int,D: int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ki(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_5797_arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kj(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,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_5798_gauss__sum,axiom,
    ! [N: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_5799_gauss__sum,axiom,
    ! [N: nat] : groups6325495683096345652atural(semiri3763490453095760265atural,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(semiri3763490453095760265atural,N)),one_one_Code_natural)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_5800_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(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_5801_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,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_5802_sum__gp__offset,axiom,
    ! [Xa: rat,M: nat,N: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(Xa)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = $ite(Xa = one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xa),M)),minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xa),aa_nat_nat(suc,N)))),minus_minus_rat(one_one_rat,Xa))) ).

% sum_gp_offset
tff(fact_5803_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_5804_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : groups6325495683096345652atural(semiri3763490453095760265atural,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(semiri3763490453095760265atural,N)),one_one_Code_natural)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_5805_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(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_5806_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,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_5807_gchoose__row__sum__weighted,axiom,
    ! [R: rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_hn(R)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)),numeral_numeral_rat(aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_5808_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_kp(Z)),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,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_5809_ex__nat__less,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ? [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),N)
          & aa_nat_o(Pa,M7) )
    <=> ? [X3: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))
          & aa_nat_o(Pa,X3) ) ) ).

% ex_nat_less
tff(fact_5810_all__nat__less,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ! [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),N)
         => aa_nat_o(Pa,M7) )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))
         => aa_nat_o(Pa,X3) ) ) ).

% all_nat_less
tff(fact_5811_bezw_Osimps,axiom,
    ! [Xa: nat,Y3: nat] :
      aa_nat4424723855478970923nt_int(bezw(Xa),Y3) = $ite(Y3 = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_nat4424723855478970923nt_int(bezw(Y3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y3)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_nat4424723855478970923nt_int(bezw(Y3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y3)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_nat4424723855478970923nt_int(bezw(Y3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y3)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3)))))) ).

% bezw.simps
tff(fact_5812_bezw_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y3: product_prod_int_int] :
      ( ( aa_nat4424723855478970923nt_int(bezw(Xa),Xaa) = Y3 )
     => ( Y3 = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_nat4424723855478970923nt_int(bezw(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_nat4424723855478970923nt_int(bezw(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_nat4424723855478970923nt_int(bezw(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_5813_bezw__non__0,axiom,
    ! [Y3: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y3)
     => ( aa_nat4424723855478970923nt_int(bezw(Xa),Y3) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_nat4424723855478970923nt_int(bezw(Y3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y3)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_nat4424723855478970923nt_int(bezw(Y3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y3)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_nat4424723855478970923nt_int(bezw(Y3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y3)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y3))))) ) ) ).

% bezw_non_0
tff(fact_5814_of__nat__id,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = N ).

% of_nat_id
tff(fact_5815_Ints__prod,axiom,
    ! [A3: set_se7855581050983116737at_nat,F: fun_se9142667792015812128at_int] :
      ( ! [X4: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(X4,A3)
         => member_int(aa_set4746584241299694929at_int(F,X4),ring_1_Ints_int) )
     => member_int(groups1212143684287452584at_int(F,A3),ring_1_Ints_int) ) ).

% Ints_prod
tff(fact_5816_Ints__prod,axiom,
    ! [A3: set_se6260736226359567993nt_int,F: fun_se9146809000320981208nt_int] :
      ( ! [X4: set_Pr958786334691620121nt_int] :
          ( member2340774599025711042nt_int(X4,A3)
         => member_int(aa_set4750725449604864009nt_int(F,X4),ring_1_Ints_int) )
     => member_int(groups1216284892592621664nt_int(F,A3),ring_1_Ints_int) ) ).

% Ints_prod
tff(fact_5817_Ints__prod,axiom,
    ! [A3: set_set_nat,F: fun_set_nat_int] :
      ( ! [X4: set_nat] :
          ( member_set_nat(X4,A3)
         => member_int(aa_set_nat_int(F,X4),ring_1_Ints_int) )
     => member_int(groups4246057289670975065at_int(F,A3),ring_1_Ints_int) ) ).

% Ints_prod
tff(fact_5818_Ints__prod,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => member_int(aa_nat_int(F,X4),ring_1_Ints_int) )
     => member_int(aa_set_nat_int(groups705719431365010083at_int(F),A3),ring_1_Ints_int) ) ).

% Ints_prod
tff(fact_5819_Ints__prod,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => member_int(aa_int_int(F,X4),ring_1_Ints_int) )
     => member_int(groups1705073143266064639nt_int(F,A3),ring_1_Ints_int) ) ).

% Ints_prod
tff(fact_5820_snd__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger,Xa: produc8923325533196201883nteger] : aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),Xa)) = aa_Cod1783974652533396728nteger(F,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Xa)) ).

% snd_apsnd
tff(fact_5821_snd__apsnd,axiom,
    ! [F: fun_int_int,Xa: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_Pro6970504127699319548nt_int(produc4463282112584876420nt_int(F),Xa)) = aa_int_int(F,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)) ).

% snd_apsnd
tff(fact_5822_snd__apsnd,axiom,
    ! [F: fun_nat_nat,Xa: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_Pro262675715343573244at_nat(produc3094765310956027504at_nat(F),Xa)) = aa_nat_nat(F,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa)) ).

% snd_apsnd
tff(fact_5823_apsnd__eq__conv,axiom,
    ! [F: fun_Co8499238673658658103nteger,Xa: produc8923325533196201883nteger,G: fun_Co8499238673658658103nteger] :
      ( ( aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),Xa) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,G),Xa) )
    <=> ( aa_Cod1783974652533396728nteger(F,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Xa)) = aa_Cod1783974652533396728nteger(G,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Xa)) ) ) ).

% apsnd_eq_conv
tff(fact_5824_prod_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_kq),A3) = one_one_nat ).

% prod.neutral_const
tff(fact_5825_prod_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_kr),A3) = one_one_int ).

% prod.neutral_const
tff(fact_5826_prod_Oneutral__const,axiom,
    ! [A3: set_int] : groups1705073143266064639nt_int(aTP_Lamm_ks,A3) = one_one_int ).

% prod.neutral_const
tff(fact_5827_of__nat__prod,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A3)) = groups1705073143266064639nt_int(aTP_Lamm_fo(F),A3) ).

% of_nat_prod
tff(fact_5828_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fq(F)),A3) ).

% of_nat_prod
tff(fact_5829_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_fp(F)),A3) ).

% of_nat_prod
tff(fact_5830_of__int__prod,axiom,
    ! [F: fun_nat_int,A3: set_nat] : aa_int_rat(ring_1_of_int_rat,aa_set_nat_int(groups705719431365010083at_int(F),A3)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_kt(F)),A3) ).

% of_int_prod
tff(fact_5831_of__int__prod,axiom,
    ! [F: fun_nat_int,A3: set_nat] : aa_int_int(ring_1_of_int_int,aa_set_nat_int(groups705719431365010083at_int(F),A3)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ku(F)),A3) ).

% of_int_prod
tff(fact_5832_of__int__prod,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A3)) = groups1072433553688619179nt_rat(aTP_Lamm_fm(F),A3) ).

% of_int_prod
tff(fact_5833_of__int__prod,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A3)) = groups1705073143266064639nt_int(aTP_Lamm_fn(F),A3) ).

% of_int_prod
tff(fact_5834_prod_Ocollapse,axiom,
    ! [Prod: produc7509114972854644512nteger] : aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,aa_Pro1361292240318994510e_term(produc3547674943397867238nteger,Prod)),aa_Pro5745608113995628733nteger(produc9082904322759093540nteger,Prod)) = Prod ).

% prod.collapse
tff(fact_5835_prod_Ocollapse,axiom,
    ! [Prod: produc9140211490168418807atural] : aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,aa_Pro1606062155721680164e_term(produc5314839617791632573atural,Prod)),aa_Pro2627516385176489788atural(produc199808700204356347atural,Prod)) = Prod ).

% prod.collapse
tff(fact_5836_prod_Ocollapse,axiom,
    ! [Prod: produc2453742332425399446nteger] : aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,Prod)),aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,Prod)) = Prod ).

% prod.collapse
tff(fact_5837_prod_Ocollapse,axiom,
    ! [Prod: produc3043004577601930755nt_int] : aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,Prod)),aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,Prod)) = Prod ).

% prod.collapse
tff(fact_5838_prod_Ocollapse,axiom,
    ! [Prod: produc4788131091073775919nt_int] : aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,Prod)),aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,Prod)) = Prod ).

% prod.collapse
tff(fact_5839_prod_Ocollapse,axiom,
    ! [Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) = Prod ).

% prod.collapse
tff(fact_5840_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] : aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) = Prod ).

% prod.collapse
tff(fact_5841_snd__comp__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger] : comp_P3428181540658402820nteger(produc6174133586879617921nteger,aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F)) = comp_C1043215000170864188nteger(F,produc6174133586879617921nteger) ).

% snd_comp_apsnd
tff(fact_5842_snd__comp__apsnd,axiom,
    ! [F: fun_int_int] : comp_P5356232565778667509nt_int(product_snd_int_int,produc4463282112584876420nt_int(F)) = comp_i4330622015127193916nt_int(F,product_snd_int_int) ).

% snd_comp_apsnd
tff(fact_5843_snd__comp__apsnd,axiom,
    ! [F: fun_nat_nat] : comp_P1567445206330693457at_nat(product_snd_nat_nat,produc3094765310956027504at_nat(F)) = comp_n2711240574518091068at_nat(F,product_snd_nat_nat) ).

% snd_comp_apsnd
tff(fact_5844_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_Code_natural,M: nat,N: nat] :
      groups2279045934846249631atural(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),one_one_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(groups2279045934846249631atural(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_Code_natural(G,aa_nat_nat(suc,N)))) ).

% prod.cl_ivl_Suc
tff(fact_5845_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N: nat] :
      aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_Code_integer(G,aa_nat_nat(suc,N)))) ).

% prod.cl_ivl_Suc
tff(fact_5846_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] :
      aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N)))) ).

% prod.cl_ivl_Suc
tff(fact_5847_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N)))) ).

% prod.cl_ivl_Suc
tff(fact_5848_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] :
      aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N)))) ).

% prod.cl_ivl_Suc
tff(fact_5849_one__mod__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,N)) ).

% one_mod_numeral
tff(fact_5850_one__mod__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,N)) ).

% one_mod_numeral
tff(fact_5851_one__mod__numeral,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,unique3479559517661332726nteger(one,N)) ).

% one_mod_numeral
tff(fact_5852_prod_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B3: set_nat,A3: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_kv(G),B3)),A3) = aa_set_nat_nat(groups708209901874060359at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_kw(G),A3)),B3) ).

% prod.swap
tff(fact_5853_prod_Oswap,axiom,
    ! [G: fun_nat_fun_nat_int,B3: set_nat,A3: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aa_set2359242532610849469at_int(aTP_Lamm_kx(G),B3)),A3) = aa_set_nat_int(groups705719431365010083at_int(aa_set2359242532610849469at_int(aTP_Lamm_kz(G),A3)),B3) ).

% prod.swap
tff(fact_5854_prod_Oswap,axiom,
    ! [G: fun_nat_fun_int_int,B3: set_int,A3: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aa_set6310687161026558745at_int(aTP_Lamm_la(G),B3)),A3) = groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_lc(G),A3),B3) ).

% prod.swap
tff(fact_5855_prod_Oswap,axiom,
    ! [G: fun_int_fun_nat_int,B3: set_nat,A3: set_int] : groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_ld(G),B3),A3) = aa_set_nat_int(groups705719431365010083at_int(aa_set6310687161026558745at_int(aTP_Lamm_lf(G),A3)),B3) ).

% prod.swap
tff(fact_5856_prod_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B3: set_int,A3: set_int] : groups1705073143266064639nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_lg(G),B3),A3) = groups1705073143266064639nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_lh(G),A3),B3) ).

% prod.swap
tff(fact_5857_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_nat,A3: set_int] :
      ( ( groups1707563613775114915nt_nat(G,A3) != one_one_nat )
     => ~ ! [A2: int] :
            ( member_int(A2,A3)
           => ( aa_int_nat(G,A2) = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_5858_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_Code_integer,A3: set_nat] :
      ( ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3) != one_one_Code_integer )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
           => ( aa_nat_Code_integer(G,A2) = one_one_Code_integer ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_5859_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_Code_integer,A3: set_int] :
      ( ( groups3827104343326376752nteger(G,A3) != one_one_Code_integer )
     => ~ ! [A2: int] :
            ( member_int(A2,A3)
           => ( aa_int_Code_integer(G,A2) = one_one_Code_integer ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_5860_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A3: set_nat] :
      ( ( aa_set_nat_rat(groups73079841787564623at_rat(G),A3) != one_one_rat )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
           => ( aa_nat_rat(G,A2) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_5861_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A3: set_int] :
      ( ( groups1072433553688619179nt_rat(G,A3) != one_one_rat )
     => ~ ! [A2: int] :
            ( member_int(A2,A3)
           => ( aa_int_rat(G,A2) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_5862_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_Code_natural,A3: set_nat] :
      ( ( groups2279045934846249631atural(G,A3) != one_one_Code_natural )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
           => ( aa_nat_Code_natural(G,A2) = one_one_Code_natural ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_5863_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_Code_natural,A3: set_int] :
      ( ( groups2650699495083094267atural(G,A3) != one_one_Code_natural )
     => ~ ! [A2: int] :
            ( member_int(A2,A3)
           => ( aa_int_Code_natural(G,A2) = one_one_Code_natural ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_5864_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_nat,A3: set_nat] :
      ( ( aa_set_nat_nat(groups708209901874060359at_nat(G),A3) != one_one_nat )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
           => ( aa_nat_nat(G,A2) = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_5865_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_int,A3: set_nat] :
      ( ( aa_set_nat_int(groups705719431365010083at_int(G),A3) != one_one_int )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
           => ( aa_nat_int(G,A2) = one_one_int ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_5866_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_int,A3: set_int] :
      ( ( groups1705073143266064639nt_int(G,A3) != one_one_int )
     => ~ ! [A2: int] :
            ( member_int(A2,A3)
           => ( aa_int_int(G,A2) = one_one_int ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_5867_prod_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => ( aa_nat_nat(G,X4) = one_one_nat ) )
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A3) = one_one_nat ) ) ).

% prod.neutral
tff(fact_5868_prod_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => ( aa_nat_int(G,X4) = one_one_int ) )
     => ( aa_set_nat_int(groups705719431365010083at_int(G),A3) = one_one_int ) ) ).

% prod.neutral
tff(fact_5869_prod_Oneutral,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => ( aa_int_int(G,X4) = one_one_int ) )
     => ( groups1705073143266064639nt_int(G,A3) = one_one_int ) ) ).

% prod.neutral
tff(fact_5870_snd__conv,axiom,
    ! [X1: fun_Co3460408610320954864e_term,X2: produc8923325533196201883nteger] : aa_Pro5745608113995628733nteger(produc9082904322759093540nteger,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X1),X2)) = X2 ).

% snd_conv
tff(fact_5871_snd__conv,axiom,
    ! [X1: fun_Pr285225112257423215e_term,X2: produc7822875418678951345atural] : aa_Pro2627516385176489788atural(produc199808700204356347atural,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,X1),X2)) = X2 ).

% snd_conv
tff(fact_5872_snd__conv,axiom,
    ! [X1: fun_Pr5098669352780829690e_term,X2: produc8923325533196201883nteger] : aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X1),X2)) = X2 ).

% snd_conv
tff(fact_5873_snd__conv,axiom,
    ! [X1: fun_Pr7763680713840462443e_term,X2: product_prod_int_int] : aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X1),X2)) = X2 ).

% snd_conv
tff(fact_5874_snd__conv,axiom,
    ! [X1: fun_in5730379987311621695e_term,X2: product_prod_int_int] : aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X1),X2)) = X2 ).

% snd_conv
tff(fact_5875_snd__conv,axiom,
    ! [X1: int,X2: int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = X2 ).

% snd_conv
tff(fact_5876_snd__conv,axiom,
    ! [X1: nat,X2: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = X2 ).

% snd_conv
tff(fact_5877_snd__eqD,axiom,
    ! [Xa: fun_Co3460408610320954864e_term,Y3: produc8923325533196201883nteger,A: produc8923325533196201883nteger] :
      ( ( aa_Pro5745608113995628733nteger(produc9082904322759093540nteger,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Xa),Y3)) = A )
     => ( Y3 = A ) ) ).

% snd_eqD
tff(fact_5878_snd__eqD,axiom,
    ! [Xa: fun_Pr285225112257423215e_term,Y3: produc7822875418678951345atural,A: produc7822875418678951345atural] :
      ( ( aa_Pro2627516385176489788atural(produc199808700204356347atural,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,Xa),Y3)) = A )
     => ( Y3 = A ) ) ).

% snd_eqD
tff(fact_5879_snd__eqD,axiom,
    ! [Xa: fun_Pr5098669352780829690e_term,Y3: produc8923325533196201883nteger,A: produc8923325533196201883nteger] :
      ( ( aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Xa),Y3)) = A )
     => ( Y3 = A ) ) ).

% snd_eqD
tff(fact_5880_snd__eqD,axiom,
    ! [Xa: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int,A: product_prod_int_int] :
      ( ( aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Xa),Y3)) = A )
     => ( Y3 = A ) ) ).

% snd_eqD
tff(fact_5881_snd__eqD,axiom,
    ! [Xa: fun_in5730379987311621695e_term,Y3: product_prod_int_int,A: product_prod_int_int] :
      ( ( aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Xa),Y3)) = A )
     => ( Y3 = A ) ) ).

% snd_eqD
tff(fact_5882_snd__eqD,axiom,
    ! [Xa: int,Y3: int,A: int] :
      ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y3)) = A )
     => ( Y3 = A ) ) ).

% snd_eqD
tff(fact_5883_snd__eqD,axiom,
    ! [Xa: nat,Y3: nat,A: nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y3)) = A )
     => ( Y3 = A ) ) ).

% snd_eqD
tff(fact_5884_sndI,axiom,
    ! [Xa: produc7509114972854644512nteger,Y3: fun_Co3460408610320954864e_term,Z: produc8923325533196201883nteger] :
      ( ( Xa = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Y3),Z) )
     => ( aa_Pro5745608113995628733nteger(produc9082904322759093540nteger,Xa) = Z ) ) ).

% sndI
tff(fact_5885_sndI,axiom,
    ! [Xa: produc9140211490168418807atural,Y3: fun_Pr285225112257423215e_term,Z: produc7822875418678951345atural] :
      ( ( Xa = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,Y3),Z) )
     => ( aa_Pro2627516385176489788atural(produc199808700204356347atural,Xa) = Z ) ) ).

% sndI
tff(fact_5886_sndI,axiom,
    ! [Xa: produc2453742332425399446nteger,Y3: fun_Pr5098669352780829690e_term,Z: produc8923325533196201883nteger] :
      ( ( Xa = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Y3),Z) )
     => ( aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,Xa) = Z ) ) ).

% sndI
tff(fact_5887_sndI,axiom,
    ! [Xa: produc3043004577601930755nt_int,Y3: fun_Pr7763680713840462443e_term,Z: product_prod_int_int] :
      ( ( Xa = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Y3),Z) )
     => ( aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,Xa) = Z ) ) ).

% sndI
tff(fact_5888_sndI,axiom,
    ! [Xa: produc4788131091073775919nt_int,Y3: fun_in5730379987311621695e_term,Z: product_prod_int_int] :
      ( ( Xa = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Y3),Z) )
     => ( aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,Xa) = Z ) ) ).

% sndI
tff(fact_5889_sndI,axiom,
    ! [Xa: product_prod_int_int,Y3: int,Z: int] :
      ( ( Xa = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y3),Z) )
     => ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa) = Z ) ) ).

% sndI
tff(fact_5890_sndI,axiom,
    ! [Xa: product_prod_nat_nat,Y3: nat,Z: nat] :
      ( ( Xa = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),Z) )
     => ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa) = Z ) ) ).

% sndI
tff(fact_5891_prod_Oexpand,axiom,
    ! [Prod: product_prod_int_int,Prod2: product_prod_int_int] :
      ( ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod) = aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod2) )
        & ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod) = aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_5892_prod_Oexpand,axiom,
    ! [Prod: product_prod_nat_nat,Prod2: product_prod_nat_nat] :
      ( ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod2) )
        & ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_5893_prod__eqI,axiom,
    ! [P: product_prod_int_int,Q2: product_prod_int_int] :
      ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,P) = aa_Pro6799258434465214121nt_int(product_fst_int_int,Q2) )
     => ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,P) = aa_Pro6799258434465214121nt_int(product_snd_int_int,Q2) )
       => ( P = Q2 ) ) ) ).

% prod_eqI
tff(fact_5894_prod__eqI,axiom,
    ! [P: product_prod_nat_nat,Q2: product_prod_nat_nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Q2) )
     => ( ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Q2) )
       => ( P = Q2 ) ) ) ).

% prod_eqI
tff(fact_5895_prod__eq__iff,axiom,
    ! [S: product_prod_int_int,T: product_prod_int_int] :
      ( ( S = T )
    <=> ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,S) = aa_Pro6799258434465214121nt_int(product_fst_int_int,T) )
        & ( aa_Pro6799258434465214121nt_int(product_snd_int_int,S) = aa_Pro6799258434465214121nt_int(product_snd_int_int,T) ) ) ) ).

% prod_eq_iff
tff(fact_5896_prod__eq__iff,axiom,
    ! [S: product_prod_nat_nat,T: product_prod_nat_nat] :
      ( ( S = T )
    <=> ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,S) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,T) )
        & ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,S) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,T) ) ) ) ).

% prod_eq_iff
tff(fact_5897_prod_Odistrib,axiom,
    ! [G: fun_nat_nat,Ha: fun_nat_nat,A3: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_li(G),Ha)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),A3)),aa_set_nat_nat(groups708209901874060359at_nat(Ha),A3)) ).

% prod.distrib
tff(fact_5898_prod_Odistrib,axiom,
    ! [G: fun_nat_int,Ha: fun_nat_int,A3: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_lj(G),Ha)),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),A3)),aa_set_nat_int(groups705719431365010083at_int(Ha),A3)) ).

% prod.distrib
tff(fact_5899_prod_Odistrib,axiom,
    ! [G: fun_int_int,Ha: fun_int_int,A3: set_int] : groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_lk(G),Ha),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,A3)),groups1705073143266064639nt_int(Ha,A3)) ).

% prod.distrib
tff(fact_5900_prod__power__distrib,axiom,
    ! [F: fun_nat_nat,A3: set_nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),N) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ll(F),N)),A3) ).

% prod_power_distrib
tff(fact_5901_prod__power__distrib,axiom,
    ! [F: fun_nat_int,A3: set_nat,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_set_nat_int(groups705719431365010083at_int(F),A3)),N) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_lm(F),N)),A3) ).

% prod_power_distrib
tff(fact_5902_prod__power__distrib,axiom,
    ! [F: fun_int_int,A3: set_int,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,groups1705073143266064639nt_int(F,A3)),N) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_ln(F),N),A3) ).

% prod_power_distrib
tff(fact_5903_mod__prod__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fr(F),A)),A3)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),A) ).

% mod_prod_eq
tff(fact_5904_mod__prod__eq,axiom,
    ! [F: fun_nat_int,A: int,A3: set_nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_lo(F),A)),A3)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_set_nat_int(groups705719431365010083at_int(F),A3)),A) ).

% mod_prod_eq
tff(fact_5905_mod__prod__eq,axiom,
    ! [F: fun_int_int,A: int,A3: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_fs(F),A),A3)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(F,A3)),A) ).

% mod_prod_eq
tff(fact_5906_snd__def,axiom,
    ! [Prod: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_lp),Prod) ).

% snd_def
tff(fact_5907_snd__def,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_lq),Prod) ).

% snd_def
tff(fact_5908_prod__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) ) ).

% prod_nonneg
tff(fact_5909_prod__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_set_nat_int(groups705719431365010083at_int(F),A3)) ) ).

% prod_nonneg
tff(fact_5910_prod__nonneg,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A3)) ) ).

% prod_nonneg
tff(fact_5911_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A3)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I2))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,I2)),aa_nat_Code_integer(G,I2)) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A3)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3)) ) ).

% prod_mono
tff(fact_5912_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I2))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I2)),aa_int_Code_integer(G,I2)) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F,A3)),groups3827104343326376752nteger(G,A3)) ) ).

% prod_mono
tff(fact_5913_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_Code_natural,G: fun_nat_Code_natural] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A3)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,I2))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(F,I2)),aa_nat_Code_natural(G,I2)) ) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups2279045934846249631atural(F,A3)),groups2279045934846249631atural(G,A3)) ) ).

% prod_mono
tff(fact_5914_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_Code_natural,G: fun_int_Code_natural] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,I2))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_int_Code_natural(F,I2)),aa_int_Code_natural(G,I2)) ) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups2650699495083094267atural(F,A3)),groups2650699495083094267atural(G,A3)) ) ).

% prod_mono
tff(fact_5915_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups73079841787564623at_rat(F),A3)),aa_set_nat_rat(groups73079841787564623at_rat(G),A3)) ) ).

% prod_mono
tff(fact_5916_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),groups1072433553688619179nt_rat(G,A3)) ) ).

% prod_mono
tff(fact_5917_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A3)),groups1707563613775114915nt_nat(G,A3)) ) ).

% prod_mono
tff(fact_5918_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,I2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I2)),aa_nat_nat(G,I2)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),aa_set_nat_nat(groups708209901874060359at_nat(G),A3)) ) ).

% prod_mono
tff(fact_5919_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2)) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups705719431365010083at_int(F),A3)),aa_set_nat_int(groups705719431365010083at_int(G),A3)) ) ).

% prod_mono
tff(fact_5920_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_int,G: fun_int_int] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,I2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,I2)),aa_int_int(G,I2)) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1705073143266064639nt_int(F,A3)),groups1705073143266064639nt_int(G,A3)) ) ).

% prod_mono
tff(fact_5921_prod__pos,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) ) ).

% prod_pos
tff(fact_5922_prod__pos,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_set_nat_int(groups705719431365010083at_int(F),A3)) ) ).

% prod_pos
tff(fact_5923_prod__pos,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A3)) ) ).

% prod_pos
tff(fact_5924_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(F,X4)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A3)) ) ).

% prod_ge_1
tff(fact_5925_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F,X4)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),groups3827104343326376752nteger(F,A3)) ) ).

% prod_ge_1
tff(fact_5926_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_Code_natural] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),aa_nat_Code_natural(F,X4)) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),groups2279045934846249631atural(F,A3)) ) ).

% prod_ge_1
tff(fact_5927_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_Code_natural] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),aa_int_Code_natural(F,X4)) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),groups2650699495083094267atural(F,A3)) ) ).

% prod_ge_1
tff(fact_5928_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X4)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_set_nat_rat(groups73079841787564623at_rat(F),A3)) ) ).

% prod_ge_1
tff(fact_5929_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X4)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_5930_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_int_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups1707563613775114915nt_nat(F,A3)) ) ).

% prod_ge_1
tff(fact_5931_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) ) ).

% prod_ge_1
tff(fact_5932_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_set_nat_int(groups705719431365010083at_int(F),A3)) ) ).

% prod_ge_1
tff(fact_5933_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),groups1705073143266064639nt_int(F,A3)) ) ).

% prod_ge_1
tff(fact_5934_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7509114972854644512nteger] : Prod = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,aa_Pro1361292240318994510e_term(produc3547674943397867238nteger,Prod)),aa_Pro5745608113995628733nteger(produc9082904322759093540nteger,Prod)) ).

% prod.exhaust_sel
tff(fact_5935_prod_Oexhaust__sel,axiom,
    ! [Prod: produc9140211490168418807atural] : Prod = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,aa_Pro1606062155721680164e_term(produc5314839617791632573atural,Prod)),aa_Pro2627516385176489788atural(produc199808700204356347atural,Prod)) ).

% prod.exhaust_sel
tff(fact_5936_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2453742332425399446nteger] : Prod = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,Prod)),aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,Prod)) ).

% prod.exhaust_sel
tff(fact_5937_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3043004577601930755nt_int] : Prod = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,Prod)),aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,Prod)) ).

% prod.exhaust_sel
tff(fact_5938_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4788131091073775919nt_int] : Prod = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,Prod)),aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,Prod)) ).

% prod.exhaust_sel
tff(fact_5939_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_int_int] : Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).

% prod.exhaust_sel
tff(fact_5940_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] : Prod = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ).

% prod.exhaust_sel
tff(fact_5941_surjective__pairing,axiom,
    ! [T: produc7509114972854644512nteger] : T = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,aa_Pro1361292240318994510e_term(produc3547674943397867238nteger,T)),aa_Pro5745608113995628733nteger(produc9082904322759093540nteger,T)) ).

% surjective_pairing
tff(fact_5942_surjective__pairing,axiom,
    ! [T: produc9140211490168418807atural] : T = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,aa_Pro1606062155721680164e_term(produc5314839617791632573atural,T)),aa_Pro2627516385176489788atural(produc199808700204356347atural,T)) ).

% surjective_pairing
tff(fact_5943_surjective__pairing,axiom,
    ! [T: produc2453742332425399446nteger] : T = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,T)),aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,T)) ).

% surjective_pairing
tff(fact_5944_surjective__pairing,axiom,
    ! [T: produc3043004577601930755nt_int] : T = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,T)),aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,T)) ).

% surjective_pairing
tff(fact_5945_surjective__pairing,axiom,
    ! [T: produc4788131091073775919nt_int] : T = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,T)),aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,T)) ).

% surjective_pairing
tff(fact_5946_surjective__pairing,axiom,
    ! [T: product_prod_int_int] : T = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,T)),aa_Pro6799258434465214121nt_int(product_snd_int_int,T)) ).

% surjective_pairing
tff(fact_5947_surjective__pairing,axiom,
    ! [T: product_prod_nat_nat] : T = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_Pro4243173770044397973at_nat(product_fst_nat_nat,T)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,T)) ).

% surjective_pairing
tff(fact_5948_conjI__realizer,axiom,
    ! [Pa: fun_fu7717421471775461377term_o,P: fun_Co3460408610320954864e_term,Q: fun_Pr7211993375558241536eger_o,Q2: produc8923325533196201883nteger] :
      ( aa_fun7659544460261446762term_o(Pa,P)
     => ( aa_Pro4926521453222654329eger_o(Q,Q2)
       => ( aa_fun7659544460261446762term_o(Pa,aa_Pro1361292240318994510e_term(produc3547674943397867238nteger,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,P),Q2)))
          & aa_Pro4926521453222654329eger_o(Q,aa_Pro5745608113995628733nteger(produc9082904322759093540nteger,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,P),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_5949_conjI__realizer,axiom,
    ! [Pa: fun_fu7188715730271225026term_o,P: fun_Pr285225112257423215e_term,Q: fun_Pr8359263388657976682ural_o,Q2: produc7822875418678951345atural] :
      ( aa_fun1221993630413146411term_o(Pa,P)
     => ( aa_Pro6073791466322389475ural_o(Q,Q2)
       => ( aa_fun1221993630413146411term_o(Pa,aa_Pro1606062155721680164e_term(produc5314839617791632573atural,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,P),Q2)))
          & aa_Pro6073791466322389475ural_o(Q,aa_Pro2627516385176489788atural(produc199808700204356347atural,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,P),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_5950_conjI__realizer,axiom,
    ! [Pa: fun_fu4736795817735658359term_o,P: fun_Pr5098669352780829690e_term,Q: fun_Pr7211993375558241536eger_o,Q2: produc8923325533196201883nteger] :
      ( aa_fun7993445754732355552term_o(Pa,P)
     => ( aa_Pro4926521453222654329eger_o(Q,Q2)
       => ( aa_fun7993445754732355552term_o(Pa,aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,P),Q2)))
          & aa_Pro4926521453222654329eger_o(Q,aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,P),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_5951_conjI__realizer,axiom,
    ! [Pa: fun_fu6378075912484669126term_o,P: fun_Pr7763680713840462443e_term,Q: fun_Pr3238374608701176162_int_o,Q2: product_prod_int_int] :
      ( aa_fun7650857933016935215term_o(Pa,P)
     => ( aa_Pro4592068669816069595_int_o(Q,Q2)
       => ( aa_fun7650857933016935215term_o(Pa,aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,P),Q2)))
          & aa_Pro4592068669816069595_int_o(Q,aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,P),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_5952_conjI__realizer,axiom,
    ! [Pa: fun_fu3265955482920998386term_o,P: fun_in5730379987311621695e_term,Q: fun_Pr3238374608701176162_int_o,Q2: product_prod_int_int] :
      ( aa_fun3020711759848831579term_o(Pa,P)
     => ( aa_Pro4592068669816069595_int_o(Q,Q2)
       => ( aa_fun3020711759848831579term_o(Pa,aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,P),Q2)))
          & aa_Pro4592068669816069595_int_o(Q,aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,P),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_5953_conjI__realizer,axiom,
    ! [Pa: fun_int_o,P: int,Q: fun_int_o,Q2: int] :
      ( aa_int_o(Pa,P)
     => ( aa_int_o(Q,Q2)
       => ( aa_int_o(Pa,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q2)))
          & aa_int_o(Q,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_5954_conjI__realizer,axiom,
    ! [Pa: fun_nat_o,P: nat,Q: fun_nat_o,Q2: nat] :
      ( aa_nat_o(Pa,P)
     => ( aa_nat_o(Q,Q2)
       => ( aa_nat_o(Pa,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(P),Q2)))
          & aa_nat_o(Q,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(P),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_5955_exI__realizer,axiom,
    ! [Pa: fun_Pr8608829099060101225term_o,Y3: produc8923325533196201883nteger,Xa: fun_Co3460408610320954864e_term] :
      ( aa_fun7659544460261446762term_o(aa_Pro4353979721973784418term_o(Pa,Y3),Xa)
     => aa_fun7659544460261446762term_o(aa_Pro4353979721973784418term_o(Pa,aa_Pro5745608113995628733nteger(produc9082904322759093540nteger,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Xa),Y3))),aa_Pro1361292240318994510e_term(produc3547674943397867238nteger,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Xa),Y3))) ) ).

% exI_realizer
tff(fact_5956_exI__realizer,axiom,
    ! [Pa: fun_Pr8801881646629708820term_o,Y3: produc7822875418678951345atural,Xa: fun_Pr285225112257423215e_term] :
      ( aa_fun1221993630413146411term_o(aa_Pro4093371377458885517term_o(Pa,Y3),Xa)
     => aa_fun1221993630413146411term_o(aa_Pro4093371377458885517term_o(Pa,aa_Pro2627516385176489788atural(produc199808700204356347atural,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,Xa),Y3))),aa_Pro1606062155721680164e_term(produc5314839617791632573atural,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,Xa),Y3))) ) ).

% exI_realizer
tff(fact_5957_exI__realizer,axiom,
    ! [Pa: fun_Pr4680327687137491167term_o,Y3: produc8923325533196201883nteger,Xa: fun_Pr5098669352780829690e_term] :
      ( aa_fun7993445754732355552term_o(aa_Pro9035772811636298456term_o(Pa,Y3),Xa)
     => aa_fun7993445754732355552term_o(aa_Pro9035772811636298456term_o(Pa,aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Xa),Y3))),aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Xa),Y3))) ) ).

% exI_realizer
tff(fact_5958_exI__realizer,axiom,
    ! [Pa: fun_Pr5526332658708858896term_o,Y3: product_prod_int_int,Xa: fun_Pr7763680713840462443e_term] :
      ( aa_fun7650857933016935215term_o(aa_Pro1023942307601852809term_o(Pa,Y3),Xa)
     => aa_fun7650857933016935215term_o(aa_Pro1023942307601852809term_o(Pa,aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Xa),Y3))),aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Xa),Y3))) ) ).

% exI_realizer
tff(fact_5959_exI__realizer,axiom,
    ! [Pa: fun_Pr7145817153658775868term_o,Y3: product_prod_int_int,Xa: fun_in5730379987311621695e_term] :
      ( aa_fun3020711759848831579term_o(aa_Pro1235490124264145077term_o(Pa,Y3),Xa)
     => aa_fun3020711759848831579term_o(aa_Pro1235490124264145077term_o(Pa,aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Xa),Y3))),aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Xa),Y3))) ) ).

% exI_realizer
tff(fact_5960_exI__realizer,axiom,
    ! [Pa: fun_int_fun_int_o,Y3: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(Pa,Y3),Xa)
     => aa_int_o(aa_int_fun_int_o(Pa,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y3))),aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y3))) ) ).

% exI_realizer
tff(fact_5961_exI__realizer,axiom,
    ! [Pa: fun_nat_fun_nat_o,Y3: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(Pa,Y3),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(Pa,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y3))),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y3))) ) ).

% exI_realizer
tff(fact_5962_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [Pa: fun_fu3422731660242565741eger_o,Xa: fun_Co3460408610320954864e_term,Y3: produc8923325533196201883nteger,A: produc7509114972854644512nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(Pa,Xa),Y3)
     => ( ( A = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Xa),Y3) )
       => aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(Pa,aa_Pro1361292240318994510e_term(produc3547674943397867238nteger,A)),aa_Pro5745608113995628733nteger(produc9082904322759093540nteger,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_5963_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [Pa: fun_fu3694549533497745878ural_o,Xa: fun_Pr285225112257423215e_term,Y3: produc7822875418678951345atural,A: produc9140211490168418807atural] :
      ( aa_Pro6073791466322389475ural_o(aa_fun9089857049899810327ural_o(Pa,Xa),Y3)
     => ( ( A = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,Xa),Y3) )
       => aa_Pro6073791466322389475ural_o(aa_fun9089857049899810327ural_o(Pa,aa_Pro1606062155721680164e_term(produc5314839617791632573atural,A)),aa_Pro2627516385176489788atural(produc199808700204356347atural,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_5964_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [Pa: fun_fu7695273699059480439eger_o,Xa: fun_Pr5098669352780829690e_term,Y3: produc8923325533196201883nteger,A: produc2453742332425399446nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(Pa,Xa),Y3)
     => ( ( A = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Xa),Y3) )
       => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(Pa,aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,A)),aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_5965_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [Pa: fun_fu6865642474282033866_int_o,Xa: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int,A: produc3043004577601930755nt_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(Pa,Xa),Y3)
     => ( ( A = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Xa),Y3) )
       => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(Pa,aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,A)),aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_5966_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [Pa: fun_fu2978539155105838238_int_o,Xa: fun_in5730379987311621695e_term,Y3: product_prod_int_int,A: produc4788131091073775919nt_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(Pa,Xa),Y3)
     => ( ( A = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Xa),Y3) )
       => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(Pa,aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,A)),aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_5967_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [Pa: fun_int_fun_int_o,Xa: int,Y3: int,A: product_prod_int_int] :
      ( aa_int_o(aa_int_fun_int_o(Pa,Xa),Y3)
     => ( ( A = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y3) )
       => aa_int_o(aa_int_fun_int_o(Pa,aa_Pro6799258434465214121nt_int(product_fst_int_int,A)),aa_Pro6799258434465214121nt_int(product_snd_int_int,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_5968_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [Pa: fun_nat_fun_nat_o,Xa: nat,Y3: nat,A: product_prod_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(Pa,Xa),Y3)
     => ( ( A = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y3) )
       => aa_nat_o(aa_nat_fun_nat_o(Pa,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,A)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_5969_prod_OatLeast__Suc__atMost__Suc__shift,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(comp_nat_nat_nat(G,suc)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_5970_prod_OatLeast__Suc__atMost__Suc__shift,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(comp_nat_int_nat(G,suc)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_5971_prod_OatLeastAtMost__shift__bounds,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(comp_nat_nat_nat(G,aa_nat_fun_nat_nat(plus_plus_nat,K))),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_5972_prod_OatLeastAtMost__shift__bounds,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(comp_nat_int_nat(G,aa_nat_fun_nat_nat(plus_plus_nat,K))),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_5973_case__prod__beta,axiom,
    ! [F: fun_na1668100129766719005at_nat,P: product_prod_nat_nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),P) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P)) ).

% case_prod_beta
tff(fact_5974_case__prod__beta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,P: product_prod_nat_nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),P) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P)) ).

% case_prod_beta
tff(fact_5975_case__prod__beta,axiom,
    ! [F: fun_in1696471567231710611nt_int,P: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),P) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P)) ).

% case_prod_beta
tff(fact_5976_case__prod__beta,axiom,
    ! [F: fun_int_fun_int_o,P: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),P)
    <=> aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P)) ) ).

% case_prod_beta
tff(fact_5977_case__prod__beta,axiom,
    ! [F: fun_int_fun_int_int,P: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),P) = aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P)) ).

% case_prod_beta
tff(fact_5978_split__beta,axiom,
    ! [F: fun_na1668100129766719005at_nat,Prod: product_prod_nat_nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),Prod) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ).

% split_beta
tff(fact_5979_split__beta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),Prod) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ).

% split_beta
tff(fact_5980_split__beta,axiom,
    ! [F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).

% split_beta
tff(fact_5981_split__beta,axiom,
    ! [F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod)
    <=> aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ) ).

% split_beta
tff(fact_5982_split__beta,axiom,
    ! [F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod) = aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).

% split_beta
tff(fact_5983_Product__Type_OCollect__case__prodD,axiom,
    ! [Xa: product_prod_nat_nat,A3: fun_nat_fun_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(A3)))
     => aa_nat_o(aa_nat_fun_nat_o(A3,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_5984_Product__Type_OCollect__case__prodD,axiom,
    ! [Xa: product_prod_int_int,A3: fun_int_fun_int_o] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Xa),collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,A3)))
     => aa_int_o(aa_int_fun_int_o(A3,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_5985_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_hv(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_5986_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_ia(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_5987_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_hw(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_5988_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_lr(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_5989_power__sum,axiom,
    ! [C: nat,F: fun_nat_nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ls(C),F)),A3) ).

% power_sum
tff(fact_5990_power__sum,axiom,
    ! [C: int,F: fun_nat_nat,A3: set_nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_int(groups705719431365010083at_int(aa_fun1613598493704395220at_int(aTP_Lamm_lt(C),F)),A3) ).

% power_sum
tff(fact_5991_power__sum,axiom,
    ! [C: int,F: fun_int_nat,A3: set_int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,C),groups4541462559716669496nt_nat(F,A3)) = groups1705073143266064639nt_int(aa_fun2328901131235339916nt_int(aTP_Lamm_lu(C),F),A3) ).

% power_sum
tff(fact_5992_case__prod__beta_H,axiom,
    ! [F: fun_na1668100129766719005at_nat,X: product_prod_nat_nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),X) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,X)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,X)) ).

% case_prod_beta'
tff(fact_5993_case__prod__beta_H,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X: product_prod_nat_nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),X) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,X)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,X)) ).

% case_prod_beta'
tff(fact_5994_case__prod__beta_H,axiom,
    ! [F: fun_in1696471567231710611nt_int,X: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),X) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X)) ).

% case_prod_beta'
tff(fact_5995_case__prod__beta_H,axiom,
    ! [F: fun_int_fun_int_o,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),X)
    <=> aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X)) ) ).

% case_prod_beta'
tff(fact_5996_case__prod__beta_H,axiom,
    ! [F: fun_int_fun_int_int,X: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),X) = aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X)) ).

% case_prod_beta'
tff(fact_5997_case__prod__unfold,axiom,
    ! [X: fun_na1668100129766719005at_nat,Xa3: product_prod_nat_nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,X),Xa3) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(X,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa3)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa3)) ).

% case_prod_unfold
tff(fact_5998_case__prod__unfold,axiom,
    ! [X: fun_na5914473681949881652_nat_o,Xa3: product_prod_nat_nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,X),Xa3) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(X,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa3)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa3)) ).

% case_prod_unfold
tff(fact_5999_case__prod__unfold,axiom,
    ! [X: fun_in1696471567231710611nt_int,Xa3: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,X),Xa3) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(X,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa3)) ).

% case_prod_unfold
tff(fact_6000_case__prod__unfold,axiom,
    ! [X: fun_int_fun_int_o,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,X),Xa3)
    <=> aa_int_o(aa_int_fun_int_o(X,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa3)) ) ).

% case_prod_unfold
tff(fact_6001_case__prod__unfold,axiom,
    ! [X: fun_int_fun_int_int,Xa3: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,X),Xa3) = aa_int_int(aa_int_fun_int_int(X,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa3)) ).

% case_prod_unfold
tff(fact_6002_exE__realizer,axiom,
    ! [Pa: fun_nat_fun_nat_o,P: product_prod_nat_nat,Q: fun_fu8391177546919232046_nat_o,F: fun_na1668100129766719005at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(Pa,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P))
     => ( ! [X4: nat,Y4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(Pa,Y4),X4)
           => aa_fun3538214411680374423_nat_o(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X4),Y4)) )
       => aa_fun3538214411680374423_nat_o(Q,aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),P)) ) ) ).

% exE_realizer
tff(fact_6003_exE__realizer,axiom,
    ! [Pa: fun_nat_fun_nat_o,P: product_prod_nat_nat,Q: fun_fu6891447142836604759at_o_o,F: fun_na5914473681949881652_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(Pa,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P))
     => ( ! [X4: nat,Y4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(Pa,Y4),X4)
           => aa_fun8408151826566693824at_o_o(Q,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X4),Y4)) )
       => aa_fun8408151826566693824at_o_o(Q,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),P)) ) ) ).

% exE_realizer
tff(fact_6004_exE__realizer,axiom,
    ! [Pa: fun_int_fun_int_o,P: product_prod_int_int,Q: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int] :
      ( aa_int_o(aa_int_fun_int_o(Pa,aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P))
     => ( ! [X4: int,Y4: int] :
            ( aa_int_o(aa_int_fun_int_o(Pa,Y4),X4)
           => aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X4),Y4)) )
       => aa_Pro4592068669816069595_int_o(Q,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),P)) ) ) ).

% exE_realizer
tff(fact_6005_exE__realizer,axiom,
    ! [Pa: fun_int_fun_int_o,P: product_prod_int_int,Q: fun_o_o,F: fun_int_fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(Pa,aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P))
     => ( ! [X4: int,Y4: int] :
            ( aa_int_o(aa_int_fun_int_o(Pa,Y4),X4)
           => aa_o_o(Q,aa_int_o(aa_int_fun_int_o(F,X4),Y4)) )
       => aa_o_o(Q,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),P)) ) ) ).

% exE_realizer
tff(fact_6006_exE__realizer,axiom,
    ! [Pa: fun_int_fun_int_o,P: product_prod_int_int,Q: fun_int_o,F: fun_int_fun_int_int] :
      ( aa_int_o(aa_int_fun_int_o(Pa,aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P))
     => ( ! [X4: int,Y4: int] :
            ( aa_int_o(aa_int_fun_int_o(Pa,Y4),X4)
           => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(F,X4),Y4)) )
       => aa_int_o(Q,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),P)) ) ) ).

% exE_realizer
tff(fact_6007_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X4))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X4)),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A3)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_6008_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X4))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X4)),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F,A3)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_6009_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_Code_natural] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,X4))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(F,X4)),one_one_Code_natural) ) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups2279045934846249631atural(F,A3)),one_one_Code_natural) ) ).

% prod_le_1
tff(fact_6010_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_Code_natural] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,X4))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_int_Code_natural(F,X4)),one_one_Code_natural) ) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups2650699495083094267atural(F,A3)),one_one_Code_natural) ) ).

% prod_le_1
tff(fact_6011_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups73079841787564623at_rat(F),A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_6012_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X4)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_6013_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A3)),one_one_nat) ) ).

% prod_le_1
tff(fact_6014_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),one_one_nat) ) ).

% prod_le_1
tff(fact_6015_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),one_one_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups705719431365010083at_int(F),A3)),one_one_int) ) ).

% prod_le_1
tff(fact_6016_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,A3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X4)),one_one_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1705073143266064639nt_int(F,A3)),one_one_int) ) ).

% prod_le_1
tff(fact_6017_divides__aux__def,axiom,
    ! [Qr: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,Qr)
    <=> ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Qr) = zero_zero_nat ) ) ).

% divides_aux_def
tff(fact_6018_divides__aux__def,axiom,
    ! [Qr: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,Qr)
    <=> ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Qr) = zero_zero_int ) ) ).

% divides_aux_def
tff(fact_6019_divides__aux__def,axiom,
    ! [Qr: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(unique5706413561485394159nteger,Qr)
    <=> ( aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Qr) = zero_z3403309356797280102nteger ) ) ).

% divides_aux_def
tff(fact_6020_prod_Osplit__sel__asm,axiom,
    ! [Pa: fun_fu8391177546919232046_nat_o,F: fun_na1668100129766719005at_nat,Prod: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(Pa,aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),Prod))
    <=> ~ ( ( Prod = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
          & ~ aa_fun3538214411680374423_nat_o(Pa,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_6021_prod_Osplit__sel__asm,axiom,
    ! [Pa: fun_fu6891447142836604759at_o_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(Pa,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),Prod))
    <=> ~ ( ( Prod = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
          & ~ aa_fun8408151826566693824at_o_o(Pa,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_6022_prod_Osplit__sel__asm,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Pa,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod))
    <=> ~ ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
          & ~ aa_Pro4592068669816069595_int_o(Pa,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_6023_prod_Osplit__sel__asm,axiom,
    ! [Pa: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(Pa,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod))
    <=> ~ ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
          & ~ aa_o_o(Pa,aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_6024_prod_Osplit__sel__asm,axiom,
    ! [Pa: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(Pa,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod))
    <=> ~ ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
          & ~ aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_6025_prod_Osplit__sel,axiom,
    ! [Pa: fun_fu8391177546919232046_nat_o,F: fun_na1668100129766719005at_nat,Prod: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(Pa,aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),Prod))
    <=> ( ( Prod = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
       => aa_fun3538214411680374423_nat_o(Pa,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel
tff(fact_6026_prod_Osplit__sel,axiom,
    ! [Pa: fun_fu6891447142836604759at_o_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(Pa,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),Prod))
    <=> ( ( Prod = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
       => aa_fun8408151826566693824at_o_o(Pa,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel
tff(fact_6027_prod_Osplit__sel,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Pa,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod))
    <=> ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
       => aa_Pro4592068669816069595_int_o(Pa,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel
tff(fact_6028_prod_Osplit__sel,axiom,
    ! [Pa: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(Pa,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod))
    <=> ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
       => aa_o_o(Pa,aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel
tff(fact_6029_prod_Osplit__sel,axiom,
    ! [Pa: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(Pa,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod))
    <=> ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
       => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel
tff(fact_6030_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_hx(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% prod.atLeastAtMost_rev
tff(fact_6031_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_lv(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% prod.atLeastAtMost_rev
tff(fact_6032_case__prod__comp,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,Xa: produc3346328345660746739nteger] : produc4281629032016041417nteger(comp_f6739083541385432444nteger(F,G),Xa) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,produc1861665112190112313nteger(Xa))),produc7955729907432593271nteger(Xa)) ).

% case_prod_comp
tff(fact_6033_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_Code_integer(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6034_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6035_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_6036_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_6037_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,zero_zero_nat)),aa_set871585252387937313nteger(groups3455450783089532116nteger(comp_n3898172953802868194er_nat(G,suc)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_6038_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups73079841787564623at_rat(comp_nat_rat_nat(G,suc)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_6039_prod_OatLeast0__atMost__Suc__shift,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_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(comp_nat_nat_nat(G,suc)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_6040_prod_OatLeast0__atMost__Suc__shift,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_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups705719431365010083at_int(comp_nat_int_nat(G,suc)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_6041_snd__diag__snd,axiom,
    comp_P5356232565778667509nt_int(product_snd_int_int,comp_i5941103828406351439nt_int(aTP_Lamm_hg,product_snd_int_int)) = product_snd_int_int ).

% snd_diag_snd
tff(fact_6042_snd__diag__snd,axiom,
    comp_P1567445206330693457at_nat(product_snd_nat_nat,comp_n829915685812334579at_nat(aTP_Lamm_hf,product_snd_nat_nat)) = product_snd_nat_nat ).

% snd_diag_snd
tff(fact_6043_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_Code_integer(G,aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,M)),aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_lw(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6044_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,M)),aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_hz(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6045_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_hv(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6046_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_ia(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6047_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,zero_zero_nat)),aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_lw(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_6048_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_hz(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_6049_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_hv(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_6050_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_ia(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_6051_prod_OatLeast__atMost__pred__shift,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(comp_nat_nat_nat(G,aTP_Lamm_ih)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.atLeast_atMost_pred_shift
tff(fact_6052_prod_OatLeast__atMost__pred__shift,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : aa_set_nat_int(groups705719431365010083at_int(comp_nat_int_nat(G,aTP_Lamm_ih)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.atLeast_atMost_pred_shift
tff(fact_6053_fact__prod,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N))) ).

% fact_prod
tff(fact_6054_fact__prod,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N))) ).

% fact_prod
tff(fact_6055_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_Code_natural,A: nat,B: nat] : groups2279045934846249631atural(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo9131927060563008058atural(aTP_Lamm_lx(F),A,B,one_one_Code_natural) ).

% prod_atLeastAtMost_code
tff(fact_6056_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_Code_integer,A: nat,B: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_ly(F),A,B),one_one_Code_integer) ).

% prod_atLeastAtMost_code
tff(fact_6057_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : aa_set_nat_rat(groups73079841787564623at_rat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo1949268297981939178at_rat(aTP_Lamm_lz(F),A,B,one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_6058_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups708209901874060359at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_ma(F),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_6059_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : aa_set_nat_int(groups705719431365010083at_int(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_mb(F),A),B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_6060_fst__snd__flip,axiom,
    ! [Xy: produc3611471421753121338e_term] : produc4337129453304281928e_term(Xy) = comp_P4516649579262022418e_term(produc9082904322759093540nteger,produc2742967963346599297nteger(aTP_Lamm_mc),Xy) ).

% fst_snd_flip
tff(fact_6061_fst__snd__flip,axiom,
    ! [Xy: produc7223384021016661839e_term] : produc2007160776547428317e_term(Xy) = comp_P812235655064240294e_term(produc199808700204356347atural,produc5705453634168500973atural(aTP_Lamm_md),Xy) ).

% fst_snd_flip
tff(fact_6062_fst__snd__flip,axiom,
    ! [Xy: produc592931078481472324e_term] : produc3658391496133417298e_term(Xy) = comp_P2694320290640119590e_term(produc6589491691745714842nteger,produc1137639668546780417nteger(aTP_Lamm_me),Xy) ).

% fst_snd_flip
tff(fact_6063_fst__snd__flip,axiom,
    ! [Xy: produc4155239895104388691e_term] : produc8655982770948792545e_term(Xy) = comp_P8873622951661117094e_term(produc1557020426977177351nt_int,produc4605791130040086525nt_int(aTP_Lamm_mf),Xy) ).

% fst_snd_flip
tff(fact_6064_fst__snd__flip,axiom,
    ! [Xy: produc5464280974217902119e_term] : produc7504962776628944053e_term(Xy) = comp_P3263061180182656590e_term(produc7629796867588368947nt_int,produc8624921108698535677nt_int(aTP_Lamm_mg),Xy) ).

% fst_snd_flip
tff(fact_6065_fst__snd__flip,axiom,
    ! [Xy: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xy) = aa_Pro4243173770044397973at_nat(comp_P1567445206330693457at_nat(product_snd_nat_nat,produc2626176000494625587at_nat(aTP_Lamm_mh)),Xy) ).

% fst_snd_flip
tff(fact_6066_fst__snd__flip,axiom,
    ! [Xy: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,Xy) = aa_Pro6799258434465214121nt_int(comp_P5356232565778667509nt_int(product_snd_int_int,aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_mi)),Xy) ).

% fst_snd_flip
tff(fact_6067_snd__fst__flip,axiom,
    ! [Xy: produc3611471421753121338e_term] : produc648986795810732422e_term(Xy) = comp_P6784595103646691119e_term(produc3547674943397867238nteger,produc2742967963346599297nteger(aTP_Lamm_mc),Xy) ).

% snd_fst_flip
tff(fact_6068_snd__fst__flip,axiom,
    ! [Xy: produc7223384021016661839e_term] : produc6115501895814927899e_term(Xy) = comp_P823182482625092890e_term(produc5314839617791632573atural,produc5705453634168500973atural(aTP_Lamm_md),Xy) ).

% snd_fst_flip
tff(fact_6069_snd__fst__flip,axiom,
    ! [Xy: produc592931078481472324e_term] : produc7766732615400916880e_term(Xy) = comp_P9209726524401416377e_term(produc2481150572478215260nteger,produc1137639668546780417nteger(aTP_Lamm_me),Xy) ).

% snd_fst_flip
tff(fact_6070_snd__fst__flip,axiom,
    ! [Xy: produc4155239895104388691e_term] : produc7082206506123769631e_term(Xy) = comp_P4791264009094562086e_term(produc3130796691802200265nt_int,produc4605791130040086525nt_int(aTP_Lamm_mf),Xy) ).

% snd_fst_flip
tff(fact_6071_snd__fst__flip,axiom,
    ! [Xy: produc5464280974217902119e_term] : produc1448379345012327155e_term(Xy) = comp_P7738237167614049274e_term(produc4463008262350210037nt_int,produc8624921108698535677nt_int(aTP_Lamm_mg),Xy) ).

% snd_fst_flip
tff(fact_6072_snd__fst__flip,axiom,
    ! [Xy: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xy) = aa_Pro4243173770044397973at_nat(comp_P1567445206330693457at_nat(product_fst_nat_nat,produc2626176000494625587at_nat(aTP_Lamm_mh)),Xy) ).

% snd_fst_flip
tff(fact_6073_snd__fst__flip,axiom,
    ! [Xy: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,Xy) = aa_Pro6799258434465214121nt_int(comp_P5356232565778667509nt_int(product_fst_int_int,aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_mi)),Xy) ).

% snd_fst_flip
tff(fact_6074_prod_OatLeastAtMost__shift__0,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_set_nat_nat(groups708209901874060359at_nat(comp_nat_nat_nat(G,aa_nat_fun_nat_nat(plus_plus_nat,M))),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_6075_prod_OatLeastAtMost__shift__0,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_set_nat_int(groups705719431365010083at_int(comp_nat_int_nat(G,aa_nat_fun_nat_nat(plus_plus_nat,M))),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_6076_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_Code_integer,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set871585252387937313nteger(groups3455450783089532116nteger(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),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_6077_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_rat(groups73079841787564623at_rat(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),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_6078_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),P))) = 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),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_6079_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),P))) = 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),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_6080_fst__diag__snd,axiom,
    comp_P5356232565778667509nt_int(product_fst_int_int,comp_i5941103828406351439nt_int(aTP_Lamm_hg,product_snd_int_int)) = product_snd_int_int ).

% fst_diag_snd
tff(fact_6081_fst__diag__snd,axiom,
    comp_P1567445206330693457at_nat(product_fst_nat_nat,comp_n829915685812334579at_nat(aTP_Lamm_hf,product_snd_nat_nat)) = product_snd_nat_nat ).

% fst_diag_snd
tff(fact_6082_snd__diag__fst,axiom,
    comp_P5356232565778667509nt_int(product_snd_int_int,comp_i5941103828406351439nt_int(aTP_Lamm_hg,product_fst_int_int)) = product_fst_int_int ).

% snd_diag_fst
tff(fact_6083_snd__diag__fst,axiom,
    comp_P1567445206330693457at_nat(product_snd_nat_nat,comp_n829915685812334579at_nat(aTP_Lamm_hf,product_fst_nat_nat)) = product_fst_nat_nat ).

% snd_diag_fst
tff(fact_6084_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_6085_pochhammer__Suc__prod,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_mj(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_6086_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_mk,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_6087_pochhammer__Suc__prod,axiom,
    ! [A: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N)) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_ml,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_6088_pochhammer__prod__rev,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_mm(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_6089_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_mn(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_6090_pochhammer__prod__rev,axiom,
    ! [A: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_mo(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_6091_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_av),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_6092_prod_Oin__pairs,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_mp(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_6093_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] : aa_set_nat_rat(groups73079841787564623at_rat(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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_mq(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_6094_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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_mr(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_6095_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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ms(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_6096_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_mp(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_6097_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_mq(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_6098_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_mr(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_6099_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ms(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_6100_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_mm(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_6101_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_mn(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_6102_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_mo(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_6103_prod_Ozero__middle,axiom,
    ! [P: nat,K: nat,G: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P)
       => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_mt(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_jq(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6104_prod_Ozero__middle,axiom,
    ! [P: nat,K: nat,G: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P)
       => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_mu(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_js(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6105_prod_Ozero__middle,axiom,
    ! [P: nat,K: nat,G: fun_nat_Code_natural,Ha: fun_nat_Code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P)
       => ( groups2279045934846249631atural(aa_fun56224542555504168atural(aa_fun1651958757531339674atural(aTP_Lamm_mv(K),G),Ha),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups2279045934846249631atural(aa_fun56224542555504168atural(aa_fun1651958757531339674atural(aTP_Lamm_ju(K),G),Ha),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6106_prod_Ozero__middle,axiom,
    ! [P: nat,K: nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_mw(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_jw(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6107_prod_Ozero__middle,axiom,
    ! [P: nat,K: nat,G: fun_nat_int,Ha: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P)
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_mx(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_jo(K),G),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6108_gbinomial__Suc,axiom,
    ! [A: code_integer,K: nat] : aa_nat_Code_integer(gbinom8545251970709558553nteger(A),aa_nat_nat(suc,K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_my(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_6109_gbinomial__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = divide_divide_rat(aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_mz(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_6110_gbinomial__Suc,axiom,
    ! [A: code_natural,K: nat] : aa_nat_Code_natural(gbinom7368847122466276068atural(A),aa_nat_nat(suc,K)) = divide5121882707175180666atural(groups2279045934846249631atural(aTP_Lamm_na(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)),aa_nat_Code_natural(semiri2447717529341329178atural,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_6111_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_nb,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_6112_gbinomial__Suc,axiom,
    ! [A: int,K: nat] : aa_nat_int(aa_int_fun_nat_int(gbinomial_int,A),aa_nat_nat(suc,K)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_nc,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_6113_size__prod__simp,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,P: product_prod_int_int] : basic_1872990034501187214nt_int(F,G,P) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,P))),aa_int_nat(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,P)))),aa_nat_nat(suc,zero_zero_nat)) ).

% size_prod_simp
tff(fact_6114_size__prod__simp,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,P: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(basic_876126793109182934at_nat(F,G),P) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P))),aa_nat_nat(G,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P)))),aa_nat_nat(suc,zero_zero_nat)) ).

% size_prod_simp
tff(fact_6115_bezw_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y3: product_prod_int_int] :
      ( ( aa_nat4424723855478970923nt_int(bezw(Xa),Xaa) = Y3 )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa))
       => ~ ( ( Y3 = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_nat4424723855478970923nt_int(bezw(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_nat4424723855478970923nt_int(bezw(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_nat4424723855478970923nt_int(bezw(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Xaa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_6116_normalize__def,axiom,
    ! [P: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(normalize,P) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),
        $let(
          a2: int,
          a2:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),a2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),a2)) ),
        $ite(
          aa_Pro6799258434465214121nt_int(product_snd_int_int,P) = zero_zero_int,
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),
          $let(
            a2: int,
            a2:= aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P))),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),a2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),a2)) ) ) ) ).

% normalize_def
tff(fact_6117_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : aa_set4749074711808745205at_nat(groups977919841031483927at_nat(produc6842872674320459806at_nat(G)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_nd(N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.triangle_reindex
tff(fact_6118_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),N)
     => ( groups4538972089207619220nt_int(aTP_Lamm_cn,set_or1266510415728281911st_int(M,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_6119_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_av),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,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_6120_gcd__right__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ).

% gcd_right_idem
tff(fact_6121_gcd__right__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ).

% gcd_right_idem
tff(fact_6122_gcd__left__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ).

% gcd_left_idem
tff(fact_6123_gcd__left__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ).

% gcd_left_idem
tff(fact_6124_gcd__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_eq_0_iff
tff(fact_6125_gcd__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & ( B = zero_zero_int ) ) ) ).

% gcd_eq_0_iff
tff(fact_6126_gcd__add1,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ).

% gcd_add1
tff(fact_6127_gcd__add1,axiom,
    ! [M: int,N: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N)),N) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N) ).

% gcd_add1
tff(fact_6128_gcd__add2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ).

% gcd_add2
tff(fact_6129_gcd__add2,axiom,
    ! [M: int,N: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N) ).

% gcd_add2
tff(fact_6130_gcd_Obottom__left__bottom,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,one_one_nat),A) = one_one_nat ).

% gcd.bottom_left_bottom
tff(fact_6131_gcd_Obottom__left__bottom,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,one_one_int),A) = one_one_int ).

% gcd.bottom_left_bottom
tff(fact_6132_gcd_Obottom__right__bottom,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),one_one_nat) = one_one_nat ).

% gcd.bottom_right_bottom
tff(fact_6133_gcd_Obottom__right__bottom,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),one_one_int) = one_one_int ).

% gcd.bottom_right_bottom
tff(fact_6134_gcd__neg2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ).

% gcd_neg2
tff(fact_6135_gcd__neg1,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ).

% gcd_neg1
tff(fact_6136_gcd__greatest__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,B),C))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% gcd_greatest_iff
tff(fact_6137_gcd__greatest__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% gcd_greatest_iff
tff(fact_6138_gcd__dvd2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B) ).

% gcd_dvd2
tff(fact_6139_gcd__dvd2,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ).

% gcd_dvd2
tff(fact_6140_gcd__dvd1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),A) ).

% gcd_dvd1
tff(fact_6141_gcd__dvd1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A) ).

% gcd_dvd1
tff(fact_6142_gcd__exp,axiom,
    ! [A: nat,N: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),N) ).

% gcd_exp
tff(fact_6143_gcd__exp,axiom,
    ! [A: int,N: nat,B: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),N) ).

% gcd_exp
tff(fact_6144_gcd__1__int,axiom,
    ! [M: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_6145_gcd__neg2__int,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),aa_int_int(uminus_uminus_int,Y3)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3) ).

% gcd_neg2_int
tff(fact_6146_gcd__neg1__int,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xa)),Y3) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3) ).

% gcd_neg1_int
tff(fact_6147_abs__gcd__int,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3) ).

% abs_gcd_int
tff(fact_6148_gcd__abs1__int,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(abs_abs_int,Xa)),Y3) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3) ).

% gcd_abs1_int
tff(fact_6149_gcd__abs2__int,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),aa_int_int(abs_abs_int,Y3)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3) ).

% gcd_abs2_int
tff(fact_6150_gcd__neg__numeral__1,axiom,
    ! [N: num,A: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),A) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,N)),A) ).

% gcd_neg_numeral_1
tff(fact_6151_gcd__neg__numeral__2,axiom,
    ! [A: int,N: num] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),aa_num_int(numeral_numeral_int,N)) ).

% gcd_neg_numeral_2
tff(fact_6152_is__unit__gcd__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),one_one_nat)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = one_one_nat ) ) ).

% is_unit_gcd_iff
tff(fact_6153_is__unit__gcd__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),one_one_int)
    <=> ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) = one_one_int ) ) ).

% is_unit_gcd_iff
tff(fact_6154_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N))
    <=> ( ( M != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_6155_gcd__neg__numeral__1__int,axiom,
    ! [N: num,Xa: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),Xa) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,N)),Xa) ).

% gcd_neg_numeral_1_int
tff(fact_6156_gcd__neg__numeral__2__int,axiom,
    ! [Xa: int,N: num] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),aa_num_int(numeral_numeral_int,N)) ).

% gcd_neg_numeral_2_int
tff(fact_6157_gcd__0__int,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),zero_zero_int) = aa_int_int(abs_abs_int,Xa) ).

% gcd_0_int
tff(fact_6158_gcd__0__left__int,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,zero_zero_int),Xa) = aa_int_int(abs_abs_int,Xa) ).

% gcd_0_left_int
tff(fact_6159_gcd__proj2__if__dvd__int,axiom,
    ! [Y3: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y3),Xa)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3) = aa_int_int(abs_abs_int,Y3) ) ) ).

% gcd_proj2_if_dvd_int
tff(fact_6160_gcd__proj1__if__dvd__int,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),Y3)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3) = aa_int_int(abs_abs_int,Xa) ) ) ).

% gcd_proj1_if_dvd_int
tff(fact_6161_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_rat(G,N)) ).

% sum.lessThan_Suc
tff(fact_6162_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_6163_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_6164_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_int(G,N))) ).

% sum.op_ivl_Suc
tff(fact_6165_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N: nat] :
      aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_Code_integer(G,N))) ).

% sum.op_ivl_Suc
tff(fact_6166_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_rat(G,N))) ).

% sum.op_ivl_Suc
tff(fact_6167_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_Code_natural,M: nat,N: nat] :
      groups6325495683096345652atural(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),zero_z2226904508553997617atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,groups6325495683096345652atural(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_Code_natural(G,N))) ).

% sum.op_ivl_Suc
tff(fact_6168_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_nat(G,N))) ).

% sum.op_ivl_Suc
tff(fact_6169_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_Code_natural,M: nat,N: nat] :
      groups2279045934846249631atural(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(groups2279045934846249631atural(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_Code_natural(G,N))) ).

% prod.op_ivl_Suc
tff(fact_6170_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N: nat] :
      aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_Code_integer(G,N))) ).

% prod.op_ivl_Suc
tff(fact_6171_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] :
      aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_rat(G,N))) ).

% prod.op_ivl_Suc
tff(fact_6172_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_nat(G,N))) ).

% prod.op_ivl_Suc
tff(fact_6173_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] :
      aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_int(G,N))) ).

% prod.op_ivl_Suc
tff(fact_6174_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_6175_int__prod,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A3)) = groups1705073143266064639nt_int(aTP_Lamm_fo(F),A3) ).

% int_prod
tff(fact_6176_int__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_fp(F)),A3) ).

% int_prod
tff(fact_6177_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_6178_gcd_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,B),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),C)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,B),C)) ).

% gcd.left_commute
tff(fact_6179_gcd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C)) ).

% gcd.left_commute
tff(fact_6180_gcd_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,B),A) ).

% gcd.commute
tff(fact_6181_gcd_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),A) ).

% gcd.commute
tff(fact_6182_gcd_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,B),C)) ).

% gcd.assoc
tff(fact_6183_gcd_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C)) ).

% gcd.assoc
tff(fact_6184_gcd__diff2,axiom,
    ! [N: int,M: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),M)),N) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N) ).

% gcd_diff2
tff(fact_6185_gcd__diff1,axiom,
    ! [M: int,N: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N)),N) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N) ).

% gcd_diff1
tff(fact_6186_gcd__dvd__antisym,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,C),D))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,C),D)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B))
       => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,C),D) ) ) ) ).

% gcd_dvd_antisym
tff(fact_6187_gcd__dvd__antisym,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,C),D))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,C),D) ) ) ) ).

% gcd_dvd_antisym
tff(fact_6188_gcd__greatest,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)) ) ) ).

% gcd_greatest
tff(fact_6189_gcd__greatest,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ) ).

% gcd_greatest
tff(fact_6190_gcd__dvdI2,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),C) ) ).

% gcd_dvdI2
tff(fact_6191_gcd__dvdI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C) ) ).

% gcd_dvdI2
tff(fact_6192_gcd__dvdI1,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),C) ) ).

% gcd_dvdI1
tff(fact_6193_gcd__dvdI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C) ) ).

% gcd_dvdI1
tff(fact_6194_dvd__gcdD2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,B),C))
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_gcdD2
tff(fact_6195_dvd__gcdD2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_gcdD2
tff(fact_6196_dvd__gcdD1,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,B),C))
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_gcdD1
tff(fact_6197_dvd__gcdD1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_gcdD1
tff(fact_6198_gcd__mono,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),D)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,C),D)) ) ) ).

% gcd_mono
tff(fact_6199_gcd__mono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,C),D)) ) ) ).

% gcd_mono
tff(fact_6200_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_cn,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_6201_gcd__idem__int,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Xa) = aa_int_int(abs_abs_int,Xa) ).

% gcd_idem_int
tff(fact_6202_gcd__red__int,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y3),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),Y3)) ).

% gcd_red_int
tff(fact_6203_effect__heapE,axiom,
    ! [F: fun_he8889679731534893401it_nat,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: a,N: nat] :
      ( aa_nat_o(heap_Time_effect_a(heap_Time_heap_a(F),Ha,H,R),N)
     => ~ ( ( H = produc2838810277069100634it_nat(produc6461318759014699077it_nat(aa_hea5645563107516622602it_nat(F,Ha))) )
         => ( ( N = produc2594868675584459928it_nat(produc6461318759014699077it_nat(aa_hea5645563107516622602it_nat(F,Ha))) )
           => ( R != produc4481111470393856007it_nat(aa_hea5645563107516622602it_nat(F,Ha)) ) ) ) ) ).

% effect_heapE
tff(fact_6204_effect__heapI,axiom,
    ! [N: nat,F: fun_he8889679731534893401it_nat,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: a] :
      ( ( N = produc2594868675584459928it_nat(produc6461318759014699077it_nat(aa_hea5645563107516622602it_nat(F,Ha))) )
     => ( ( H = produc2838810277069100634it_nat(produc6461318759014699077it_nat(aa_hea5645563107516622602it_nat(F,Ha))) )
       => ( ( R = produc4481111470393856007it_nat(aa_hea5645563107516622602it_nat(F,Ha)) )
         => aa_nat_o(heap_Time_effect_a(heap_Time_heap_a(F),Ha,H,R),N) ) ) ) ).

% effect_heapI
tff(fact_6205_effect__guardE,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,F: fun_he8889679731534893401it_nat,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: a,N: nat] :
      ( aa_nat_o(heap_Time_effect_a(heap_Time_guard_a(Pa,F),Ha,H,R),N)
     => ~ ( ( H = produc2838810277069100634it_nat(produc6461318759014699077it_nat(aa_hea5645563107516622602it_nat(F,Ha))) )
         => ( ( N = produc2594868675584459928it_nat(produc6461318759014699077it_nat(aa_hea5645563107516622602it_nat(F,Ha))) )
           => ( ( R = produc4481111470393856007it_nat(aa_hea5645563107516622602it_nat(F,Ha)) )
             => ~ aa_hea7994211679605476167unit_o(Pa,Ha) ) ) ) ) ).

% effect_guardE
tff(fact_6206_effect__guardI,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,F: fun_he8889679731534893401it_nat,N: nat,R: a] :
      ( aa_hea7994211679605476167unit_o(Pa,Ha)
     => ( ( H = produc2838810277069100634it_nat(produc6461318759014699077it_nat(aa_hea5645563107516622602it_nat(F,Ha))) )
       => ( ( N = produc2594868675584459928it_nat(produc6461318759014699077it_nat(aa_hea5645563107516622602it_nat(F,Ha))) )
         => ( ( R = produc4481111470393856007it_nat(aa_hea5645563107516622602it_nat(F,Ha)) )
           => aa_nat_o(heap_Time_effect_a(heap_Time_guard_a(Pa,F),Ha,H,R),N) ) ) ) ) ).

% effect_guardI
tff(fact_6207_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [G: fun_int_nat,M: nat,N: nat] : groups1707563613775114915nt_nat(G,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_set_nat_nat(groups708209901874060359at_nat(comp_int_nat_nat(G,semiri1314217659103216013at_int)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_6208_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [G: fun_int_int,M: nat,N: nat] : groups1705073143266064639nt_int(G,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_set_nat_int(groups705719431365010083at_int(comp_int_int_nat(G,semiri1314217659103216013at_int)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_6209_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_cn,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_6210_sum_Onat__group,axiom,
    ! [G: fun_nat_nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ne(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))) ).

% sum.nat_group
tff(fact_6211_prod_Onat__group,axiom,
    ! [G: fun_nat_nat,K: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nf(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))) ).

% prod.nat_group
tff(fact_6212_prod_Onat__group,axiom,
    ! [G: fun_nat_int,K: nat,N: nat] : aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ng(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))) ).

% prod.nat_group
tff(fact_6213_effectE,axiom,
    ! [C: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: a,N: nat] :
      ( aa_nat_o(heap_Time_effect_a(C,Ha,H,R),N)
     => ~ ( ( R = produc4481111470393856007it_nat(the_Pr3616839707675578981it_nat(aa_hea4724376117875161946it_nat(heap_Time_execute_a(C),Ha))) )
         => ( ( H = produc2838810277069100634it_nat(produc6461318759014699077it_nat(the_Pr3616839707675578981it_nat(aa_hea4724376117875161946it_nat(heap_Time_execute_a(C),Ha)))) )
           => ( ( N = produc2594868675584459928it_nat(produc6461318759014699077it_nat(the_Pr3616839707675578981it_nat(aa_hea4724376117875161946it_nat(heap_Time_execute_a(C),Ha)))) )
             => ~ aa_hea7994211679605476167unit_o(heap_Time_success_a(C),Ha) ) ) ) ) ).

% effectE
tff(fact_6214_lessThan__def,axiom,
    ! [U: set_Pr1261947904930325089at_nat] : set_or5243998310507771012at_nat(U) = collec5514110066124741708at_nat(aTP_Lamm_nh(U)) ).

% lessThan_def
tff(fact_6215_lessThan__def,axiom,
    ! [U: set_Pr958786334691620121nt_int] : set_or4940836740269066044nt_int(U) = collec5210948495886036740nt_int(aTP_Lamm_ni(U)) ).

% lessThan_def
tff(fact_6216_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aTP_Lamm_nj(U)) ).

% lessThan_def
tff(fact_6217_lessThan__def,axiom,
    ! [U: literal] : set_or2436086697161274615iteral(U) = collect_literal(aTP_Lamm_nk(U)) ).

% lessThan_def
tff(fact_6218_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_nl,U)) ).

% lessThan_def
tff(fact_6219_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aTP_Lamm_nm(U)) ).

% lessThan_def
tff(fact_6220_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_nn,U)) ).

% lessThan_def
tff(fact_6221_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fl,U)) ).

% lessThan_def
tff(fact_6222_gcd__add__mult,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ).

% gcd_add_mult
tff(fact_6223_gcd__add__mult,axiom,
    ! [M: int,K: int,N: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),N)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N) ).

% gcd_add_mult
tff(fact_6224_gcd__dvd__prod,axiom,
    ! [A: nat,B: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),B)) ).

% gcd_dvd_prod
tff(fact_6225_gcd__dvd__prod,axiom,
    ! [A: int,B: int,K: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,K),B)) ).

% gcd_dvd_prod
tff(fact_6226_ex__nat__less__eq,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ? [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M7),N)
          & aa_nat_o(Pa,M7) )
    <=> ? [X3: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))
          & aa_nat_o(Pa,X3) ) ) ).

% ex_nat_less_eq
tff(fact_6227_all__nat__less__eq,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ! [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M7),N)
         => aa_nat_o(Pa,M7) )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))
         => aa_nat_o(Pa,X3) ) ) ).

% all_nat_less_eq
tff(fact_6228_gcd__ge__0__int,axiom,
    ! [Xa: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3)) ).

% gcd_ge_0_int
tff(fact_6229_bezout__int,axiom,
    ! [Xa: int,Y3: int] :
    ? [U2: int,V4: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U2),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,V4),Y3)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3) ).

% bezout_int
tff(fact_6230_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_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)) ).

% gcd_mult_distrib_int
tff(fact_6231_execute__bind__success,axiom,
    ! [F: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,G: fun_a_6093291656547882009Heap_a] :
      ( aa_hea7994211679605476167unit_o(heap_Time_success_a(F),Ha)
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_a_a(F,G)),Ha) = heap_T7616092557645711335rame_a(produc2594868675584459928it_nat(produc6461318759014699077it_nat(the_Pr3616839707675578981it_nat(aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha)))),aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(G,produc4481111470393856007it_nat(the_Pr3616839707675578981it_nat(aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha))))),produc2838810277069100634it_nat(produc6461318759014699077it_nat(the_Pr3616839707675578981it_nat(aa_hea4724376117875161946it_nat(heap_Time_execute_a(F),Ha)))))) ) ) ).

% execute_bind_success
tff(fact_6232_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hv(G)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% sum.shift_bounds_Suc_ivl
tff(fact_6233_sum_Oshift__bounds__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_or4665077453230672383an_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_hw(G),K)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% sum.shift_bounds_nat_ivl
tff(fact_6234_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_hv(G)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% prod.shift_bounds_Suc_ivl
tff(fact_6235_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ia(G)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% prod.shift_bounds_Suc_ivl
tff(fact_6236_prod_Oshift__bounds__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_or4665077453230672383an_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_hw(G),K)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% prod.shift_bounds_nat_ivl
tff(fact_6237_prod_Oshift__bounds__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_or4665077453230672383an_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_lr(G),K)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% prod.shift_bounds_nat_ivl
tff(fact_6238_gcd__mult__unit2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C) ) ) ).

% gcd_mult_unit2
tff(fact_6239_gcd__mult__unit2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,B),C) ) ) ).

% gcd_mult_unit2
tff(fact_6240_gcd__mult__unit1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C) ) ) ).

% gcd_mult_unit1
tff(fact_6241_gcd__mult__unit1,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,B),C) ) ) ).

% gcd_mult_unit1
tff(fact_6242_gcd__div__unit2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,B),C) ) ) ).

% gcd_div_unit2
tff(fact_6243_gcd__div__unit2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C) ) ) ).

% gcd_div_unit2
tff(fact_6244_gcd__div__unit1,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),C) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,B),C) ) ) ).

% gcd_div_unit1
tff(fact_6245_gcd__div__unit1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C) ) ) ).

% gcd_div_unit1
tff(fact_6246_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B) ) ).

% gcd_le2_int
tff(fact_6247_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),A) ) ).

% gcd_le1_int
tff(fact_6248_gcd__cases__int,axiom,
    ! [Xa: int,Y3: int,Pa: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
         => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),zero_zero_int)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),aa_int_int(uminus_uminus_int,Y3))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xa)),Y3)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),zero_zero_int)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xa)),aa_int_int(uminus_uminus_int,Y3))) ) )
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3)) ) ) ) ) ).

% gcd_cases_int
tff(fact_6249_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),P)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),P))) = aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),P)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_6250_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),P)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),P))) = aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),P)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_6251_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),P)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),P))) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),P)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_6252_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),A)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),B) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),D) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).

% gcd_unique_int
tff(fact_6253_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [G: fun_int_nat,M: nat,N: nat] : groups4541462559716669496nt_nat(G,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(comp_int_nat_nat(G,semiri1314217659103216013at_int)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_6254_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [G: fun_int_int,M: nat,N: nat] : groups4538972089207619220nt_int(G,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_set_nat_int(groups3539618377306564664at_int(comp_int_int_nat(G,semiri1314217659103216013at_int)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_6255_gcd__non__0__int,axiom,
    ! [Y3: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Y3)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y3),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),Y3)) ) ) ).

% gcd_non_0_int
tff(fact_6256_gcd__code__int,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K),L) = aa_int_int(abs_abs_int,
        $ite(L = zero_zero_int,K,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))))) ).

% gcd_code_int
tff(fact_6257_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_6258_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_6259_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(G,suc)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_6260_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_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(comp_nat_nat_nat(G,aa_nat_fun_nat_nat(plus_plus_nat,K))),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_6261_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups708209901874060359at_nat(comp_nat_nat_nat(G,suc)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_6262_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_int(groups705719431365010083at_int(comp_nat_int_nat(G,suc)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_6263_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_no(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_6264_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_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(comp_nat_nat_nat(G,aa_nat_fun_nat_nat(plus_plus_nat,K))),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_6265_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_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(comp_nat_int_nat(G,aa_nat_fun_nat_nat(plus_plus_nat,K))),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_6266_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_no(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_6267_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_np(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_6268_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : aa_set4749074711808745205at_nat(groups4077766827762148844at_nat(produc6842872674320459806at_nat(G)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_nd(N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_nq(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.triangle_reindex
tff(fact_6269_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_int,N: nat] : aa_set4746584241299694929at_int(groups4075276357253098568at_int(produc6840382203811409530at_int(G)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_nd(N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ns(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.triangle_reindex
tff(fact_6270_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_6271_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_Code_integer,K: nat] :
      ( ( aa_nat_Code_integer(F,zero_zero_nat) = zero_z3403309356797280102nteger )
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_6272_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_rat(groups2906978787729119204at_rat(F),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_6273_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_Code_natural,K: nat] :
      ( ( aa_nat_Code_natural(F,zero_zero_nat) = zero_z2226904508553997617atural )
     => ( groups6325495683096345652atural(F,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups6325495683096345652atural(F,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_6274_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_6275_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_rat(G,N)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_6276_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_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_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_int(G,N)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_6277_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_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_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_nat(G,N)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_6278_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,M)),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_6279_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_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_or4665077453230672383an_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_6280_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_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_or4665077453230672383an_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_6281_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_rat(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_6282_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_int(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_6283_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_nat(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_6284_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_Code_integer(G,N)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_6285_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_rat(G,N)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_6286_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_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_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_nat(G,N)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_6287_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_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_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_int(G,N)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_6288_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,N)),aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) ) ) ).

% sum.last_plus
tff(fact_6289_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(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,N)),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) ) ) ).

% sum.last_plus
tff(fact_6290_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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,N)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) ) ) ).

% sum.last_plus
tff(fact_6291_fact__prod__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% fact_prod_Suc
tff(fact_6292_fact__prod__Suc,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% fact_prod_Suc
tff(fact_6293_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_hz(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6294_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_ia(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6295_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_hv(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6296_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_ie(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_6297_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_Code_integer,M: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_if(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(F,M)),aa_nat_Code_integer(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6298_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ig(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = minus_minus_rat(aa_nat_rat(F,M),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6299_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_ib(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_6300_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_Code_integer,M: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_ic(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(F,zero_zero_nat)),aa_nat_Code_integer(F,M)) ).

% sum_lessThan_telescope'
tff(fact_6301_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_id(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = minus_minus_rat(aa_nat_rat(F,zero_zero_nat),aa_nat_rat(F,M)) ).

% sum_lessThan_telescope'
tff(fact_6302_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,zero_zero_nat)),aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_lw(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_6303_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_hz(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_6304_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_hv(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_6305_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_ia(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_6306_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,Pa: fun_nat_nat,N: nat] :
      ( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q,X4)),aa_nat_nat(Pa,X4))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Pa),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_he(Q),Pa)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ) ).

% sum_diff_distrib
tff(fact_6307_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_hv(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_6308_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_hv(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_6309_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_ia(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_6310_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hv(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_6311_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nt(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nu(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nested_swap'
tff(fact_6312_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_nv(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nw(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nested_swap'
tff(fact_6313_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_nx(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ny(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nested_swap'
tff(fact_6314_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_ie(F)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_6315_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_if(F)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(F,N)),aa_nat_Code_integer(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_6316_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_ig(F)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = minus_minus_rat(aa_nat_rat(F,N),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_6317_sum_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_nz(G),N),M)),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) ).

% sum.atLeastLessThan_rev
tff(fact_6318_sum_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_oa(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nu(A),N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% sum.nested_swap
tff(fact_6319_prod_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_nz(G),N),M)),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) ).

% prod.atLeastLessThan_rev
tff(fact_6320_prod_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_int,N: nat,M: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ob(G),N),M)),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) ).

% prod.atLeastLessThan_rev
tff(fact_6321_prod_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_oc(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nw(A),N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% prod.nested_swap
tff(fact_6322_prod_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_od(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ny(A),N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% prod.nested_swap
tff(fact_6323_fact__prod__rev,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% fact_prod_rev
tff(fact_6324_fact__prod__rev,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% fact_prod_rev
tff(fact_6325_sum_Ohead__if,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_int(G,N))) ).

% sum.head_if
tff(fact_6326_sum_Ohead__if,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N: nat] :
      aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_Code_integer(G,N))) ).

% sum.head_if
tff(fact_6327_sum_Ohead__if,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_rat(G,N))) ).

% sum.head_if
tff(fact_6328_sum_Ohead__if,axiom,
    ! [G: fun_nat_Code_natural,M: nat,N: nat] :
      groups6325495683096345652atural(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),zero_z2226904508553997617atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,groups6325495683096345652atural(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_Code_natural(G,N))) ).

% sum.head_if
tff(fact_6329_sum_Ohead__if,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_nat(G,N))) ).

% sum.head_if
tff(fact_6330_prod_Ohead__if,axiom,
    ! [G: fun_nat_Code_natural,M: nat,N: nat] :
      groups2279045934846249631atural(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(groups2279045934846249631atural(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_Code_natural(G,N))) ).

% prod.head_if
tff(fact_6331_prod_Ohead__if,axiom,
    ! [G: fun_nat_Code_integer,M: nat,N: nat] :
      aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_Code_integer(G,N))) ).

% prod.head_if
tff(fact_6332_prod_Ohead__if,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] :
      aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_rat(G,N))) ).

% prod.head_if
tff(fact_6333_prod_Ohead__if,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_nat(G,N))) ).

% prod.head_if
tff(fact_6334_prod_Ohead__if,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] :
      aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_int(G,N))) ).

% prod.head_if
tff(fact_6335_power__diff__1__eq,axiom,
    ! [Xa: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),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,Xa),one_one_int)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xa)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6336_power__diff__1__eq,axiom,
    ! [Xa: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),N)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xa),one_one_Code_integer)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(Xa)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6337_power__diff__1__eq,axiom,
    ! [Xa: rat,N: nat] : minus_minus_rat(aa_nat_rat(power_power_rat(Xa),N),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Xa,one_one_rat)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(Xa)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6338_one__diff__power__eq,axiom,
    ! [Xa: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),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),Xa)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xa)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6339_one__diff__power__eq,axiom,
    ! [Xa: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(Xa),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),Xa)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(Xa)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6340_one__diff__power__eq,axiom,
    ! [Xa: rat,N: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xa),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(one_one_rat,Xa)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(Xa)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6341_geometric__sum,axiom,
    ! [Xa: rat,N: nat] :
      ( ( Xa != one_one_rat )
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(Xa)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xa),N),one_one_rat),minus_minus_rat(Xa,one_one_rat)) ) ) ).

% geometric_sum
tff(fact_6342_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups2906978787729119204at_rat(comp_nat_rat_nat(G,suc)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_6343_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_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(comp_nat_int_nat(G,suc)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_6344_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_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(comp_nat_nat_nat(G,suc)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_6345_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,zero_zero_nat)),aa_set871585252387937313nteger(groups3455450783089532116nteger(comp_n3898172953802868194er_nat(G,suc)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_6346_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups73079841787564623at_rat(comp_nat_rat_nat(G,suc)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_6347_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_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(comp_nat_nat_nat(G,suc)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_6348_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_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(comp_nat_int_nat(G,suc)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_6349_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_hz(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_6350_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_ia(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_6351_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_hv(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_6352_prod_OatMost__shift,axiom,
    ! [G: fun_nat_Code_integer,N: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,zero_zero_nat)),aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_lw(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_6353_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : aa_set_nat_rat(groups73079841787564623at_rat(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_hz(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_6354_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_hv(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_6355_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_ia(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_6356_sum_OatLeastLessThan__shift__0,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(G,aa_nat_fun_nat_nat(plus_plus_nat,M))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ).

% sum.atLeastLessThan_shift_0
tff(fact_6357_prod_OatLeastLessThan__shift__0,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_set_nat_nat(groups708209901874060359at_nat(comp_nat_nat_nat(G,aa_nat_fun_nat_nat(plus_plus_nat,M))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ).

% prod.atLeastLessThan_shift_0
tff(fact_6358_prod_OatLeastLessThan__shift__0,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_set_nat_int(groups705719431365010083at_int(comp_nat_int_nat(G,aa_nat_fun_nat_nat(plus_plus_nat,M))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ).

% prod.atLeastLessThan_shift_0
tff(fact_6359_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_hx(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M)) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_6360_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_hx(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M)) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_6361_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_int,N: nat,M: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_lv(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M)) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_6362_pochhammer__prod,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_mj(A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% pochhammer_prod
tff(fact_6363_pochhammer__prod,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_mk,A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% pochhammer_prod
tff(fact_6364_pochhammer__prod,axiom,
    ! [A: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_ml,A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% pochhammer_prod
tff(fact_6365_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(G,aTP_Lamm_ih)),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_6366_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(comp_nat_nat_nat(G,aTP_Lamm_ih)),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_6367_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : aa_set_nat_int(groups705719431365010083at_int(comp_nat_int_nat(G,aTP_Lamm_ih)),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_6368_sum__gp__strict,axiom,
    ! [Xa: rat,N: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(Xa)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = $ite(Xa = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,N),divide_divide_rat(minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xa),N)),minus_minus_rat(one_one_rat,Xa))) ).

% sum_gp_strict
tff(fact_6369_power__diff__sumr2,axiom,
    ! [Xa: int,N: nat,Y3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y3)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_oe(Xa),N),Y3)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6370_power__diff__sumr2,axiom,
    ! [Xa: code_integer,N: nat,Y3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xa),Y3)),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_of(Xa),N),Y3)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6371_power__diff__sumr2,axiom,
    ! [Xa: rat,N: nat,Y3: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(Xa),N),aa_nat_rat(power_power_rat(Y3),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Xa,Y3)),aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_og(Xa),N),Y3)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6372_diff__power__eq__sum,axiom,
    ! [Xa: int,N: nat,Y3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),aa_nat_nat(suc,N))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),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,Xa),Y3)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_oh(Xa),N),Y3)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6373_diff__power__eq__sum,axiom,
    ! [Xa: code_integer,N: nat,Y3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_nat_nat(suc,N))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xa),Y3)),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_oi(Xa),N),Y3)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6374_diff__power__eq__sum,axiom,
    ! [Xa: rat,N: nat,Y3: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(Xa),aa_nat_nat(suc,N)),aa_nat_rat(power_power_rat(Y3),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Xa,Y3)),aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_oj(Xa),N),Y3)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6375_one__diff__power__eq_H,axiom,
    ! [Xa: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xa),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),Xa)),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ok(Xa),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6376_one__diff__power__eq_H,axiom,
    ! [Xa: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(Xa),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),Xa)),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aTP_Lamm_ol(Xa),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6377_one__diff__power__eq_H,axiom,
    ! [Xa: rat,N: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xa),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(one_one_rat,Xa)),aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_om(Xa),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6378_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : aa_set4749074711808745205at_nat(groups4077766827762148844at_nat(produc6842872674320459806at_nat(G)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_iq(N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_nq(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.triangle_reindex_eq
tff(fact_6379_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_int,N: nat] : aa_set4746584241299694929at_int(groups4075276357253098568at_int(produc6840382203811409530at_int(G)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_iq(N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ns(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.triangle_reindex_eq
tff(fact_6380_gbinomial__altdef__of__nat,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_on(A),K)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ).

% gbinomial_altdef_of_nat
tff(fact_6381_binomial__altdef__of__nat,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_oo(K),N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% binomial_altdef_of_nat
tff(fact_6382_gbinomial__mult__fact_H,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_mz(A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ).

% gbinomial_mult_fact'
tff(fact_6383_gbinomial__mult__fact,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(gbinomial_rat(A),K)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_mz(A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ).

% gbinomial_mult_fact
tff(fact_6384_gbinomial__prod__rev,axiom,
    ! [A: code_integer,K: nat] : aa_nat_Code_integer(gbinom8545251970709558553nteger(A),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_my(A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K))),aa_nat_Code_integer(semiri3624122377584611663nteger,K)) ).

% gbinomial_prod_rev
tff(fact_6385_gbinomial__prod__rev,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = divide_divide_rat(aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_mz(A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_prod_rev
tff(fact_6386_gbinomial__prod__rev,axiom,
    ! [A: code_natural,K: nat] : aa_nat_Code_natural(gbinom7368847122466276068atural(A),K) = divide5121882707175180666atural(groups2279045934846249631atural(aTP_Lamm_na(A),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)),aa_nat_Code_natural(semiri2447717529341329178atural,K)) ).

% gbinomial_prod_rev
tff(fact_6387_gbinomial__prod__rev,axiom,
    ! [A: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nb,A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K))),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ).

% gbinomial_prod_rev
tff(fact_6388_gbinomial__prod__rev,axiom,
    ! [A: int,K: nat] : aa_nat_int(aa_int_fun_nat_int(gbinomial_int,A),K) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_nc,A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K))),aa_nat_int(semiri1406184849735516958ct_int,K)) ).

% gbinomial_prod_rev
tff(fact_6389_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),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,aa_num_num(bit0,one))),K)),one_one_nat) ).

% sum_power2
tff(fact_6390_prod__decode__aux_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y3: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(Xa),Xaa) = Y3 )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa))
       => ~ ( ( Y3 = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xa),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Xaa)),aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xa)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_6391_sum__zero__power_H,axiom,
    ! [C: fun_nat_rat,D: fun_nat_rat,A3: set_nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_op(C),D)),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),A3) ),
        divide_divide_rat(aa_nat_rat(C,zero_zero_nat),aa_nat_rat(D,zero_zero_nat)),
        zero_zero_rat ) ).

% sum_zero_power'
tff(fact_6392_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(bit_un7362597486090784418nd_num(aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_ep),aa_num_option_num(bit_un7362597486090784418nd_num(M),N)) ).

% and_num.simps(9)
tff(fact_6393_or__numerals_I4_J,axiom,
    ! [Xa: num,Y3: num] : bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Xa)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y3))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,Xa),aa_num_Code_natural(numera5444537566228673987atural,Y3)))) ).

% or_numerals(4)
tff(fact_6394_or__numerals_I4_J,axiom,
    ! [Xa: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y3)))) ).

% or_numerals(4)
tff(fact_6395_or__numerals_I4_J,axiom,
    ! [Xa: num,Y3: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Xa))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y3))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa)),aa_num_Code_integer(numera6620942414471956472nteger,Y3)))) ).

% or_numerals(4)
tff(fact_6396_or__numerals_I4_J,axiom,
    ! [Xa: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xa)),aa_num_nat(numeral_numeral_nat,Y3)))) ).

% or_numerals(4)
tff(fact_6397_or__numerals_I6_J,axiom,
    ! [Xa: num,Y3: num] : bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Xa)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Y3))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,Xa),aa_num_Code_natural(numera5444537566228673987atural,Y3)))) ).

% or_numerals(6)
tff(fact_6398_or__numerals_I6_J,axiom,
    ! [Xa: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y3)))) ).

% or_numerals(6)
tff(fact_6399_or__numerals_I6_J,axiom,
    ! [Xa: num,Y3: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Xa))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y3))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa)),aa_num_Code_integer(numera6620942414471956472nteger,Y3)))) ).

% or_numerals(6)
tff(fact_6400_or__numerals_I6_J,axiom,
    ! [Xa: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xa)),aa_num_nat(numeral_numeral_nat,Y3)))) ).

% or_numerals(6)
tff(fact_6401_or__numerals_I7_J,axiom,
    ! [Xa: num,Y3: num] : bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Xa)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y3))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,Xa),aa_num_Code_natural(numera5444537566228673987atural,Y3)))) ).

% or_numerals(7)
tff(fact_6402_or__numerals_I7_J,axiom,
    ! [Xa: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y3)))) ).

% or_numerals(7)
tff(fact_6403_or__numerals_I7_J,axiom,
    ! [Xa: num,Y3: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Xa))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y3))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xa)),aa_num_Code_integer(numera6620942414471956472nteger,Y3)))) ).

% or_numerals(7)
tff(fact_6404_or__numerals_I7_J,axiom,
    ! [Xa: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xa)),aa_num_nat(numeral_numeral_nat,Y3)))) ).

% or_numerals(7)
tff(fact_6405_gcd__0__left__nat,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),Xa) = Xa ).

% gcd_0_left_nat
tff(fact_6406_gcd__0__nat,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),zero_zero_nat) = Xa ).

% gcd_0_nat
tff(fact_6407_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),zero_zero_nat) = A ).

% gcd_nat.right_neutral
tff(fact_6408_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_6409_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),A) = A ).

% gcd_nat.left_neutral
tff(fact_6410_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_6411_gcd__nat_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).

% gcd_nat.absorb1
tff(fact_6412_gcd__nat_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).

% gcd_nat.absorb2
tff(fact_6413_gcd__nat_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% gcd_nat.bounded_iff
tff(fact_6414_gcd__proj1__if__dvd__nat,axiom,
    ! [Xa: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),Y3)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Y3) = Xa ) ) ).

% gcd_proj1_if_dvd_nat
tff(fact_6415_gcd__proj2__if__dvd__nat,axiom,
    ! [Y3: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y3),Xa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Y3) = Y3 ) ) ).

% gcd_proj2_if_dvd_nat
tff(fact_6416_gcd__1__nat,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_6417_gcd__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_6418_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N))
    <=> ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_6419_or_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,zero_zero_int),A) = A ).

% or.left_neutral
tff(fact_6420_or_Oleft__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,zero_z3403309356797280102nteger),A) = A ).

% or.left_neutral
tff(fact_6421_or_Oleft__neutral,axiom,
    ! [A: code_natural] : bit_se9127793120404214118atural(zero_z2226904508553997617atural,A) = A ).

% or.left_neutral
tff(fact_6422_or_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,zero_zero_nat),A) = A ).

% or.left_neutral
tff(fact_6423_or_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),zero_zero_int) = A ).

% or.right_neutral
tff(fact_6424_or_Oright__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),zero_z3403309356797280102nteger) = A ).

% or.right_neutral
tff(fact_6425_or_Oright__neutral,axiom,
    ! [A: code_natural] : bit_se9127793120404214118atural(A,zero_z2226904508553997617atural) = A ).

% or.right_neutral
tff(fact_6426_or_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),zero_zero_nat) = A ).

% or.right_neutral
tff(fact_6427_gcd__int__int__eq,axiom,
    ! [M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N)) ).

% gcd_int_int_eq
tff(fact_6428_sum__eq__0__iff,axiom,
    ! [F4: set_int,F: fun_int_nat] :
      ( finite_finite_int(F4)
     => ( ( groups4541462559716669496nt_nat(F,F4) = zero_zero_nat )
      <=> ! [X3: int] :
            ( member_int(X3,F4)
           => ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6429_sum__eq__0__iff,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
     => ( ( aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),F4) = zero_zero_nat )
      <=> ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),F4)
           => ( aa_Pro4243173770044397973at_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6430_sum__eq__0__iff,axiom,
    ! [F4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F4) = zero_zero_nat )
      <=> ! [X3: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),F4)
           => ( aa_nat_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6431_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_nat] :
      ( ~ finite_finite_int(A3)
     => ( groups4541462559716669496nt_nat(G,A3) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_6432_sum_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_int] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),A3) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_6433_sum_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_Code_integer] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),A3) = zero_z3403309356797280102nteger ) ) ).

% sum.infinite
tff(fact_6434_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_Code_integer] :
      ( ~ finite_finite_int(A3)
     => ( groups7873554091576472773nteger(G,A3) = zero_z3403309356797280102nteger ) ) ).

% sum.infinite
tff(fact_6435_sum_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_6436_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( ~ finite_finite_int(A3)
     => ( groups3906332499630173760nt_rat(G,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_6437_sum_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_Code_natural] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( groups6325495683096345652atural(G,A3) = zero_z2226904508553997617atural ) ) ).

% sum.infinite
tff(fact_6438_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_Code_natural] :
      ( ~ finite_finite_int(A3)
     => ( groups6697149243333190288atural(G,A3) = zero_z2226904508553997617atural ) ) ).

% sum.infinite
tff(fact_6439_sum_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_6440_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( ~ finite_finite_int(A3)
     => ( groups4538972089207619220nt_int(G,A3) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_6441_prod__zero__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( finite_finite_int(A3)
     => ( ( groups1707563613775114915nt_nat(F,A3) = zero_zero_nat )
      <=> ? [X3: int] :
            ( member_int(X3,A3)
            & ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_6442_prod__zero__iff,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A3) = zero_z3403309356797280102nteger )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
            & ( aa_nat_Code_integer(F,X3) = zero_z3403309356797280102nteger ) ) ) ) ).

% prod_zero_iff
tff(fact_6443_prod__zero__iff,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( finite_finite_int(A3)
     => ( ( groups3827104343326376752nteger(F,A3) = zero_z3403309356797280102nteger )
      <=> ? [X3: int] :
            ( member_int(X3,A3)
            & ( aa_int_Code_integer(F,X3) = zero_z3403309356797280102nteger ) ) ) ) ).

% prod_zero_iff
tff(fact_6444_prod__zero__iff,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set_nat_rat(groups73079841787564623at_rat(F),A3) = zero_zero_rat )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
            & ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_6445_prod__zero__iff,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( finite_finite_int(A3)
     => ( ( groups1072433553688619179nt_rat(F,A3) = zero_zero_rat )
      <=> ? [X3: int] :
            ( member_int(X3,A3)
            & ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_6446_prod__zero__iff,axiom,
    ! [A3: set_nat,F: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( groups2279045934846249631atural(F,A3) = zero_z2226904508553997617atural )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
            & ( aa_nat_Code_natural(F,X3) = zero_z2226904508553997617atural ) ) ) ) ).

% prod_zero_iff
tff(fact_6447_prod__zero__iff,axiom,
    ! [A3: set_int,F: fun_int_Code_natural] :
      ( finite_finite_int(A3)
     => ( ( groups2650699495083094267atural(F,A3) = zero_z2226904508553997617atural )
      <=> ? [X3: int] :
            ( member_int(X3,A3)
            & ( aa_int_Code_natural(F,X3) = zero_z2226904508553997617atural ) ) ) ) ).

% prod_zero_iff
tff(fact_6448_prod__zero__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set_nat_nat(groups708209901874060359at_nat(F),A3) = zero_zero_nat )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
            & ( aa_nat_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_6449_prod__zero__iff,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set_nat_int(groups705719431365010083at_int(F),A3) = zero_zero_int )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
            & ( aa_nat_int(F,X3) = zero_zero_int ) ) ) ) ).

% prod_zero_iff
tff(fact_6450_prod__zero__iff,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( finite_finite_int(A3)
     => ( ( groups1705073143266064639nt_int(F,A3) = zero_zero_int )
      <=> ? [X3: int] :
            ( member_int(X3,A3)
            & ( aa_int_int(F,X3) = zero_zero_int ) ) ) ) ).

% prod_zero_iff
tff(fact_6451_prod_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_nat] :
      ( ~ finite_finite_int(A3)
     => ( groups1707563613775114915nt_nat(G,A3) = one_one_nat ) ) ).

% prod.infinite
tff(fact_6452_prod_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_Code_integer] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3) = one_one_Code_integer ) ) ).

% prod.infinite
tff(fact_6453_prod_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_Code_integer] :
      ( ~ finite_finite_int(A3)
     => ( groups3827104343326376752nteger(G,A3) = one_one_Code_integer ) ) ).

% prod.infinite
tff(fact_6454_prod_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),A3) = one_one_rat ) ) ).

% prod.infinite
tff(fact_6455_prod_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( ~ finite_finite_int(A3)
     => ( groups1072433553688619179nt_rat(G,A3) = one_one_rat ) ) ).

% prod.infinite
tff(fact_6456_prod_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_Code_natural] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( groups2279045934846249631atural(G,A3) = one_one_Code_natural ) ) ).

% prod.infinite
tff(fact_6457_prod_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_Code_natural] :
      ( ~ finite_finite_int(A3)
     => ( groups2650699495083094267atural(G,A3) = one_one_Code_natural ) ) ).

% prod.infinite
tff(fact_6458_prod_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A3) = one_one_nat ) ) ).

% prod.infinite
tff(fact_6459_prod_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_int] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),A3) = one_one_int ) ) ).

% prod.infinite
tff(fact_6460_prod_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( ~ finite_finite_int(A3)
     => ( groups1705073143266064639nt_int(G,A3) = one_one_int ) ) ).

% prod.infinite
tff(fact_6461_bit_Odisj__one__right,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xa),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_one_right
tff(fact_6462_bit_Odisj__one__right,axiom,
    ! [Xa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,Xa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_one_right
tff(fact_6463_bit_Odisj__one__left,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,one_one_int)),Xa) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_one_left
tff(fact_6464_bit_Odisj__one__left,axiom,
    ! [Xa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Xa) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_one_left
tff(fact_6465_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_nat] :
      ( finite_finite_int(S2)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_oq(A),B),S2) = $ite(member_int(A,S2),aa_int_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_6466_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_or(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_int(B,A),zero_zero_int) ) ) ).

% sum.delta'
tff(fact_6467_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_os(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta'
tff(fact_6468_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_Code_integer] :
      ( finite_finite_int(S2)
     => ( groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_ot(A),B),S2) = $ite(member_int(A,S2),aa_int_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta'
tff(fact_6469_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ou(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6470_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( finite_finite_int(S2)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ov(A),B),S2) = $ite(member_int(A,S2),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6471_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups6325495683096345652atural(aa_fun56224542555504168atural(aTP_Lamm_ow(A),B),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_Code_natural(B,A),zero_z2226904508553997617atural) ) ) ).

% sum.delta'
tff(fact_6472_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_Code_natural] :
      ( finite_finite_int(S2)
     => ( groups6697149243333190288atural(aa_fun4771645977350521568atural(aTP_Lamm_ox(A),B),S2) = $ite(member_int(A,S2),aa_int_Code_natural(B,A),zero_z2226904508553997617atural) ) ) ).

% sum.delta'
tff(fact_6473_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_oy(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_6474_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_int] :
      ( finite_finite_int(S2)
     => ( groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_oz(A),B),S2) = $ite(member_int(A,S2),aa_int_int(B,A),zero_zero_int) ) ) ).

% sum.delta'
tff(fact_6475_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_nat] :
      ( finite_finite_int(S2)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pa(A),B),S2) = $ite(member_int(A,S2),aa_int_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_6476_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_pb(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_int(B,A),zero_zero_int) ) ) ).

% sum.delta
tff(fact_6477_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_pc(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta
tff(fact_6478_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_Code_integer] :
      ( finite_finite_int(S2)
     => ( groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_pd(A),B),S2) = $ite(member_int(A,S2),aa_int_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta
tff(fact_6479_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_pe(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6480_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( finite_finite_int(S2)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_pf(A),B),S2) = $ite(member_int(A,S2),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6481_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups6325495683096345652atural(aa_fun56224542555504168atural(aTP_Lamm_pg(A),B),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_Code_natural(B,A),zero_z2226904508553997617atural) ) ) ).

% sum.delta
tff(fact_6482_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_Code_natural] :
      ( finite_finite_int(S2)
     => ( groups6697149243333190288atural(aa_fun4771645977350521568atural(aTP_Lamm_ph(A),B),S2) = $ite(member_int(A,S2),aa_int_Code_natural(B,A),zero_z2226904508553997617atural) ) ) ).

% sum.delta
tff(fact_6483_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pi(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_6484_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_int] :
      ( finite_finite_int(S2)
     => ( groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_pj(A),B),S2) = $ite(member_int(A,S2),aa_int_int(B,A),zero_zero_int) ) ) ).

% sum.delta
tff(fact_6485_prod__eq__1__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( finite_finite_int(A3)
     => ( ( groups1707563613775114915nt_nat(F,A3) = one_one_nat )
      <=> ! [X3: int] :
            ( member_int(X3,A3)
           => ( aa_int_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_6486_prod__eq__1__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( ( aa_set4749074711808745205at_nat(groups4077766827762148844at_nat(F),A3) = one_one_nat )
      <=> ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A3)
           => ( aa_Pro4243173770044397973at_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_6487_prod__eq__1__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set_nat_nat(groups708209901874060359at_nat(F),A3) = one_one_nat )
      <=> ! [X3: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
           => ( aa_nat_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_6488_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_nat] :
      ( finite_finite_int(S2)
     => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pk(A),B),S2) = $ite(member_int(A,S2),aa_int_nat(B,A),one_one_nat) ) ) ).

% prod.delta
tff(fact_6489_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aTP_Lamm_pl(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta
tff(fact_6490_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_Code_integer] :
      ( finite_finite_int(S2)
     => ( groups3827104343326376752nteger(aa_fun5304498221708291146nteger(aTP_Lamm_pm(A),B),S2) = $ite(member_int(A,S2),aa_int_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta
tff(fact_6491_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_pn(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_6492_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( finite_finite_int(S2)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_po(A),B),S2) = $ite(member_int(A,S2),aa_int_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_6493_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups2279045934846249631atural(aa_fun56224542555504168atural(aTP_Lamm_pp(A),B),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_Code_natural(B,A),one_one_Code_natural) ) ) ).

% prod.delta
tff(fact_6494_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_Code_natural] :
      ( finite_finite_int(S2)
     => ( groups2650699495083094267atural(aa_fun4771645977350521568atural(aTP_Lamm_pq(A),B),S2) = $ite(member_int(A,S2),aa_int_Code_natural(B,A),one_one_Code_natural) ) ) ).

% prod.delta
tff(fact_6495_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pr(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_nat(B,A),one_one_nat) ) ) ).

% prod.delta
tff(fact_6496_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_ps(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_int(B,A),one_one_int) ) ) ).

% prod.delta
tff(fact_6497_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_int] :
      ( finite_finite_int(S2)
     => ( groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_pt(A),B),S2) = $ite(member_int(A,S2),aa_int_int(B,A),one_one_int) ) ) ).

% prod.delta
tff(fact_6498_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_nat] :
      ( finite_finite_int(S2)
     => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pu(A),B),S2) = $ite(member_int(A,S2),aa_int_nat(B,A),one_one_nat) ) ) ).

% prod.delta'
tff(fact_6499_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aTP_Lamm_pv(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta'
tff(fact_6500_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_Code_integer] :
      ( finite_finite_int(S2)
     => ( groups3827104343326376752nteger(aa_fun5304498221708291146nteger(aTP_Lamm_pw(A),B),S2) = $ite(member_int(A,S2),aa_int_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta'
tff(fact_6501_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_px(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_6502_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( finite_finite_int(S2)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_py(A),B),S2) = $ite(member_int(A,S2),aa_int_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_6503_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups2279045934846249631atural(aa_fun56224542555504168atural(aTP_Lamm_pz(A),B),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_Code_natural(B,A),one_one_Code_natural) ) ) ).

% prod.delta'
tff(fact_6504_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_Code_natural] :
      ( finite_finite_int(S2)
     => ( groups2650699495083094267atural(aa_fun4771645977350521568atural(aTP_Lamm_qa(A),B),S2) = $ite(member_int(A,S2),aa_int_Code_natural(B,A),one_one_Code_natural) ) ) ).

% prod.delta'
tff(fact_6505_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qb(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_nat(B,A),one_one_nat) ) ) ).

% prod.delta'
tff(fact_6506_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_qc(A),B)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_int(B,A),one_one_int) ) ) ).

% prod.delta'
tff(fact_6507_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_int] :
      ( finite_finite_int(S2)
     => ( groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_qd(A),B),S2) = $ite(member_int(A,S2),aa_int_int(B,A),one_one_int) ) ) ).

% prod.delta'
tff(fact_6508_or__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% or_nat_numerals(4)
tff(fact_6509_or__nat__numerals_I2_J,axiom,
    ! [Y3: 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,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% or_nat_numerals(2)
tff(fact_6510_or__numerals_I2_J,axiom,
    ! [Y3: num] : bit_se9127793120404214118atural(one_one_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y3))) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y3)) ).

% or_numerals(2)
tff(fact_6511_or__numerals_I2_J,axiom,
    ! [Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3)) ).

% or_numerals(2)
tff(fact_6512_or__numerals_I2_J,axiom,
    ! [Y3: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y3))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y3)) ).

% or_numerals(2)
tff(fact_6513_or__numerals_I2_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% or_numerals(2)
tff(fact_6514_or__numerals_I8_J,axiom,
    ! [Xa: num] : bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Xa)),one_one_Code_natural) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Xa)) ).

% or_numerals(8)
tff(fact_6515_or__numerals_I8_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa)) ).

% or_numerals(8)
tff(fact_6516_or__numerals_I8_J,axiom,
    ! [Xa: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Xa))),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Xa)) ).

% or_numerals(8)
tff(fact_6517_or__numerals_I8_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% or_numerals(8)
tff(fact_6518_prod__pos__nat__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( finite_finite_int(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F,A3))
      <=> ! [X3: int] :
            ( member_int(X3,A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,X3)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6519_prod__pos__nat__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set4749074711808745205at_nat(groups4077766827762148844at_nat(F),A3))
      <=> ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Pro4243173770044397973at_nat(F,X3)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6520_prod__pos__nat__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3))
      <=> ! [X3: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6521_gcd__nat__abs__left__eq,axiom,
    ! [K: int,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% gcd_nat_abs_left_eq
tff(fact_6522_gcd__nat__abs__right__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ).

% gcd_nat_abs_right_eq
tff(fact_6523_or__nat__numerals_I1_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% or_nat_numerals(1)
tff(fact_6524_or__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% or_nat_numerals(3)
tff(fact_6525_or__numerals_I1_J,axiom,
    ! [Y3: num] : bit_se9127793120404214118atural(one_one_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Y3))) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y3)) ).

% or_numerals(1)
tff(fact_6526_or__numerals_I1_J,axiom,
    ! [Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y3))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3)) ).

% or_numerals(1)
tff(fact_6527_or__numerals_I1_J,axiom,
    ! [Y3: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y3))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y3)) ).

% or_numerals(1)
tff(fact_6528_or__numerals_I1_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% or_numerals(1)
tff(fact_6529_or__numerals_I5_J,axiom,
    ! [Xa: num] : bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Xa)),one_one_Code_natural) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Xa)) ).

% or_numerals(5)
tff(fact_6530_or__numerals_I5_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xa))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa)) ).

% or_numerals(5)
tff(fact_6531_or__numerals_I5_J,axiom,
    ! [Xa: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Xa))),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Xa)) ).

% or_numerals(5)
tff(fact_6532_or__numerals_I5_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% or_numerals(5)
tff(fact_6533_sum__zero__power,axiom,
    ! [C: fun_nat_rat,A3: set_nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qe(C)),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),A3) ),
        aa_nat_rat(C,zero_zero_nat),
        zero_zero_rat ) ).

% sum_zero_power
tff(fact_6534_or__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & ( B = zero_zero_int ) ) ) ).

% or_eq_0_iff
tff(fact_6535_or__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),B) = zero_z3403309356797280102nteger )
    <=> ( ( A = zero_z3403309356797280102nteger )
        & ( B = zero_z3403309356797280102nteger ) ) ) ).

% or_eq_0_iff
tff(fact_6536_or__eq__0__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( bit_se9127793120404214118atural(A,B) = zero_z2226904508553997617atural )
    <=> ( ( A = zero_z2226904508553997617atural )
        & ( B = zero_z2226904508553997617atural ) ) ) ).

% or_eq_0_iff
tff(fact_6537_or__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% or_eq_0_iff
tff(fact_6538_bit_Odisj__zero__right,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xa),zero_zero_int) = Xa ).

% bit.disj_zero_right
tff(fact_6539_bit_Odisj__zero__right,axiom,
    ! [Xa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,Xa),zero_z3403309356797280102nteger) = Xa ).

% bit.disj_zero_right
tff(fact_6540_gcd__mult__distrib__nat,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(gcd_gcd_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_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)) ).

% gcd_mult_distrib_nat
tff(fact_6541_gcd__nat_Omono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,C),D)) ) ) ).

% gcd_nat.mono
tff(fact_6542_gcd__nat_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).

% gcd_nat.orderE
tff(fact_6543_gcd__nat_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% gcd_nat.orderI
tff(fact_6544_gcd__nat_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).

% gcd_nat.absorb3
tff(fact_6545_gcd__nat_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
        & ( B != A ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).

% gcd_nat.absorb4
tff(fact_6546_gcd__nat_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
         => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% gcd_nat.boundedE
tff(fact_6547_gcd__nat_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C)) ) ) ).

% gcd_nat.boundedI
tff(fact_6548_gcd__nat_Oorder__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).

% gcd_nat.order_iff
tff(fact_6549_gcd__nat_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A) ).

% gcd_nat.cobounded1
tff(fact_6550_gcd__nat_Ocobounded2,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ).

% gcd_nat.cobounded2
tff(fact_6551_gcd__nat_Oabsorb__iff1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).

% gcd_nat.absorb_iff1
tff(fact_6552_gcd__nat_Oabsorb__iff2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).

% gcd_nat.absorb_iff2
tff(fact_6553_gcd__nat_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C) ) ).

% gcd_nat.coboundedI1
tff(fact_6554_gcd__nat_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C) ) ).

% gcd_nat.coboundedI2
tff(fact_6555_gcd__nat_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C))
        & ( A != aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C) ) )
     => ~ ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
            & ( A != B ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
              & ( A != C ) ) ) ) ).

% gcd_nat.strict_boundedE
tff(fact_6556_gcd__nat_Ostrict__order__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
        & ( A != B ) ) ) ).

% gcd_nat.strict_order_iff
tff(fact_6557_gcd__nat_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
        & ( A != C ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) != C ) ) ) ).

% gcd_nat.strict_coboundedI1
tff(fact_6558_gcd__nat_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
        & ( B != C ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) != C ) ) ) ).

% gcd_nat.strict_coboundedI2
tff(fact_6559_gcd__unique__nat,axiom,
    ! [D: nat,A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
        & ! [E3: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,E3),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,E3),B) )
           => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,E3),D) ) )
    <=> ( D = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).

% gcd_unique_nat
tff(fact_6560_gcd__red__nat,axiom,
    ! [Xa: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Y3) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y3)) ).

% gcd_red_nat
tff(fact_6561_finite__M__bounded__by__nat,axiom,
    ! [Pa: fun_nat_o,I: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_qf(Pa),I))) ).

% finite_M_bounded_by_nat
tff(fact_6562_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(F,N2))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_qg(F),U))) ) ).

% finite_less_ub
tff(fact_6563_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B3: set_nat,G: fun_int_fun_nat_nat,R4: fun_int_fun_nat_o] :
      ( finite_finite_int(A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_qi(B3),G),R4),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_ql(A3),G),R4)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6564_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_int,G: fun_nat_fun_int_int,R4: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( finite_finite_int(B3)
       => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_qn(B3),G),R4)),A3) = groups4538972089207619220nt_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_qp(A3),G),R4),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6565_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_int,G: fun_nat_fun_int_nat,R4: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( finite_finite_int(B3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_qq(B3),G),R4)),A3) = groups4541462559716669496nt_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_qs(A3),G),R4),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6566_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_fun_nat_nat,R4: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_qu(B3),G),R4)),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_qw(A3),G),R4)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6567_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B3: set_nat,G: fun_int_fun_nat_int,R4: fun_int_fun_nat_o] :
      ( finite_finite_int(A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups4538972089207619220nt_int(aa_fun472946340410393319nt_int(aa_fun6062826860762256214nt_int(aTP_Lamm_qx(B3),G),R4),A3) = aa_set_nat_int(groups3539618377306564664at_int(aa_fun8696964665364114571at_int(aa_fun3875905286069234426at_int(aTP_Lamm_qy(A3),G),R4)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6568_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_fun_int_int,R4: fun_int_fun_int_o] :
      ( finite_finite_int(A3)
     => ( finite_finite_int(B3)
       => ( groups4538972089207619220nt_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_ra(B3),G),R4),A3) = groups4538972089207619220nt_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_rc(A3),G),R4),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6569_sum_Oswap__restrict,axiom,
    ! [A3: set_set_nat,B3: set_nat,G: fun_se1179353343636554400at_nat,R4: fun_se382663106265794036_nat_o] :
      ( finite1152437895449049373et_nat(A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups8294997508430121362at_nat(aa_fun6238879321244136459at_nat(aa_fun1036415571424436988at_nat(aTP_Lamm_re(B3),G),R4),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun1153650177998607445at_nat(aa_fun5868473082678171206at_nat(aTP_Lamm_rh(A3),G),R4)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6570_sum_Oswap__restrict,axiom,
    ! [A3: set_set_nat,B3: set_int,G: fun_se876191773397849432nt_int,R4: fun_se7360890175450123928_int_o] :
      ( finite1152437895449049373et_nat(A3)
     => ( finite_finite_int(B3)
       => ( groups8292507037921071086at_int(aa_fun5327250345464651019at_int(aa_fun3157859007690329412at_int(aTP_Lamm_rj(B3),G),R4),A3) = groups4538972089207619220nt_int(aa_fun7007818992012977073nt_int(aa_fun7130641361752856810nt_int(aTP_Lamm_rm(A3),G),R4),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6571_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_set_nat,G: fun_na106483476154090400at_nat,R4: fun_na8563630686737515764_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( finite1152437895449049373et_nat(B3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7187098123162309461at_nat(aa_fun585184524491904070at_nat(aTP_Lamm_ro(B3),G),R4)),A3) = groups8294997508430121362at_nat(aa_fun255155394271802123at_nat(aa_fun6284103964554420476at_nat(aTP_Lamm_rr(A3),G),R4),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6572_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B3: set_set_nat,G: fun_in6555721293707029592at_int,R4: fun_in7136481858371106712_nat_o] :
      ( finite_finite_int(A3)
     => ( finite1152437895449049373et_nat(B3)
       => ( groups4538972089207619220nt_int(aa_fun1461796484577889457nt_int(aa_fun1283830802835827434nt_int(aTP_Lamm_rt(B3),G),R4),A3) = groups8292507037921071086at_int(aa_fun5805668902077597195at_int(aa_fun5576367408028688196at_int(aTP_Lamm_rw(A3),G),R4),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_6573_prod_Oswap__restrict,axiom,
    ! [A3: set_int,B3: set_nat,G: fun_int_fun_nat_nat,R4: fun_int_fun_nat_o] :
      ( finite_finite_int(A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups1707563613775114915nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_rx(B3),G),R4),A3) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_ry(A3),G),R4)),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6574_prod_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_int,G: fun_nat_fun_int_nat,R4: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( finite_finite_int(B3)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_rz(B3),G),R4)),A3) = groups1707563613775114915nt_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_sa(A3),G),R4),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6575_prod_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_fun_nat_nat,R4: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_sb(B3),G),R4)),A3) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_sc(A3),G),R4)),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6576_prod_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_nat,G: fun_nat_fun_nat_int,R4: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun4679828425598226919at_int(aa_fun5734951013428723122at_int(aTP_Lamm_sd(B3),G),R4)),A3) = aa_set_nat_int(groups705719431365010083at_int(aa_fun4679828425598226919at_int(aa_fun5734951013428723122at_int(aTP_Lamm_se(A3),G),R4)),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6577_prod_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_int,G: fun_nat_fun_int_int,R4: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( finite_finite_int(B3)
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_sf(B3),G),R4)),A3) = groups1705073143266064639nt_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_sg(A3),G),R4),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6578_prod_Oswap__restrict,axiom,
    ! [A3: set_int,B3: set_nat,G: fun_int_fun_nat_int,R4: fun_int_fun_nat_o] :
      ( finite_finite_int(A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups1705073143266064639nt_int(aa_fun472946340410393319nt_int(aa_fun6062826860762256214nt_int(aTP_Lamm_sh(B3),G),R4),A3) = aa_set_nat_int(groups705719431365010083at_int(aa_fun8696964665364114571at_int(aa_fun3875905286069234426at_int(aTP_Lamm_si(A3),G),R4)),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6579_prod_Oswap__restrict,axiom,
    ! [A3: set_int,B3: set_int,G: fun_int_fun_int_int,R4: fun_int_fun_int_o] :
      ( finite_finite_int(A3)
     => ( finite_finite_int(B3)
       => ( groups1705073143266064639nt_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_sj(B3),G),R4),A3) = groups1705073143266064639nt_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_sk(A3),G),R4),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6580_prod_Oswap__restrict,axiom,
    ! [A3: set_set_nat,B3: set_nat,G: fun_se1179353343636554400at_nat,R4: fun_se382663106265794036_nat_o] :
      ( finite1152437895449049373et_nat(A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups4248547760180025341at_nat(aa_fun6238879321244136459at_nat(aa_fun1036415571424436988at_nat(aTP_Lamm_sl(B3),G),R4),A3) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun1153650177998607445at_nat(aa_fun5868473082678171206at_nat(aTP_Lamm_sm(A3),G),R4)),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6581_prod_Oswap__restrict,axiom,
    ! [A3: set_set_nat,B3: set_nat,G: fun_se7912642235559603452at_int,R4: fun_se382663106265794036_nat_o] :
      ( finite1152437895449049373et_nat(A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups4246057289670975065at_int(aa_fun2061028301734939751at_int(aa_fun8609273137425597436at_int(aTP_Lamm_sn(B3),G),R4),A3) = aa_set_nat_int(groups705719431365010083at_int(aa_fun6199171195344186545at_int(aa_fun2727794920753907526at_int(aTP_Lamm_sp(A3),G),R4)),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6582_prod_Oswap__restrict,axiom,
    ! [A3: set_set_nat,B3: set_int,G: fun_se876191773397849432nt_int,R4: fun_se7360890175450123928_int_o] :
      ( finite1152437895449049373et_nat(A3)
     => ( finite_finite_int(B3)
       => ( groups4246057289670975065at_int(aa_fun5327250345464651019at_int(aa_fun3157859007690329412at_int(aTP_Lamm_sq(B3),G),R4),A3) = groups1705073143266064639nt_int(aa_fun7007818992012977073nt_int(aa_fun7130641361752856810nt_int(aTP_Lamm_sr(A3),G),R4),B3) ) ) ) ).

% prod.swap_restrict
tff(fact_6583_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ) ).

% gcd_le2_nat
tff(fact_6584_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A) ) ).

% gcd_le1_nat
tff(fact_6585_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

% gcd_diff2_nat
tff(fact_6586_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

% gcd_diff1_nat
tff(fact_6587_gcd__nat_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Xaa) = Y3 )
     => ( Y3 = $ite(Xaa = zero_zero_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_6588_gcd__nat_Osimps,axiom,
    ! [Xa: nat,Y3: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Y3) = $ite(Y3 = zero_zero_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y3))) ).

% gcd_nat.simps
tff(fact_6589_gcd__non__0__nat,axiom,
    ! [Y3: nat,Xa: nat] :
      ( ( Y3 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Y3) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y3)) ) ) ).

% gcd_non_0_nat
tff(fact_6590_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T3: set_int,G: fun_int_nat,Ha: fun_int_nat] :
      ( finite_finite_int(S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_int_int(I,aa_int_int(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => member_int(aa_int_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_int_int(J,aa_int_int(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => member_int(aa_int_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_nat(G,A2) = zero_zero_nat ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_nat(Ha,B2) = zero_zero_nat ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_int_nat(Ha,aa_int_int(J,A2)) = aa_int_nat(G,A2) ) )
                     => ( groups4541462559716669496nt_nat(G,S2) = groups4541462559716669496nt_nat(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6591_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T3: set_int,G: fun_int_Code_integer,Ha: fun_int_Code_integer] :
      ( finite_finite_int(S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_int_int(I,aa_int_int(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => member_int(aa_int_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_int_int(J,aa_int_int(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => member_int(aa_int_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_Code_integer(G,A2) = zero_z3403309356797280102nteger ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_Code_integer(Ha,B2) = zero_z3403309356797280102nteger ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_int_Code_integer(Ha,aa_int_int(J,A2)) = aa_int_Code_integer(G,A2) ) )
                     => ( groups7873554091576472773nteger(G,S2) = groups7873554091576472773nteger(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6592_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T3: set_int,G: fun_int_rat,Ha: fun_int_rat] :
      ( finite_finite_int(S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_int_int(I,aa_int_int(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => member_int(aa_int_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_int_int(J,aa_int_int(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => member_int(aa_int_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_rat(G,A2) = zero_zero_rat ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_rat(Ha,B2) = zero_zero_rat ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_int_rat(Ha,aa_int_int(J,A2)) = aa_int_rat(G,A2) ) )
                     => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6593_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T3: set_int,G: fun_int_Code_natural,Ha: fun_int_Code_natural] :
      ( finite_finite_int(S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_int_int(I,aa_int_int(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => member_int(aa_int_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_int_int(J,aa_int_int(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => member_int(aa_int_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_Code_natural(G,A2) = zero_z2226904508553997617atural ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_Code_natural(Ha,B2) = zero_z2226904508553997617atural ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_int_Code_natural(Ha,aa_int_int(J,A2)) = aa_int_Code_natural(G,A2) ) )
                     => ( groups6697149243333190288atural(G,S2) = groups6697149243333190288atural(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6594_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_nat,S2: set_int,I: fun_nat_int,J: fun_int_nat,T3: set_nat,G: fun_int_Code_integer,Ha: fun_nat_Code_integer] :
      ( finite_finite_int(S5)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_nat_int(I,aa_int_nat(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(J,A2)),minus_minus_set_nat(T3,T2)) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(T3,T2))
                 => ( aa_int_nat(J,aa_nat_int(I,B2)) = B2 ) )
             => ( ! [B2: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(T3,T2))
                   => member_int(aa_nat_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_Code_integer(G,A2) = zero_z3403309356797280102nteger ) )
                 => ( ! [B2: nat] :
                        ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),T2)
                       => ( aa_nat_Code_integer(Ha,B2) = zero_z3403309356797280102nteger ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_nat_Code_integer(Ha,aa_int_nat(J,A2)) = aa_int_Code_integer(G,A2) ) )
                     => ( groups7873554091576472773nteger(G,S2) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6595_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_nat,S2: set_int,I: fun_nat_int,J: fun_int_nat,T3: set_nat,G: fun_int_rat,Ha: fun_nat_rat] :
      ( finite_finite_int(S5)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_nat_int(I,aa_int_nat(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(J,A2)),minus_minus_set_nat(T3,T2)) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(T3,T2))
                 => ( aa_int_nat(J,aa_nat_int(I,B2)) = B2 ) )
             => ( ! [B2: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(T3,T2))
                   => member_int(aa_nat_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_rat(G,A2) = zero_zero_rat ) )
                 => ( ! [B2: nat] :
                        ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),T2)
                       => ( aa_nat_rat(Ha,B2) = zero_zero_rat ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_nat_rat(Ha,aa_int_nat(J,A2)) = aa_int_rat(G,A2) ) )
                     => ( groups3906332499630173760nt_rat(G,S2) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6596_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_nat,S2: set_int,I: fun_nat_int,J: fun_int_nat,T3: set_nat,G: fun_int_Code_natural,Ha: fun_nat_Code_natural] :
      ( finite_finite_int(S5)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_nat_int(I,aa_int_nat(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(J,A2)),minus_minus_set_nat(T3,T2)) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(T3,T2))
                 => ( aa_int_nat(J,aa_nat_int(I,B2)) = B2 ) )
             => ( ! [B2: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(T3,T2))
                   => member_int(aa_nat_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_Code_natural(G,A2) = zero_z2226904508553997617atural ) )
                 => ( ! [B2: nat] :
                        ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),T2)
                       => ( aa_nat_Code_natural(Ha,B2) = zero_z2226904508553997617atural ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_nat_Code_natural(Ha,aa_int_nat(J,A2)) = aa_int_Code_natural(G,A2) ) )
                     => ( groups6697149243333190288atural(G,S2) = groups6325495683096345652atural(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6597_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_nat,T2: set_int,S2: set_nat,I: fun_int_nat,J: fun_nat_int,T3: set_int,G: fun_nat_Code_integer,Ha: fun_int_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(S2,S5))
             => ( aa_int_nat(I,aa_nat_int(J,A2)) = A2 ) )
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(S2,S5))
               => member_int(aa_nat_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_nat_int(J,aa_int_nat(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(I,B2)),minus_minus_set_nat(S2,S5)) )
               => ( ! [A2: nat] :
                      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),S5)
                     => ( aa_nat_Code_integer(G,A2) = zero_z3403309356797280102nteger ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_Code_integer(Ha,B2) = zero_z3403309356797280102nteger ) )
                   => ( ! [A2: nat] :
                          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),S2)
                         => ( aa_int_Code_integer(Ha,aa_nat_int(J,A2)) = aa_nat_Code_integer(G,A2) ) )
                     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),S2) = groups7873554091576472773nteger(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6598_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_nat,T2: set_int,S2: set_nat,I: fun_int_nat,J: fun_nat_int,T3: set_int,G: fun_nat_rat,Ha: fun_int_rat] :
      ( aa_set_nat_o(finite_finite_nat,S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(S2,S5))
             => ( aa_int_nat(I,aa_nat_int(J,A2)) = A2 ) )
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(S2,S5))
               => member_int(aa_nat_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_nat_int(J,aa_int_nat(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(I,B2)),minus_minus_set_nat(S2,S5)) )
               => ( ! [A2: nat] :
                      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),S5)
                     => ( aa_nat_rat(G,A2) = zero_zero_rat ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_rat(Ha,B2) = zero_zero_rat ) )
                   => ( ! [A2: nat] :
                          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),S2)
                         => ( aa_int_rat(Ha,aa_nat_int(J,A2)) = aa_nat_rat(G,A2) ) )
                     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),S2) = groups3906332499630173760nt_rat(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6599_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_nat,T2: set_int,S2: set_nat,I: fun_int_nat,J: fun_nat_int,T3: set_int,G: fun_nat_Code_natural,Ha: fun_int_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(S2,S5))
             => ( aa_int_nat(I,aa_nat_int(J,A2)) = A2 ) )
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(S2,S5))
               => member_int(aa_nat_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_nat_int(J,aa_int_nat(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(I,B2)),minus_minus_set_nat(S2,S5)) )
               => ( ! [A2: nat] :
                      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),S5)
                     => ( aa_nat_Code_natural(G,A2) = zero_z2226904508553997617atural ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_Code_natural(Ha,B2) = zero_z2226904508553997617atural ) )
                   => ( ! [A2: nat] :
                          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),S2)
                         => ( aa_int_Code_natural(Ha,aa_nat_int(J,A2)) = aa_nat_Code_natural(G,A2) ) )
                     => ( groups6325495683096345652atural(G,S2) = groups6697149243333190288atural(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6600_prod__zero,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( finite_finite_int(A3)
     => ( ? [X: int] :
            ( member_int(X,A3)
            & ( aa_int_nat(F,X) = zero_zero_nat ) )
       => ( groups1707563613775114915nt_nat(F,A3) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_6601_prod__zero,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ? [X: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A3)
            & ( aa_nat_Code_integer(F,X) = zero_z3403309356797280102nteger ) )
       => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A3) = zero_z3403309356797280102nteger ) ) ) ).

% prod_zero
tff(fact_6602_prod__zero,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( finite_finite_int(A3)
     => ( ? [X: int] :
            ( member_int(X,A3)
            & ( aa_int_Code_integer(F,X) = zero_z3403309356797280102nteger ) )
       => ( groups3827104343326376752nteger(F,A3) = zero_z3403309356797280102nteger ) ) ) ).

% prod_zero
tff(fact_6603_prod__zero,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ? [X: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A3)
            & ( aa_nat_rat(F,X) = zero_zero_rat ) )
       => ( aa_set_nat_rat(groups73079841787564623at_rat(F),A3) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_6604_prod__zero,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( finite_finite_int(A3)
     => ( ? [X: int] :
            ( member_int(X,A3)
            & ( aa_int_rat(F,X) = zero_zero_rat ) )
       => ( groups1072433553688619179nt_rat(F,A3) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_6605_prod__zero,axiom,
    ! [A3: set_nat,F: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ? [X: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A3)
            & ( aa_nat_Code_natural(F,X) = zero_z2226904508553997617atural ) )
       => ( groups2279045934846249631atural(F,A3) = zero_z2226904508553997617atural ) ) ) ).

% prod_zero
tff(fact_6606_prod__zero,axiom,
    ! [A3: set_int,F: fun_int_Code_natural] :
      ( finite_finite_int(A3)
     => ( ? [X: int] :
            ( member_int(X,A3)
            & ( aa_int_Code_natural(F,X) = zero_z2226904508553997617atural ) )
       => ( groups2650699495083094267atural(F,A3) = zero_z2226904508553997617atural ) ) ) ).

% prod_zero
tff(fact_6607_prod__zero,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ? [X: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A3)
            & ( aa_nat_nat(F,X) = zero_zero_nat ) )
       => ( aa_set_nat_nat(groups708209901874060359at_nat(F),A3) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_6608_prod__zero,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ? [X: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X),A3)
            & ( aa_nat_int(F,X) = zero_zero_int ) )
       => ( aa_set_nat_int(groups705719431365010083at_int(F),A3) = zero_zero_int ) ) ) ).

% prod_zero
tff(fact_6609_prod__zero,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( finite_finite_int(A3)
     => ( ? [X: int] :
            ( member_int(X,A3)
            & ( aa_int_int(F,X) = zero_zero_int ) )
       => ( groups1705073143266064639nt_int(F,A3) = zero_zero_int ) ) ) ).

% prod_zero
tff(fact_6610_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T3: set_int,G: fun_int_nat,Ha: fun_int_nat] :
      ( finite_finite_int(S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_int_int(I,aa_int_int(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => member_int(aa_int_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_int_int(J,aa_int_int(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => member_int(aa_int_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_nat(G,A2) = one_one_nat ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_nat(Ha,B2) = one_one_nat ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_int_nat(Ha,aa_int_int(J,A2)) = aa_int_nat(G,A2) ) )
                     => ( groups1707563613775114915nt_nat(G,S2) = groups1707563613775114915nt_nat(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6611_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T3: set_int,G: fun_int_Code_integer,Ha: fun_int_Code_integer] :
      ( finite_finite_int(S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_int_int(I,aa_int_int(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => member_int(aa_int_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_int_int(J,aa_int_int(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => member_int(aa_int_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_Code_integer(G,A2) = one_one_Code_integer ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_Code_integer(Ha,B2) = one_one_Code_integer ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_int_Code_integer(Ha,aa_int_int(J,A2)) = aa_int_Code_integer(G,A2) ) )
                     => ( groups3827104343326376752nteger(G,S2) = groups3827104343326376752nteger(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6612_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T3: set_int,G: fun_int_rat,Ha: fun_int_rat] :
      ( finite_finite_int(S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_int_int(I,aa_int_int(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => member_int(aa_int_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_int_int(J,aa_int_int(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => member_int(aa_int_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_rat(G,A2) = one_one_rat ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_rat(Ha,B2) = one_one_rat ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_int_rat(Ha,aa_int_int(J,A2)) = aa_int_rat(G,A2) ) )
                     => ( groups1072433553688619179nt_rat(G,S2) = groups1072433553688619179nt_rat(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6613_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T3: set_int,G: fun_int_Code_natural,Ha: fun_int_Code_natural] :
      ( finite_finite_int(S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_int_int(I,aa_int_int(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => member_int(aa_int_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_int_int(J,aa_int_int(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => member_int(aa_int_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_Code_natural(G,A2) = one_one_Code_natural ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_Code_natural(Ha,B2) = one_one_Code_natural ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_int_Code_natural(Ha,aa_int_int(J,A2)) = aa_int_Code_natural(G,A2) ) )
                     => ( groups2650699495083094267atural(G,S2) = groups2650699495083094267atural(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6614_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_nat,S2: set_int,I: fun_nat_int,J: fun_int_nat,T3: set_nat,G: fun_int_Code_integer,Ha: fun_nat_Code_integer] :
      ( finite_finite_int(S5)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_nat_int(I,aa_int_nat(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(J,A2)),minus_minus_set_nat(T3,T2)) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(T3,T2))
                 => ( aa_int_nat(J,aa_nat_int(I,B2)) = B2 ) )
             => ( ! [B2: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(T3,T2))
                   => member_int(aa_nat_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_Code_integer(G,A2) = one_one_Code_integer ) )
                 => ( ! [B2: nat] :
                        ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),T2)
                       => ( aa_nat_Code_integer(Ha,B2) = one_one_Code_integer ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_nat_Code_integer(Ha,aa_int_nat(J,A2)) = aa_int_Code_integer(G,A2) ) )
                     => ( groups3827104343326376752nteger(G,S2) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6615_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_nat,S2: set_int,I: fun_nat_int,J: fun_int_nat,T3: set_nat,G: fun_int_rat,Ha: fun_nat_rat] :
      ( finite_finite_int(S5)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_nat_int(I,aa_int_nat(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(J,A2)),minus_minus_set_nat(T3,T2)) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(T3,T2))
                 => ( aa_int_nat(J,aa_nat_int(I,B2)) = B2 ) )
             => ( ! [B2: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(T3,T2))
                   => member_int(aa_nat_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_rat(G,A2) = one_one_rat ) )
                 => ( ! [B2: nat] :
                        ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),T2)
                       => ( aa_nat_rat(Ha,B2) = one_one_rat ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_nat_rat(Ha,aa_int_nat(J,A2)) = aa_int_rat(G,A2) ) )
                     => ( groups1072433553688619179nt_rat(G,S2) = aa_set_nat_rat(groups73079841787564623at_rat(Ha),T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6616_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T2: set_nat,S2: set_int,I: fun_nat_int,J: fun_int_nat,T3: set_nat,G: fun_int_Code_natural,Ha: fun_nat_Code_natural] :
      ( finite_finite_int(S5)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ! [A2: int] :
              ( member_int(A2,minus_minus_set_int(S2,S5))
             => ( aa_nat_int(I,aa_int_nat(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(S2,S5))
               => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(J,A2)),minus_minus_set_nat(T3,T2)) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(T3,T2))
                 => ( aa_int_nat(J,aa_nat_int(I,B2)) = B2 ) )
             => ( ! [B2: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(T3,T2))
                   => member_int(aa_nat_int(I,B2),minus_minus_set_int(S2,S5)) )
               => ( ! [A2: int] :
                      ( member_int(A2,S5)
                     => ( aa_int_Code_natural(G,A2) = one_one_Code_natural ) )
                 => ( ! [B2: nat] :
                        ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),T2)
                       => ( aa_nat_Code_natural(Ha,B2) = one_one_Code_natural ) )
                   => ( ! [A2: int] :
                          ( member_int(A2,S2)
                         => ( aa_nat_Code_natural(Ha,aa_int_nat(J,A2)) = aa_int_Code_natural(G,A2) ) )
                     => ( groups2650699495083094267atural(G,S2) = groups2279045934846249631atural(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6617_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_nat,T2: set_int,S2: set_nat,I: fun_int_nat,J: fun_nat_int,T3: set_int,G: fun_nat_Code_integer,Ha: fun_int_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(S2,S5))
             => ( aa_int_nat(I,aa_nat_int(J,A2)) = A2 ) )
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(S2,S5))
               => member_int(aa_nat_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_nat_int(J,aa_int_nat(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(I,B2)),minus_minus_set_nat(S2,S5)) )
               => ( ! [A2: nat] :
                      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),S5)
                     => ( aa_nat_Code_integer(G,A2) = one_one_Code_integer ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_Code_integer(Ha,B2) = one_one_Code_integer ) )
                   => ( ! [A2: nat] :
                          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),S2)
                         => ( aa_int_Code_integer(Ha,aa_nat_int(J,A2)) = aa_nat_Code_integer(G,A2) ) )
                     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S2) = groups3827104343326376752nteger(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6618_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_nat,T2: set_int,S2: set_nat,I: fun_int_nat,J: fun_nat_int,T3: set_int,G: fun_nat_rat,Ha: fun_int_rat] :
      ( aa_set_nat_o(finite_finite_nat,S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(S2,S5))
             => ( aa_int_nat(I,aa_nat_int(J,A2)) = A2 ) )
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(S2,S5))
               => member_int(aa_nat_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_nat_int(J,aa_int_nat(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(I,B2)),minus_minus_set_nat(S2,S5)) )
               => ( ! [A2: nat] :
                      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),S5)
                     => ( aa_nat_rat(G,A2) = one_one_rat ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_rat(Ha,B2) = one_one_rat ) )
                   => ( ! [A2: nat] :
                          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),S2)
                         => ( aa_int_rat(Ha,aa_nat_int(J,A2)) = aa_nat_rat(G,A2) ) )
                     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),S2) = groups1072433553688619179nt_rat(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6619_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_nat,T2: set_int,S2: set_nat,I: fun_int_nat,J: fun_nat_int,T3: set_int,G: fun_nat_Code_natural,Ha: fun_int_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,S5)
     => ( finite_finite_int(T2)
       => ( ! [A2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(S2,S5))
             => ( aa_int_nat(I,aa_nat_int(J,A2)) = A2 ) )
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(S2,S5))
               => member_int(aa_nat_int(J,A2),minus_minus_set_int(T3,T2)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T3,T2))
                 => ( aa_nat_int(J,aa_int_nat(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T3,T2))
                   => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_int_nat(I,B2)),minus_minus_set_nat(S2,S5)) )
               => ( ! [A2: nat] :
                      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),S5)
                     => ( aa_nat_Code_natural(G,A2) = one_one_Code_natural ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T2)
                       => ( aa_int_Code_natural(Ha,B2) = one_one_Code_natural ) )
                   => ( ! [A2: nat] :
                          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),S2)
                         => ( aa_int_Code_natural(Ha,aa_nat_int(J,A2)) = aa_nat_Code_natural(G,A2) ) )
                     => ( groups2279045934846249631atural(G,S2) = groups2650699495083094267atural(Ha,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_6620_gcd__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,Xa),Xaa)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Code_integer_int(code_int_of_integer,Xa)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% gcd_integer.rep_eq
tff(fact_6621_gcd__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Xb)) ).

% gcd_integer.abs_eq
tff(fact_6622_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))
     => aa_set_nat_o(finite_finite_nat,N5) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_6623_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))
     => aa_set_nat_o(finite_finite_nat,N5) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_6624_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_6625_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_nat,Y3: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_ss(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_ss(I5),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_st(I5),Xa),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_6626_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_nat,Y3: fun_int_nat] :
      ( finite_finite_int(collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_su(I5),Xa)))
     => ( finite_finite_int(collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_su(I5),Y3)))
       => finite_finite_int(collect_int(aa_fun5425778157628386638_int_o(aa_fun8703205213358249326_int_o(aTP_Lamm_sv(I5),Xa),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_6627_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_int,Y3: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2180201864171524266_nat_o(aTP_Lamm_sw(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2180201864171524266_nat_o(aTP_Lamm_sw(I5),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2180201864171524266_nat_o(aa_fun8566290670789036234_nat_o(aTP_Lamm_sx(I5),Xa),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_6628_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_int,Y3: fun_int_int] :
      ( finite_finite_int(collect_int(aa_fun8155382727227203058_int_o(aTP_Lamm_sy(I5),Xa)))
     => ( finite_finite_int(collect_int(aa_fun8155382727227203058_int_o(aTP_Lamm_sy(I5),Y3)))
       => finite_finite_int(collect_int(aa_fun8155382727227203058_int_o(aa_fun1697523107123627702_int_o(aTP_Lamm_sz(I5),Xa),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_6629_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_Code_integer,Y3: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_ta(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_ta(I5),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aa_fun2745573882733218024_nat_o(aTP_Lamm_tb(I5),Xa),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_6630_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_Code_integer,Y3: fun_int_Code_integer] :
      ( finite_finite_int(collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_tc(I5),Xa)))
     => ( finite_finite_int(collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_tc(I5),Y3)))
       => finite_finite_int(collect_int(aa_fun3336537925265328129_int_o(aa_fun8818272713621706452_int_o(aTP_Lamm_td(I5),Xa),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_6631_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_rat,Y3: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_te(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_te(I5),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_tf(I5),Xa),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_6632_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_rat,Y3: fun_int_rat] :
      ( finite_finite_int(collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_tg(I5),Xa)))
     => ( finite_finite_int(collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_tg(I5),Y3)))
       => finite_finite_int(collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_th(I5),Xa),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_6633_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_Code_natural,Y3: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun440078198722963374_nat_o(aTP_Lamm_ti(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun440078198722963374_nat_o(aTP_Lamm_ti(I5),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun440078198722963374_nat_o(aa_fun7611442044815192786_nat_o(aTP_Lamm_tj(I5),Xa),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_6634_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_Code_natural,Y3: fun_int_Code_natural] :
      ( finite_finite_int(collect_int(aa_fun296549085072024822_int_o(aTP_Lamm_tk(I5),Xa)))
     => ( finite_finite_int(collect_int(aa_fun296549085072024822_int_o(aTP_Lamm_tk(I5),Y3)))
       => finite_finite_int(collect_int(aa_fun296549085072024822_int_o(aa_fun4460768838848905406_int_o(aTP_Lamm_tl(I5),Xa),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_6635_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_Code_natural,Y3: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun440078198722963374_nat_o(aTP_Lamm_tm(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun440078198722963374_nat_o(aTP_Lamm_tm(I5),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun440078198722963374_nat_o(aa_fun7611442044815192786_nat_o(aTP_Lamm_tn(I5),Xa),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_6636_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_Code_natural,Y3: fun_int_Code_natural] :
      ( finite_finite_int(collect_int(aa_fun296549085072024822_int_o(aTP_Lamm_to(I5),Xa)))
     => ( finite_finite_int(collect_int(aa_fun296549085072024822_int_o(aTP_Lamm_to(I5),Y3)))
       => finite_finite_int(collect_int(aa_fun296549085072024822_int_o(aa_fun4460768838848905406_int_o(aTP_Lamm_tp(I5),Xa),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_6637_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_nat,Y3: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_tq(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_tq(I5),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_tr(I5),Xa),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_6638_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_nat,Y3: fun_int_nat] :
      ( finite_finite_int(collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_ts(I5),Xa)))
     => ( finite_finite_int(collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_ts(I5),Y3)))
       => finite_finite_int(collect_int(aa_fun5425778157628386638_int_o(aa_fun8703205213358249326_int_o(aTP_Lamm_tt(I5),Xa),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_6639_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_int,Y3: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2180201864171524266_nat_o(aTP_Lamm_tu(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2180201864171524266_nat_o(aTP_Lamm_tu(I5),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2180201864171524266_nat_o(aa_fun8566290670789036234_nat_o(aTP_Lamm_tv(I5),Xa),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_6640_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_int,Y3: fun_int_int] :
      ( finite_finite_int(collect_int(aa_fun8155382727227203058_int_o(aTP_Lamm_tw(I5),Xa)))
     => ( finite_finite_int(collect_int(aa_fun8155382727227203058_int_o(aTP_Lamm_tw(I5),Y3)))
       => finite_finite_int(collect_int(aa_fun8155382727227203058_int_o(aa_fun1697523107123627702_int_o(aTP_Lamm_tx(I5),Xa),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_6641_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_Code_integer,Y3: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_ty(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_ty(I5),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aa_fun2745573882733218024_nat_o(aTP_Lamm_tz(I5),Xa),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_6642_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_Code_integer,Y3: fun_int_Code_integer] :
      ( finite_finite_int(collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_ua(I5),Xa)))
     => ( finite_finite_int(collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_ua(I5),Y3)))
       => finite_finite_int(collect_int(aa_fun3336537925265328129_int_o(aa_fun8818272713621706452_int_o(aTP_Lamm_ub(I5),Xa),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_6643_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_rat,Y3: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_uc(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_uc(I5),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_ud(I5),Xa),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_6644_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_rat,Y3: fun_int_rat] :
      ( finite_finite_int(collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ue(I5),Xa)))
     => ( finite_finite_int(collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ue(I5),Y3)))
       => finite_finite_int(collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_uf(I5),Xa),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_6645_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_nat,Pa: fun_int_o] :
      ( finite_finite_int(A3)
     => ( groups4541462559716669496nt_nat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(A3),Pa))) = groups4541462559716669496nt_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_ug(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6646_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_int,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(A3),Pa))) = aa_set_nat_int(groups3539618377306564664at_int(aa_fun27009930614080094at_int(aTP_Lamm_uh(G),Pa)),A3) ) ) ).

% sum.inter_filter
tff(fact_6647_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_Code_integer,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(A3),Pa))) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun3969837373441324047nteger(aTP_Lamm_ui(G),Pa)),A3) ) ) ).

% sum.inter_filter
tff(fact_6648_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_Code_integer,Pa: fun_int_o] :
      ( finite_finite_int(A3)
     => ( groups7873554091576472773nteger(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(A3),Pa))) = groups7873554091576472773nteger(aa_fun6019004165459918095nteger(aTP_Lamm_uj(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6649_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_rat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(A3),Pa))) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_uk(G),Pa)),A3) ) ) ).

% sum.inter_filter
tff(fact_6650_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_rat,Pa: fun_int_o] :
      ( finite_finite_int(A3)
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(A3),Pa))) = groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_ul(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6651_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_Code_natural,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( groups6325495683096345652atural(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(A3),Pa))) = groups6325495683096345652atural(aa_fun8428287193952513114atural(aTP_Lamm_um(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6652_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_Code_natural,Pa: fun_int_o] :
      ( finite_finite_int(A3)
     => ( groups6697149243333190288atural(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(A3),Pa))) = groups6697149243333190288atural(aa_fun1254081949116331354atural(aTP_Lamm_un(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6653_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(A3),Pa))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_uo(G),Pa)),A3) ) ) ).

% sum.inter_filter
tff(fact_6654_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_int,Pa: fun_int_o] :
      ( finite_finite_int(A3)
     => ( groups4538972089207619220nt_int(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(A3),Pa))) = groups4538972089207619220nt_int(aa_fun8822586851003501406nt_int(aTP_Lamm_up(G),Pa),A3) ) ) ).

% sum.inter_filter
tff(fact_6655_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_nat] :
      ( finite_finite_int(A3)
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_uq(G)))) = groups4541462559716669496nt_nat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6656_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_Code_integer] :
      ( finite_finite_int(A3)
     => ( groups7873554091576472773nteger(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_ur(G)))) = groups7873554091576472773nteger(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6657_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( finite_finite_int(A3)
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_us(G)))) = groups3906332499630173760nt_rat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6658_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_Code_natural] :
      ( finite_finite_int(A3)
     => ( groups6697149243333190288atural(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_ut(G)))) = groups6697149243333190288atural(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6659_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),minus_minus_set_nat(A3,collect_nat(aTP_Lamm_uu(G)))) = aa_set_nat_int(groups3539618377306564664at_int(G),A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6660_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),minus_minus_set_nat(A3,collect_nat(aTP_Lamm_uv(G)))) = aa_set871585252387937313nteger(groups7501900531339628137nteger(G),A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6661_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),minus_minus_set_nat(A3,collect_nat(aTP_Lamm_uw(G)))) = aa_set_nat_rat(groups2906978787729119204at_rat(G),A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6662_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( groups6325495683096345652atural(G,minus_minus_set_nat(A3,collect_nat(aTP_Lamm_ux(G)))) = groups6325495683096345652atural(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6663_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),minus_minus_set_nat(A3,collect_nat(aTP_Lamm_uy(G)))) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6664_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( finite_finite_int(A3)
     => ( groups4538972089207619220nt_int(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_uz(G)))) = groups4538972089207619220nt_int(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6665_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_nat] :
      ( finite_finite_int(A3)
     => ( groups1707563613775114915nt_nat(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_va(G)))) = groups1707563613775114915nt_nat(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6666_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_Code_integer] :
      ( finite_finite_int(A3)
     => ( groups3827104343326376752nteger(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_vb(G)))) = groups3827104343326376752nteger(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6667_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( finite_finite_int(A3)
     => ( groups1072433553688619179nt_rat(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_vc(G)))) = groups1072433553688619179nt_rat(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6668_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_Code_natural] :
      ( finite_finite_int(A3)
     => ( groups2650699495083094267atural(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_vd(G)))) = groups2650699495083094267atural(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6669_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),minus_minus_set_nat(A3,collect_nat(aTP_Lamm_ve(G)))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6670_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),minus_minus_set_nat(A3,collect_nat(aTP_Lamm_vf(G)))) = aa_set_nat_rat(groups73079841787564623at_rat(G),A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6671_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( groups2279045934846249631atural(G,minus_minus_set_nat(A3,collect_nat(aTP_Lamm_vg(G)))) = groups2279045934846249631atural(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6672_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),minus_minus_set_nat(A3,collect_nat(aTP_Lamm_vh(G)))) = aa_set_nat_nat(groups708209901874060359at_nat(G),A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6673_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),minus_minus_set_nat(A3,collect_nat(aTP_Lamm_vi(G)))) = aa_set_nat_int(groups705719431365010083at_int(G),A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6674_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( finite_finite_int(A3)
     => ( groups1705073143266064639nt_int(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_vj(G)))) = groups1705073143266064639nt_int(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_6675_prod_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_nat,Pa: fun_int_o] :
      ( finite_finite_int(A3)
     => ( groups1707563613775114915nt_nat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(A3),Pa))) = groups1707563613775114915nt_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_vk(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6676_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_Code_integer,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(A3),Pa))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun3969837373441324047nteger(aTP_Lamm_vl(G),Pa)),A3) ) ) ).

% prod.inter_filter
tff(fact_6677_prod_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_Code_integer,Pa: fun_int_o] :
      ( finite_finite_int(A3)
     => ( groups3827104343326376752nteger(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(A3),Pa))) = groups3827104343326376752nteger(aa_fun6019004165459918095nteger(aTP_Lamm_vm(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6678_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_rat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(A3),Pa))) = aa_set_nat_rat(groups73079841787564623at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_vn(G),Pa)),A3) ) ) ).

% prod.inter_filter
tff(fact_6679_prod_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_rat,Pa: fun_int_o] :
      ( finite_finite_int(A3)
     => ( groups1072433553688619179nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(A3),Pa))) = groups1072433553688619179nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_vo(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6680_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_Code_natural,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( groups2279045934846249631atural(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(A3),Pa))) = groups2279045934846249631atural(aa_fun8428287193952513114atural(aTP_Lamm_vp(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6681_prod_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_Code_natural,Pa: fun_int_o] :
      ( finite_finite_int(A3)
     => ( groups2650699495083094267atural(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(A3),Pa))) = groups2650699495083094267atural(aa_fun1254081949116331354atural(aTP_Lamm_vq(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6682_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(A3),Pa))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_vr(G),Pa)),A3) ) ) ).

% prod.inter_filter
tff(fact_6683_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_int,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(A3),Pa))) = aa_set_nat_int(groups705719431365010083at_int(aa_fun27009930614080094at_int(aTP_Lamm_vs(G),Pa)),A3) ) ) ).

% prod.inter_filter
tff(fact_6684_prod_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_int,Pa: fun_int_o] :
      ( finite_finite_int(A3)
     => ( groups1705073143266064639nt_int(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(A3),Pa))) = groups1705073143266064639nt_int(aa_fun8822586851003501406nt_int(aTP_Lamm_vt(G),Pa),A3) ) ) ).

% prod.inter_filter
tff(fact_6685_finite__int__segment,axiom,
    ! [A: rat,B: rat] : finite_finite_rat(collect_rat(aa_rat_fun_rat_o(aTP_Lamm_vu(A),B))) ).

% finite_int_segment
tff(fact_6686_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X4: nat,Y4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_6687_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X4: nat,Y4: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_6688_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X4)) )
       => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3) = zero_z3403309356797280102nteger )
        <=> ! [X3: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
             => ( aa_nat_Code_integer(F,X3) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6689_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( finite_finite_int(A3)
     => ( ! [X4: int] :
            ( member_int(X4,A3)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X4)) )
       => ( ( groups7873554091576472773nteger(F,A3) = zero_z3403309356797280102nteger )
        <=> ! [X3: int] :
              ( member_int(X3,A3)
             => ( aa_int_Code_integer(F,X3) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6690_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,X4)) )
       => ( ( groups6325495683096345652atural(F,A3) = zero_z2226904508553997617atural )
        <=> ! [X3: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
             => ( aa_nat_Code_natural(F,X3) = zero_z2226904508553997617atural ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6691_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_Code_natural] :
      ( finite_finite_int(A3)
     => ( ! [X4: int] :
            ( member_int(X4,A3)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,X4)) )
       => ( ( groups6697149243333190288atural(F,A3) = zero_z2226904508553997617atural )
        <=> ! [X3: int] :
              ( member_int(X3,A3)
             => ( aa_int_Code_natural(F,X3) = zero_z2226904508553997617atural ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6692_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4)) )
       => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(F),A3) = zero_zero_rat )
        <=> ! [X3: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
             => ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6693_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( finite_finite_int(A3)
     => ( ! [X4: int] :
            ( member_int(X4,A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)) )
       => ( ( groups3906332499630173760nt_rat(F,A3) = zero_zero_rat )
        <=> ! [X3: int] :
              ( member_int(X3,A3)
             => ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6694_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( finite_finite_int(A3)
     => ( ! [X4: int] :
            ( member_int(X4,A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)) )
       => ( ( groups4541462559716669496nt_nat(F,A3) = zero_zero_nat )
        <=> ! [X3: int] :
              ( member_int(X3,A3)
             => ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6695_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4)) )
       => ( ( aa_set_nat_int(groups3539618377306564664at_int(F),A3) = zero_zero_int )
        <=> ! [X3: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
             => ( aa_nat_int(F,X3) = zero_zero_int ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6696_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4)) )
       => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = zero_zero_nat )
        <=> ! [X3: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
             => ( aa_nat_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6697_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( finite_finite_int(A3)
     => ( ! [X4: int] :
            ( member_int(X4,A3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4)) )
       => ( ( groups4538972089207619220nt_int(F,A3) = zero_zero_int )
        <=> ! [X3: int] :
              ( member_int(X3,A3)
             => ( aa_int_int(F,X3) = zero_zero_int ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6698_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: fun_nat_Code_integer,I: fun_nat_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),T)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S)
               => ? [Xa3: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),T)
                    & ( aa_nat_nat(I,Xa3) = X4 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X4)),aa_nat_Code_integer(G,Xa3)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(F),S)),aa_set871585252387937313nteger(groups7501900531339628137nteger(G),T)) ) ) ) ) ).

% sum_le_included
tff(fact_6699_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: fun_int_Code_integer,I: fun_int_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( finite_finite_int(T)
       => ( ! [X4: int] :
              ( member_int(X4,T)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S)
               => ? [Xa3: int] :
                    ( member_int(Xa3,T)
                    & ( aa_int_nat(I,Xa3) = X4 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X4)),aa_int_Code_integer(G,Xa3)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(F),S)),groups7873554091576472773nteger(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_6700_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: fun_nat_Code_integer,I: fun_nat_int,F: fun_int_Code_integer] :
      ( finite_finite_int(S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),T)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(G,X4)) )
         => ( ! [X4: int] :
                ( member_int(X4,S)
               => ? [Xa3: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),T)
                    & ( aa_nat_int(I,Xa3) = X4 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X4)),aa_nat_Code_integer(G,Xa3)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(F,S)),aa_set871585252387937313nteger(groups7501900531339628137nteger(G),T)) ) ) ) ) ).

% sum_le_included
tff(fact_6701_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: fun_int_Code_integer,I: fun_int_int,F: fun_int_Code_integer] :
      ( finite_finite_int(S)
     => ( finite_finite_int(T)
       => ( ! [X4: int] :
              ( member_int(X4,T)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(G,X4)) )
         => ( ! [X4: int] :
                ( member_int(X4,S)
               => ? [Xa3: int] :
                    ( member_int(Xa3,T)
                    & ( aa_int_int(I,Xa3) = X4 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X4)),aa_int_Code_integer(G,Xa3)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(F,S)),groups7873554091576472773nteger(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_6702_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: fun_nat_Code_natural,I: fun_nat_nat,F: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),T)
             => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S)
               => ? [Xa3: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),T)
                    & ( aa_nat_nat(I,Xa3) = X4 )
                    & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(F,X4)),aa_nat_Code_natural(G,Xa3)) ) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups6325495683096345652atural(F,S)),groups6325495683096345652atural(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_6703_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: fun_int_Code_natural,I: fun_int_nat,F: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( finite_finite_int(T)
       => ( ! [X4: int] :
              ( member_int(X4,T)
             => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S)
               => ? [Xa3: int] :
                    ( member_int(Xa3,T)
                    & ( aa_int_nat(I,Xa3) = X4 )
                    & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(F,X4)),aa_int_Code_natural(G,Xa3)) ) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups6325495683096345652atural(F,S)),groups6697149243333190288atural(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_6704_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: fun_nat_Code_natural,I: fun_nat_int,F: fun_int_Code_natural] :
      ( finite_finite_int(S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),T)
             => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(G,X4)) )
         => ( ! [X4: int] :
                ( member_int(X4,S)
               => ? [Xa3: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),T)
                    & ( aa_nat_int(I,Xa3) = X4 )
                    & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_int_Code_natural(F,X4)),aa_nat_Code_natural(G,Xa3)) ) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups6697149243333190288atural(F,S)),groups6325495683096345652atural(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_6705_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: fun_int_Code_natural,I: fun_int_int,F: fun_int_Code_natural] :
      ( finite_finite_int(S)
     => ( finite_finite_int(T)
       => ( ! [X4: int] :
              ( member_int(X4,T)
             => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(G,X4)) )
         => ( ! [X4: int] :
                ( member_int(X4,S)
               => ? [Xa3: int] :
                    ( member_int(Xa3,T)
                    & ( aa_int_int(I,Xa3) = X4 )
                    & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_int_Code_natural(F,X4)),aa_int_Code_natural(G,Xa3)) ) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups6697149243333190288atural(F,S)),groups6697149243333190288atural(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_6706_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),T)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S)
               => ? [Xa3: nat] :
                    ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),T)
                    & ( aa_nat_nat(I,Xa3) = X4 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,Xa3)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),S)),aa_set_nat_rat(groups2906978787729119204at_rat(G),T)) ) ) ) ) ).

% sum_le_included
tff(fact_6707_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: fun_int_rat,I: fun_int_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( finite_finite_int(T)
       => ( ! [X4: int] :
              ( member_int(X4,T)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S)
               => ? [Xa3: int] :
                    ( member_int(Xa3,T)
                    & ( aa_int_nat(I,Xa3) = X4 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_int_rat(G,Xa3)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),S)),groups3906332499630173760nt_rat(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_6708_sum_Orelated,axiom,
    ! [R4: fun_nat_fun_nat_o,S2: set_int,Ha: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R4,zero_zero_nat),zero_zero_nat)
     => ( ! [X12: nat,Y12: nat,X22: nat,Y22: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R4,X12),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R4,Y12),Y22) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X12),Y12)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y22)) )
       => ( finite_finite_int(S2)
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => aa_nat_o(aa_nat_fun_nat_o(R4,aa_int_nat(Ha,X4)),aa_int_nat(G,X4)) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,groups4541462559716669496nt_nat(Ha,S2)),groups4541462559716669496nt_nat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_6709_sum_Orelated,axiom,
    ! [R4: fun_int_fun_int_o,S2: set_nat,Ha: fun_nat_int,G: fun_nat_int] :
      ( aa_int_o(aa_int_fun_int_o(R4,zero_zero_int),zero_zero_int)
     => ( ! [X12: int,Y12: int,X22: int,Y22: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R4,X12),X22)
              & aa_int_o(aa_int_fun_int_o(R4,Y12),Y22) )
           => aa_int_o(aa_int_fun_int_o(R4,aa_int_int(aa_int_fun_int_int(plus_plus_int,X12),Y12)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => aa_int_o(aa_int_fun_int_o(R4,aa_nat_int(Ha,X4)),aa_nat_int(G,X4)) )
           => aa_int_o(aa_int_fun_int_o(R4,aa_set_nat_int(groups3539618377306564664at_int(Ha),S2)),aa_set_nat_int(groups3539618377306564664at_int(G),S2)) ) ) ) ) ).

% sum.related
tff(fact_6710_sum_Orelated,axiom,
    ! [R4: fun_Co2081214437782444048eger_o,S2: set_nat,Ha: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)
     => ( ! [X12: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,X12),X22)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,Y12),Y22) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,aa_nat_Code_integer(Ha,X4)),aa_nat_Code_integer(G,X4)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),S2)),aa_set871585252387937313nteger(groups7501900531339628137nteger(G),S2)) ) ) ) ) ).

% sum.related
tff(fact_6711_sum_Orelated,axiom,
    ! [R4: fun_Co2081214437782444048eger_o,S2: set_int,Ha: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)
     => ( ! [X12: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,X12),X22)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,Y12),Y22) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X22),Y22)) )
       => ( finite_finite_int(S2)
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,aa_int_Code_integer(Ha,X4)),aa_int_Code_integer(G,X4)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,groups7873554091576472773nteger(Ha,S2)),groups7873554091576472773nteger(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_6712_sum_Orelated,axiom,
    ! [R4: fun_rat_fun_rat_o,S2: set_nat,Ha: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R4,zero_zero_rat),zero_zero_rat)
     => ( ! [X12: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R4,X12),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R4,Y12),Y22) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X12),Y12)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R4,aa_nat_rat(Ha,X4)),aa_nat_rat(G,X4)) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,aa_set_nat_rat(groups2906978787729119204at_rat(Ha),S2)),aa_set_nat_rat(groups2906978787729119204at_rat(G),S2)) ) ) ) ) ).

% sum.related
tff(fact_6713_sum_Orelated,axiom,
    ! [R4: fun_rat_fun_rat_o,S2: set_int,Ha: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R4,zero_zero_rat),zero_zero_rat)
     => ( ! [X12: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R4,X12),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R4,Y12),Y22) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X12),Y12)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X22),Y22)) )
       => ( finite_finite_int(S2)
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => aa_rat_o(aa_rat_fun_rat_o(R4,aa_int_rat(Ha,X4)),aa_int_rat(G,X4)) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,groups3906332499630173760nt_rat(Ha,S2)),groups3906332499630173760nt_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_6714_sum_Orelated,axiom,
    ! [R4: fun_Co7302518231560758906ural_o,S2: set_nat,Ha: fun_nat_Code_natural,G: fun_nat_Code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,zero_z2226904508553997617atural),zero_z2226904508553997617atural)
     => ( ! [X12: code_natural,Y12: code_natural,X22: code_natural,Y22: code_natural] :
            ( ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,X12),X22)
              & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,Y12),Y22) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X12),Y12)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,aa_nat_Code_natural(Ha,X4)),aa_nat_Code_natural(G,X4)) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,groups6325495683096345652atural(Ha,S2)),groups6325495683096345652atural(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_6715_sum_Orelated,axiom,
    ! [R4: fun_Co7302518231560758906ural_o,S2: set_int,Ha: fun_int_Code_natural,G: fun_int_Code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,zero_z2226904508553997617atural),zero_z2226904508553997617atural)
     => ( ! [X12: code_natural,Y12: code_natural,X22: code_natural,Y22: code_natural] :
            ( ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,X12),X22)
              & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,Y12),Y22) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X12),Y12)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X22),Y22)) )
       => ( finite_finite_int(S2)
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,aa_int_Code_natural(Ha,X4)),aa_int_Code_natural(G,X4)) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,groups6697149243333190288atural(Ha,S2)),groups6697149243333190288atural(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_6716_sum_Orelated,axiom,
    ! [R4: fun_nat_fun_nat_o,S2: set_nat,Ha: fun_nat_nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R4,zero_zero_nat),zero_zero_nat)
     => ( ! [X12: nat,Y12: nat,X22: nat,Y22: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R4,X12),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R4,Y12),Y22) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X12),Y12)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R4,aa_nat_nat(Ha,X4)),aa_nat_nat(G,X4)) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,aa_set_nat_nat(groups3542108847815614940at_nat(Ha),S2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),S2)) ) ) ) ) ).

% sum.related
tff(fact_6717_sum_Orelated,axiom,
    ! [R4: fun_int_fun_int_o,S2: set_int,Ha: fun_int_int,G: fun_int_int] :
      ( aa_int_o(aa_int_fun_int_o(R4,zero_zero_int),zero_zero_int)
     => ( ! [X12: int,Y12: int,X22: int,Y22: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R4,X12),X22)
              & aa_int_o(aa_int_fun_int_o(R4,Y12),Y22) )
           => aa_int_o(aa_int_fun_int_o(R4,aa_int_int(aa_int_fun_int_int(plus_plus_int,X12),Y12)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X22),Y22)) )
       => ( finite_finite_int(S2)
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => aa_int_o(aa_int_fun_int_o(R4,aa_int_int(Ha,X4)),aa_int_int(G,X4)) )
           => aa_int_o(aa_int_fun_int_o(R4,groups4538972089207619220nt_int(Ha,S2)),groups4538972089207619220nt_int(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_6718_sum_Osame__carrier,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_nat,Ha: fun_int_nat] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_nat(G,A2) = zero_zero_nat ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_nat(Ha,B2) = zero_zero_nat ) )
             => ( ( groups4541462559716669496nt_nat(G,A3) = groups4541462559716669496nt_nat(Ha,B3) )
              <=> ( groups4541462559716669496nt_nat(G,C5) = groups4541462559716669496nt_nat(Ha,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6719_sum_Osame__carrier,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_Code_integer,Ha: fun_int_Code_integer] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_Code_integer(G,A2) = zero_z3403309356797280102nteger ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_Code_integer(Ha,B2) = zero_z3403309356797280102nteger ) )
             => ( ( groups7873554091576472773nteger(G,A3) = groups7873554091576472773nteger(Ha,B3) )
              <=> ( groups7873554091576472773nteger(G,C5) = groups7873554091576472773nteger(Ha,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6720_sum_Osame__carrier,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_rat,Ha: fun_int_rat] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_rat(Ha,B2) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,A3) = groups3906332499630173760nt_rat(Ha,B3) )
              <=> ( groups3906332499630173760nt_rat(G,C5) = groups3906332499630173760nt_rat(Ha,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6721_sum_Osame__carrier,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_Code_natural,Ha: fun_int_Code_natural] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_Code_natural(G,A2) = zero_z2226904508553997617atural ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_Code_natural(Ha,B2) = zero_z2226904508553997617atural ) )
             => ( ( groups6697149243333190288atural(G,A3) = groups6697149243333190288atural(Ha,B3) )
              <=> ( groups6697149243333190288atural(G,C5) = groups6697149243333190288atural(Ha,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6722_sum_Osame__carrier,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_int,Ha: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_int(G,A2) = zero_zero_int ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_int(Ha,B2) = zero_zero_int ) )
             => ( ( aa_set_nat_int(groups3539618377306564664at_int(G),A3) = aa_set_nat_int(groups3539618377306564664at_int(Ha),B3) )
              <=> ( aa_set_nat_int(groups3539618377306564664at_int(G),C5) = aa_set_nat_int(groups3539618377306564664at_int(Ha),C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6723_sum_Osame__carrier,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_Code_integer(G,A2) = zero_z3403309356797280102nteger ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_Code_integer(Ha,B2) = zero_z3403309356797280102nteger ) )
             => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),A3) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),B3) )
              <=> ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),C5) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6724_sum_Osame__carrier,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_rat(Ha,B2) = zero_zero_rat ) )
             => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(G),A3) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),B3) )
              <=> ( aa_set_nat_rat(groups2906978787729119204at_rat(G),C5) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6725_sum_Osame__carrier,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_Code_natural,Ha: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_Code_natural(G,A2) = zero_z2226904508553997617atural ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_Code_natural(Ha,B2) = zero_z2226904508553997617atural ) )
             => ( ( groups6325495683096345652atural(G,A3) = groups6325495683096345652atural(Ha,B3) )
              <=> ( groups6325495683096345652atural(G,C5) = groups6325495683096345652atural(Ha,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6726_sum_Osame__carrier,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_nat(G,A2) = zero_zero_nat ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_nat(Ha,B2) = zero_zero_nat ) )
             => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(Ha),B3) )
              <=> ( aa_set_nat_nat(groups3542108847815614940at_nat(G),C5) = aa_set_nat_nat(groups3542108847815614940at_nat(Ha),C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6727_sum_Osame__carrier,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_int,Ha: fun_int_int] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_int(G,A2) = zero_zero_int ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_int(Ha,B2) = zero_zero_int ) )
             => ( ( groups4538972089207619220nt_int(G,A3) = groups4538972089207619220nt_int(Ha,B3) )
              <=> ( groups4538972089207619220nt_int(G,C5) = groups4538972089207619220nt_int(Ha,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6728_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_nat,Ha: fun_int_nat] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_nat(G,A2) = zero_zero_nat ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_nat(Ha,B2) = zero_zero_nat ) )
             => ( ( groups4541462559716669496nt_nat(G,C5) = groups4541462559716669496nt_nat(Ha,C5) )
               => ( groups4541462559716669496nt_nat(G,A3) = groups4541462559716669496nt_nat(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6729_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_Code_integer,Ha: fun_int_Code_integer] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_Code_integer(G,A2) = zero_z3403309356797280102nteger ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_Code_integer(Ha,B2) = zero_z3403309356797280102nteger ) )
             => ( ( groups7873554091576472773nteger(G,C5) = groups7873554091576472773nteger(Ha,C5) )
               => ( groups7873554091576472773nteger(G,A3) = groups7873554091576472773nteger(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6730_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_rat,Ha: fun_int_rat] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_rat(Ha,B2) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,C5) = groups3906332499630173760nt_rat(Ha,C5) )
               => ( groups3906332499630173760nt_rat(G,A3) = groups3906332499630173760nt_rat(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6731_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_Code_natural,Ha: fun_int_Code_natural] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_Code_natural(G,A2) = zero_z2226904508553997617atural ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_Code_natural(Ha,B2) = zero_z2226904508553997617atural ) )
             => ( ( groups6697149243333190288atural(G,C5) = groups6697149243333190288atural(Ha,C5) )
               => ( groups6697149243333190288atural(G,A3) = groups6697149243333190288atural(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6732_sum_Osame__carrierI,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_int,Ha: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_int(G,A2) = zero_zero_int ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_int(Ha,B2) = zero_zero_int ) )
             => ( ( aa_set_nat_int(groups3539618377306564664at_int(G),C5) = aa_set_nat_int(groups3539618377306564664at_int(Ha),C5) )
               => ( aa_set_nat_int(groups3539618377306564664at_int(G),A3) = aa_set_nat_int(groups3539618377306564664at_int(Ha),B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6733_sum_Osame__carrierI,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_Code_integer(G,A2) = zero_z3403309356797280102nteger ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_Code_integer(Ha,B2) = zero_z3403309356797280102nteger ) )
             => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),C5) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),C5) )
               => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),A3) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6734_sum_Osame__carrierI,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_rat(Ha,B2) = zero_zero_rat ) )
             => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(G),C5) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),C5) )
               => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),A3) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6735_sum_Osame__carrierI,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_Code_natural,Ha: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_Code_natural(G,A2) = zero_z2226904508553997617atural ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_Code_natural(Ha,B2) = zero_z2226904508553997617atural ) )
             => ( ( groups6325495683096345652atural(G,C5) = groups6325495683096345652atural(Ha,C5) )
               => ( groups6325495683096345652atural(G,A3) = groups6325495683096345652atural(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6736_sum_Osame__carrierI,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_nat(G,A2) = zero_zero_nat ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_nat(Ha,B2) = zero_zero_nat ) )
             => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(G),C5) = aa_set_nat_nat(groups3542108847815614940at_nat(Ha),C5) )
               => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(Ha),B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6737_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_int,Ha: fun_int_int] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_int(G,A2) = zero_zero_int ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_int(Ha,B2) = zero_zero_int ) )
             => ( ( groups4538972089207619220nt_int(G,C5) = groups4538972089207619220nt_int(Ha,C5) )
               => ( groups4538972089207619220nt_int(G,A3) = groups4538972089207619220nt_int(Ha,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6738_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_nat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_nat(G,X4) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,S2) = groups4541462559716669496nt_nat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6739_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_Code_integer] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_integer(G,X4) = zero_z3403309356797280102nteger ) )
         => ( groups7873554091576472773nteger(G,S2) = groups7873554091576472773nteger(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6740_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_rat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6741_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_Code_natural] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_natural(G,X4) = zero_z2226904508553997617atural ) )
         => ( groups6697149243333190288atural(G,S2) = groups6697149243333190288atural(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6742_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(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_6743_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_integer(G,X4) = zero_z3403309356797280102nteger ) )
         => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),S2) = aa_set871585252387937313nteger(groups7501900531339628137nteger(G),T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6744_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
         => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),S2) = aa_set_nat_rat(groups2906978787729119204at_rat(G),T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6745_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_natural(G,X4) = zero_z2226904508553997617atural ) )
         => ( groups6325495683096345652atural(G,S2) = groups6325495683096345652atural(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6746_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),S2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6747_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_int] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_int(G,X4) = zero_zero_int ) )
         => ( groups4538972089207619220nt_int(G,S2) = groups4538972089207619220nt_int(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6748_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_nat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_nat(G,X4) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,T3) = groups4541462559716669496nt_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6749_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_Code_integer] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_integer(G,X4) = zero_z3403309356797280102nteger ) )
         => ( groups7873554091576472773nteger(G,T3) = groups7873554091576472773nteger(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6750_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_rat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T3) = groups3906332499630173760nt_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6751_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_Code_natural] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_natural(G,X4) = zero_z2226904508553997617atural ) )
         => ( groups6697149243333190288atural(G,T3) = groups6697149243333190288atural(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6752_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(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_6753_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_integer(G,X4) = zero_z3403309356797280102nteger ) )
         => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),T3) = aa_set871585252387937313nteger(groups7501900531339628137nteger(G),S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6754_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
         => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),T3) = aa_set_nat_rat(groups2906978787729119204at_rat(G),S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6755_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_natural(G,X4) = zero_z2226904508553997617atural ) )
         => ( groups6325495683096345652atural(G,T3) = groups6325495683096345652atural(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6756_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),T3) = aa_set_nat_nat(groups3542108847815614940at_nat(G),S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6757_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_int] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_int(G,X4) = zero_zero_int ) )
         => ( groups4538972089207619220nt_int(G,T3) = groups4538972089207619220nt_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6758_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S2: set_int,Ha: fun_int_nat,G: fun_int_nat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_nat(Ha,X4) = zero_zero_nat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_nat(G,X4) = aa_int_nat(Ha,X4) ) )
           => ( groups4541462559716669496nt_nat(G,S2) = groups4541462559716669496nt_nat(Ha,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6759_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S2: set_int,Ha: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_integer(Ha,X4) = zero_z3403309356797280102nteger ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_Code_integer(G,X4) = aa_int_Code_integer(Ha,X4) ) )
           => ( groups7873554091576472773nteger(G,S2) = groups7873554091576472773nteger(Ha,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6760_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S2: set_int,Ha: fun_int_rat,G: fun_int_rat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_rat(Ha,X4) = zero_zero_rat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_rat(G,X4) = aa_int_rat(Ha,X4) ) )
           => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(Ha,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6761_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S2: set_int,Ha: fun_int_Code_natural,G: fun_int_Code_natural] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_natural(Ha,X4) = zero_z2226904508553997617atural ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_Code_natural(G,X4) = aa_int_Code_natural(Ha,X4) ) )
           => ( groups6697149243333190288atural(G,S2) = groups6697149243333190288atural(Ha,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6762_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,Ha: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_int(Ha,X4) = zero_zero_int ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_int(G,X4) = aa_nat_int(Ha,X4) ) )
           => ( aa_set_nat_int(groups3539618377306564664at_int(G),S2) = aa_set_nat_int(groups3539618377306564664at_int(Ha),T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6763_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,Ha: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_integer(Ha,X4) = zero_z3403309356797280102nteger ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_Code_integer(G,X4) = aa_nat_Code_integer(Ha,X4) ) )
           => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),S2) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6764_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,Ha: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_rat(Ha,X4) = zero_zero_rat ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_rat(G,X4) = aa_nat_rat(Ha,X4) ) )
           => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),S2) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6765_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,Ha: fun_nat_Code_natural,G: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_natural(Ha,X4) = zero_z2226904508553997617atural ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_Code_natural(G,X4) = aa_nat_Code_natural(Ha,X4) ) )
           => ( groups6325495683096345652atural(G,S2) = groups6325495683096345652atural(Ha,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6766_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,Ha: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_nat(Ha,X4) = zero_zero_nat ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_nat(G,X4) = aa_nat_nat(Ha,X4) ) )
           => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),S2) = aa_set_nat_nat(groups3542108847815614940at_nat(Ha),T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6767_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S2: set_int,Ha: fun_int_int,G: fun_int_int] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_int(Ha,X4) = zero_zero_int ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_int(G,X4) = aa_int_int(Ha,X4) ) )
           => ( groups4538972089207619220nt_int(G,S2) = groups4538972089207619220nt_int(Ha,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6768_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_nat,Ha: fun_int_nat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_nat(G,X4) = aa_int_nat(Ha,X4) ) )
           => ( groups4541462559716669496nt_nat(G,T3) = groups4541462559716669496nt_nat(Ha,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6769_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_Code_integer,Ha: fun_int_Code_integer] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_integer(G,X4) = zero_z3403309356797280102nteger ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_Code_integer(G,X4) = aa_int_Code_integer(Ha,X4) ) )
           => ( groups7873554091576472773nteger(G,T3) = groups7873554091576472773nteger(Ha,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6770_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_rat,Ha: fun_int_rat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_rat(G,X4) = aa_int_rat(Ha,X4) ) )
           => ( groups3906332499630173760nt_rat(G,T3) = groups3906332499630173760nt_rat(Ha,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6771_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_Code_natural,Ha: fun_int_Code_natural] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_natural(G,X4) = zero_z2226904508553997617atural ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_Code_natural(G,X4) = aa_int_Code_natural(Ha,X4) ) )
           => ( groups6697149243333190288atural(G,T3) = groups6697149243333190288atural(Ha,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6772_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_int,Ha: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_int(G,X4) = zero_zero_int ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_int(G,X4) = aa_nat_int(Ha,X4) ) )
           => ( aa_set_nat_int(groups3539618377306564664at_int(G),T3) = aa_set_nat_int(groups3539618377306564664at_int(Ha),S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6773_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_integer(G,X4) = zero_z3403309356797280102nteger ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_Code_integer(G,X4) = aa_nat_Code_integer(Ha,X4) ) )
           => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(G),T3) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6774_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_rat(G,X4) = aa_nat_rat(Ha,X4) ) )
           => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),T3) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6775_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_Code_natural,Ha: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_natural(G,X4) = zero_z2226904508553997617atural ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_Code_natural(G,X4) = aa_nat_Code_natural(Ha,X4) ) )
           => ( groups6325495683096345652atural(G,T3) = groups6325495683096345652atural(Ha,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6776_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_nat(G,X4) = aa_nat_nat(Ha,X4) ) )
           => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),T3) = aa_set_nat_nat(groups3542108847815614940at_nat(Ha),S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6777_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_int,Ha: fun_int_int] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_int(G,X4) = zero_zero_int ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_int(G,X4) = aa_int_int(Ha,X4) ) )
           => ( groups4538972089207619220nt_int(G,T3) = groups4538972089207619220nt_int(Ha,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6778_prod_Orelated,axiom,
    ! [R4: fun_Co7302518231560758906ural_o,S2: set_nat,Ha: fun_nat_Code_natural,G: fun_nat_Code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,one_one_Code_natural),one_one_Code_natural)
     => ( ! [X12: code_natural,Y12: code_natural,X22: code_natural,Y22: code_natural] :
            ( ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,X12),X22)
              & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,Y12),Y22) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,aa_Cod4272041906116450446atural(times_2397367101498566445atural(X12),Y12)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,aa_nat_Code_natural(Ha,X4)),aa_nat_Code_natural(G,X4)) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,groups2279045934846249631atural(Ha,S2)),groups2279045934846249631atural(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_6779_prod_Orelated,axiom,
    ! [R4: fun_Co7302518231560758906ural_o,S2: set_int,Ha: fun_int_Code_natural,G: fun_int_Code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,one_one_Code_natural),one_one_Code_natural)
     => ( ! [X12: code_natural,Y12: code_natural,X22: code_natural,Y22: code_natural] :
            ( ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,X12),X22)
              & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,Y12),Y22) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,aa_Cod4272041906116450446atural(times_2397367101498566445atural(X12),Y12)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(X22),Y22)) )
       => ( finite_finite_int(S2)
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,aa_int_Code_natural(Ha,X4)),aa_int_Code_natural(G,X4)) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R4,groups2650699495083094267atural(Ha,S2)),groups2650699495083094267atural(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_6780_prod_Orelated,axiom,
    ! [R4: fun_nat_fun_nat_o,S2: set_int,Ha: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R4,one_one_nat),one_one_nat)
     => ( ! [X12: nat,Y12: nat,X22: nat,Y22: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R4,X12),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R4,Y12),Y22) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X12),Y12)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X22),Y22)) )
       => ( finite_finite_int(S2)
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => aa_nat_o(aa_nat_fun_nat_o(R4,aa_int_nat(Ha,X4)),aa_int_nat(G,X4)) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,groups1707563613775114915nt_nat(Ha,S2)),groups1707563613775114915nt_nat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_6781_prod_Orelated,axiom,
    ! [R4: fun_Co2081214437782444048eger_o,S2: set_nat,Ha: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,one_one_Code_integer),one_one_Code_integer)
     => ( ! [X12: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,X12),X22)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,Y12),Y22) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,aa_nat_Code_integer(Ha,X4)),aa_nat_Code_integer(G,X4)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),S2)),aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S2)) ) ) ) ) ).

% prod.related
tff(fact_6782_prod_Orelated,axiom,
    ! [R4: fun_Co2081214437782444048eger_o,S2: set_int,Ha: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,one_one_Code_integer),one_one_Code_integer)
     => ( ! [X12: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,X12),X22)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,Y12),Y22) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X22),Y22)) )
       => ( finite_finite_int(S2)
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,aa_int_Code_integer(Ha,X4)),aa_int_Code_integer(G,X4)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R4,groups3827104343326376752nteger(Ha,S2)),groups3827104343326376752nteger(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_6783_prod_Orelated,axiom,
    ! [R4: fun_rat_fun_rat_o,S2: set_nat,Ha: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R4,one_one_rat),one_one_rat)
     => ( ! [X12: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R4,X12),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R4,Y12),Y22) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X12),Y12)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R4,aa_nat_rat(Ha,X4)),aa_nat_rat(G,X4)) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,aa_set_nat_rat(groups73079841787564623at_rat(Ha),S2)),aa_set_nat_rat(groups73079841787564623at_rat(G),S2)) ) ) ) ) ).

% prod.related
tff(fact_6784_prod_Orelated,axiom,
    ! [R4: fun_rat_fun_rat_o,S2: set_int,Ha: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R4,one_one_rat),one_one_rat)
     => ( ! [X12: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R4,X12),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R4,Y12),Y22) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X12),Y12)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X22),Y22)) )
       => ( finite_finite_int(S2)
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => aa_rat_o(aa_rat_fun_rat_o(R4,aa_int_rat(Ha,X4)),aa_int_rat(G,X4)) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,groups1072433553688619179nt_rat(Ha,S2)),groups1072433553688619179nt_rat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_6785_prod_Orelated,axiom,
    ! [R4: fun_nat_fun_nat_o,S2: set_nat,Ha: fun_nat_nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R4,one_one_nat),one_one_nat)
     => ( ! [X12: nat,Y12: nat,X22: nat,Y22: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R4,X12),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R4,Y12),Y22) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X12),Y12)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R4,aa_nat_nat(Ha,X4)),aa_nat_nat(G,X4)) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,aa_set_nat_nat(groups708209901874060359at_nat(Ha),S2)),aa_set_nat_nat(groups708209901874060359at_nat(G),S2)) ) ) ) ) ).

% prod.related
tff(fact_6786_prod_Orelated,axiom,
    ! [R4: fun_int_fun_int_o,S2: set_nat,Ha: fun_nat_int,G: fun_nat_int] :
      ( aa_int_o(aa_int_fun_int_o(R4,one_one_int),one_one_int)
     => ( ! [X12: int,Y12: int,X22: int,Y22: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R4,X12),X22)
              & aa_int_o(aa_int_fun_int_o(R4,Y12),Y22) )
           => aa_int_o(aa_int_fun_int_o(R4,aa_int_int(aa_int_fun_int_int(times_times_int,X12),Y12)),aa_int_int(aa_int_fun_int_int(times_times_int,X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => aa_int_o(aa_int_fun_int_o(R4,aa_nat_int(Ha,X4)),aa_nat_int(G,X4)) )
           => aa_int_o(aa_int_fun_int_o(R4,aa_set_nat_int(groups705719431365010083at_int(Ha),S2)),aa_set_nat_int(groups705719431365010083at_int(G),S2)) ) ) ) ) ).

% prod.related
tff(fact_6787_prod_Orelated,axiom,
    ! [R4: fun_int_fun_int_o,S2: set_int,Ha: fun_int_int,G: fun_int_int] :
      ( aa_int_o(aa_int_fun_int_o(R4,one_one_int),one_one_int)
     => ( ! [X12: int,Y12: int,X22: int,Y22: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R4,X12),X22)
              & aa_int_o(aa_int_fun_int_o(R4,Y12),Y22) )
           => aa_int_o(aa_int_fun_int_o(R4,aa_int_int(aa_int_fun_int_int(times_times_int,X12),Y12)),aa_int_int(aa_int_fun_int_int(times_times_int,X22),Y22)) )
       => ( finite_finite_int(S2)
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => aa_int_o(aa_int_fun_int_o(R4,aa_int_int(Ha,X4)),aa_int_int(G,X4)) )
           => aa_int_o(aa_int_fun_int_o(R4,groups1705073143266064639nt_int(Ha,S2)),groups1705073143266064639nt_int(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_6788_sum_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: fun_int_rat] :
      ( ord_less_eq_set_int(B3,A3)
     => ( finite_finite_int(A3)
       => ( groups3906332499630173760nt_rat(G,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G,minus_minus_set_int(A3,B3))),groups3906332499630173760nt_rat(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6789_sum_Osubset__diff,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,G: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(B3),A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
       => ( aa_set4113944651722249469at_rat(groups342789780944988191at_rat(G),A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set4113944651722249469at_rat(groups342789780944988191at_rat(G),minus_1356011639430497352at_nat(A3,B3))),aa_set4113944651722249469at_rat(groups342789780944988191at_rat(G),B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6790_sum_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: fun_int_nat] :
      ( ord_less_eq_set_int(B3,A3)
     => ( finite_finite_int(A3)
       => ( groups4541462559716669496nt_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A3,B3))),groups4541462559716669496nt_nat(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6791_sum_Osubset__diff,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,G: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(B3),A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
       => ( aa_set4749074711808745205at_nat(groups977919841031483927at_nat(G),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set4749074711808745205at_nat(groups977919841031483927at_nat(G),minus_1356011639430497352at_nat(A3,B3))),aa_set4749074711808745205at_nat(groups977919841031483927at_nat(G),B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6792_sum_Osubset__diff,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,G: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(B3),A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
       => ( aa_set4746584241299694929at_int(groups975429370522433651at_int(G),A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set4746584241299694929at_int(groups975429370522433651at_int(G),minus_1356011639430497352at_nat(A3,B3))),aa_set4746584241299694929at_int(groups975429370522433651at_int(G),B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6793_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B3),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(G),minus_minus_set_nat(A3,B3))),aa_set_nat_rat(groups2906978787729119204at_rat(G),B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6794_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B3),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),minus_minus_set_nat(A3,B3))),aa_set_nat_int(groups3539618377306564664at_int(G),B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6795_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B3),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),minus_minus_set_nat(A3,B3))),aa_set_nat_nat(groups3542108847815614940at_nat(G),B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6796_sum_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: fun_int_int] :
      ( ord_less_eq_set_int(B3,A3)
     => ( finite_finite_int(A3)
       => ( groups4538972089207619220nt_int(G,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A3,B3))),groups4538972089207619220nt_int(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6797_prod_Osame__carrier,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_nat,Ha: fun_int_nat] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_nat(G,A2) = one_one_nat ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_nat(Ha,B2) = one_one_nat ) )
             => ( ( groups1707563613775114915nt_nat(G,A3) = groups1707563613775114915nt_nat(Ha,B3) )
              <=> ( groups1707563613775114915nt_nat(G,C5) = groups1707563613775114915nt_nat(Ha,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6798_prod_Osame__carrier,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_Code_integer,Ha: fun_int_Code_integer] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_Code_integer(G,A2) = one_one_Code_integer ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_Code_integer(Ha,B2) = one_one_Code_integer ) )
             => ( ( groups3827104343326376752nteger(G,A3) = groups3827104343326376752nteger(Ha,B3) )
              <=> ( groups3827104343326376752nteger(G,C5) = groups3827104343326376752nteger(Ha,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6799_prod_Osame__carrier,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_rat,Ha: fun_int_rat] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_rat(G,A2) = one_one_rat ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_rat(Ha,B2) = one_one_rat ) )
             => ( ( groups1072433553688619179nt_rat(G,A3) = groups1072433553688619179nt_rat(Ha,B3) )
              <=> ( groups1072433553688619179nt_rat(G,C5) = groups1072433553688619179nt_rat(Ha,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6800_prod_Osame__carrier,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_Code_natural,Ha: fun_int_Code_natural] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_Code_natural(G,A2) = one_one_Code_natural ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_Code_natural(Ha,B2) = one_one_Code_natural ) )
             => ( ( groups2650699495083094267atural(G,A3) = groups2650699495083094267atural(Ha,B3) )
              <=> ( groups2650699495083094267atural(G,C5) = groups2650699495083094267atural(Ha,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6801_prod_Osame__carrier,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_Code_integer(G,A2) = one_one_Code_integer ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_Code_integer(Ha,B2) = one_one_Code_integer ) )
             => ( ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),B3) )
              <=> ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),C5) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6802_prod_Osame__carrier,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_rat(G,A2) = one_one_rat ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_rat(Ha,B2) = one_one_rat ) )
             => ( ( aa_set_nat_rat(groups73079841787564623at_rat(G),A3) = aa_set_nat_rat(groups73079841787564623at_rat(Ha),B3) )
              <=> ( aa_set_nat_rat(groups73079841787564623at_rat(G),C5) = aa_set_nat_rat(groups73079841787564623at_rat(Ha),C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6803_prod_Osame__carrier,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_Code_natural,Ha: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_Code_natural(G,A2) = one_one_Code_natural ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_Code_natural(Ha,B2) = one_one_Code_natural ) )
             => ( ( groups2279045934846249631atural(G,A3) = groups2279045934846249631atural(Ha,B3) )
              <=> ( groups2279045934846249631atural(G,C5) = groups2279045934846249631atural(Ha,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6804_prod_Osame__carrier,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_nat(G,A2) = one_one_nat ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_nat(Ha,B2) = one_one_nat ) )
             => ( ( aa_set_nat_nat(groups708209901874060359at_nat(G),A3) = aa_set_nat_nat(groups708209901874060359at_nat(Ha),B3) )
              <=> ( aa_set_nat_nat(groups708209901874060359at_nat(G),C5) = aa_set_nat_nat(groups708209901874060359at_nat(Ha),C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6805_prod_Osame__carrier,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_int,Ha: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_int(G,A2) = one_one_int ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_int(Ha,B2) = one_one_int ) )
             => ( ( aa_set_nat_int(groups705719431365010083at_int(G),A3) = aa_set_nat_int(groups705719431365010083at_int(Ha),B3) )
              <=> ( aa_set_nat_int(groups705719431365010083at_int(G),C5) = aa_set_nat_int(groups705719431365010083at_int(Ha),C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6806_prod_Osame__carrier,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_int,Ha: fun_int_int] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_int(G,A2) = one_one_int ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_int(Ha,B2) = one_one_int ) )
             => ( ( groups1705073143266064639nt_int(G,A3) = groups1705073143266064639nt_int(Ha,B3) )
              <=> ( groups1705073143266064639nt_int(G,C5) = groups1705073143266064639nt_int(Ha,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6807_prod_Osame__carrierI,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_nat,Ha: fun_int_nat] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_nat(G,A2) = one_one_nat ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_nat(Ha,B2) = one_one_nat ) )
             => ( ( groups1707563613775114915nt_nat(G,C5) = groups1707563613775114915nt_nat(Ha,C5) )
               => ( groups1707563613775114915nt_nat(G,A3) = groups1707563613775114915nt_nat(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6808_prod_Osame__carrierI,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_Code_integer,Ha: fun_int_Code_integer] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_Code_integer(G,A2) = one_one_Code_integer ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_Code_integer(Ha,B2) = one_one_Code_integer ) )
             => ( ( groups3827104343326376752nteger(G,C5) = groups3827104343326376752nteger(Ha,C5) )
               => ( groups3827104343326376752nteger(G,A3) = groups3827104343326376752nteger(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6809_prod_Osame__carrierI,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_rat,Ha: fun_int_rat] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_rat(G,A2) = one_one_rat ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_rat(Ha,B2) = one_one_rat ) )
             => ( ( groups1072433553688619179nt_rat(G,C5) = groups1072433553688619179nt_rat(Ha,C5) )
               => ( groups1072433553688619179nt_rat(G,A3) = groups1072433553688619179nt_rat(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6810_prod_Osame__carrierI,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_Code_natural,Ha: fun_int_Code_natural] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_Code_natural(G,A2) = one_one_Code_natural ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_Code_natural(Ha,B2) = one_one_Code_natural ) )
             => ( ( groups2650699495083094267atural(G,C5) = groups2650699495083094267atural(Ha,C5) )
               => ( groups2650699495083094267atural(G,A3) = groups2650699495083094267atural(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6811_prod_Osame__carrierI,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_Code_integer(G,A2) = one_one_Code_integer ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_Code_integer(Ha,B2) = one_one_Code_integer ) )
             => ( ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),C5) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),C5) )
               => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),A3) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6812_prod_Osame__carrierI,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_rat(G,A2) = one_one_rat ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_rat(Ha,B2) = one_one_rat ) )
             => ( ( aa_set_nat_rat(groups73079841787564623at_rat(G),C5) = aa_set_nat_rat(groups73079841787564623at_rat(Ha),C5) )
               => ( aa_set_nat_rat(groups73079841787564623at_rat(G),A3) = aa_set_nat_rat(groups73079841787564623at_rat(Ha),B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6813_prod_Osame__carrierI,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_Code_natural,Ha: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_Code_natural(G,A2) = one_one_Code_natural ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_Code_natural(Ha,B2) = one_one_Code_natural ) )
             => ( ( groups2279045934846249631atural(G,C5) = groups2279045934846249631atural(Ha,C5) )
               => ( groups2279045934846249631atural(G,A3) = groups2279045934846249631atural(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6814_prod_Osame__carrierI,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_nat(G,A2) = one_one_nat ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_nat(Ha,B2) = one_one_nat ) )
             => ( ( aa_set_nat_nat(groups708209901874060359at_nat(G),C5) = aa_set_nat_nat(groups708209901874060359at_nat(Ha),C5) )
               => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A3) = aa_set_nat_nat(groups708209901874060359at_nat(Ha),B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6815_prod_Osame__carrierI,axiom,
    ! [C5: set_nat,A3: set_nat,B3: set_nat,G: fun_nat_int,Ha: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),minus_minus_set_nat(C5,A3))
               => ( aa_nat_int(G,A2) = one_one_int ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(C5,B3))
                 => ( aa_nat_int(Ha,B2) = one_one_int ) )
             => ( ( aa_set_nat_int(groups705719431365010083at_int(G),C5) = aa_set_nat_int(groups705719431365010083at_int(Ha),C5) )
               => ( aa_set_nat_int(groups705719431365010083at_int(G),A3) = aa_set_nat_int(groups705719431365010083at_int(Ha),B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6816_prod_Osame__carrierI,axiom,
    ! [C5: set_int,A3: set_int,B3: set_int,G: fun_int_int,Ha: fun_int_int] :
      ( finite_finite_int(C5)
     => ( ord_less_eq_set_int(A3,C5)
       => ( ord_less_eq_set_int(B3,C5)
         => ( ! [A2: int] :
                ( member_int(A2,minus_minus_set_int(C5,A3))
               => ( aa_int_int(G,A2) = one_one_int ) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(C5,B3))
                 => ( aa_int_int(Ha,B2) = one_one_int ) )
             => ( ( groups1705073143266064639nt_int(G,C5) = groups1705073143266064639nt_int(Ha,C5) )
               => ( groups1705073143266064639nt_int(G,A3) = groups1705073143266064639nt_int(Ha,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6817_prod_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_nat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_nat(G,X4) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,S2) = groups1707563613775114915nt_nat(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6818_prod_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_Code_integer] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_integer(G,X4) = one_one_Code_integer ) )
         => ( groups3827104343326376752nteger(G,S2) = groups3827104343326376752nteger(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6819_prod_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_rat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_rat(G,X4) = one_one_rat ) )
         => ( groups1072433553688619179nt_rat(G,S2) = groups1072433553688619179nt_rat(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6820_prod_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_Code_natural] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_natural(G,X4) = one_one_Code_natural ) )
         => ( groups2650699495083094267atural(G,S2) = groups2650699495083094267atural(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6821_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_integer(G,X4) = one_one_Code_integer ) )
         => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S2) = aa_set871585252387937313nteger(groups3455450783089532116nteger(G),T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6822_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_rat(G,X4) = one_one_rat ) )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(G),S2) = aa_set_nat_rat(groups73079841787564623at_rat(G),T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6823_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_natural(G,X4) = one_one_Code_natural ) )
         => ( groups2279045934846249631atural(G,S2) = groups2279045934846249631atural(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6824_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_nat(G,X4) = one_one_nat ) )
         => ( aa_set_nat_nat(groups708209901874060359at_nat(G),S2) = aa_set_nat_nat(groups708209901874060359at_nat(G),T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6825_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_int(G,X4) = one_one_int ) )
         => ( aa_set_nat_int(groups705719431365010083at_int(G),S2) = aa_set_nat_int(groups705719431365010083at_int(G),T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6826_prod_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_int] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_int(G,X4) = one_one_int ) )
         => ( groups1705073143266064639nt_int(G,S2) = groups1705073143266064639nt_int(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6827_prod_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_nat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_nat(G,X4) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,T3) = groups1707563613775114915nt_nat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6828_prod_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_Code_integer] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_integer(G,X4) = one_one_Code_integer ) )
         => ( groups3827104343326376752nteger(G,T3) = groups3827104343326376752nteger(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6829_prod_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_rat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_rat(G,X4) = one_one_rat ) )
         => ( groups1072433553688619179nt_rat(G,T3) = groups1072433553688619179nt_rat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6830_prod_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_Code_natural] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_natural(G,X4) = one_one_Code_natural ) )
         => ( groups2650699495083094267atural(G,T3) = groups2650699495083094267atural(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6831_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_integer(G,X4) = one_one_Code_integer ) )
         => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),T3) = aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6832_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_rat(G,X4) = one_one_rat ) )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(G),T3) = aa_set_nat_rat(groups73079841787564623at_rat(G),S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6833_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_natural(G,X4) = one_one_Code_natural ) )
         => ( groups2279045934846249631atural(G,T3) = groups2279045934846249631atural(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6834_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_nat(G,X4) = one_one_nat ) )
         => ( aa_set_nat_nat(groups708209901874060359at_nat(G),T3) = aa_set_nat_nat(groups708209901874060359at_nat(G),S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6835_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_int(G,X4) = one_one_int ) )
         => ( aa_set_nat_int(groups705719431365010083at_int(G),T3) = aa_set_nat_int(groups705719431365010083at_int(G),S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6836_prod_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_int] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_int(G,X4) = one_one_int ) )
         => ( groups1705073143266064639nt_int(G,T3) = groups1705073143266064639nt_int(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6837_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S2: set_int,Ha: fun_int_nat,G: fun_int_nat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_nat(Ha,X4) = one_one_nat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_nat(G,X4) = aa_int_nat(Ha,X4) ) )
           => ( groups1707563613775114915nt_nat(G,S2) = groups1707563613775114915nt_nat(Ha,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6838_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S2: set_int,Ha: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_integer(Ha,X4) = one_one_Code_integer ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_Code_integer(G,X4) = aa_int_Code_integer(Ha,X4) ) )
           => ( groups3827104343326376752nteger(G,S2) = groups3827104343326376752nteger(Ha,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6839_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S2: set_int,Ha: fun_int_rat,G: fun_int_rat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_rat(Ha,X4) = one_one_rat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_rat(G,X4) = aa_int_rat(Ha,X4) ) )
           => ( groups1072433553688619179nt_rat(G,S2) = groups1072433553688619179nt_rat(Ha,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6840_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S2: set_int,Ha: fun_int_Code_natural,G: fun_int_Code_natural] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_natural(Ha,X4) = one_one_Code_natural ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_Code_natural(G,X4) = aa_int_Code_natural(Ha,X4) ) )
           => ( groups2650699495083094267atural(G,S2) = groups2650699495083094267atural(Ha,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6841_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,Ha: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_integer(Ha,X4) = one_one_Code_integer ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_Code_integer(G,X4) = aa_nat_Code_integer(Ha,X4) ) )
           => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),S2) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6842_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,Ha: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_rat(Ha,X4) = one_one_rat ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_rat(G,X4) = aa_nat_rat(Ha,X4) ) )
           => ( aa_set_nat_rat(groups73079841787564623at_rat(G),S2) = aa_set_nat_rat(groups73079841787564623at_rat(Ha),T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6843_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,Ha: fun_nat_Code_natural,G: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_natural(Ha,X4) = one_one_Code_natural ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_Code_natural(G,X4) = aa_nat_Code_natural(Ha,X4) ) )
           => ( groups2279045934846249631atural(G,S2) = groups2279045934846249631atural(Ha,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6844_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,Ha: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_nat(Ha,X4) = one_one_nat ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_nat(G,X4) = aa_nat_nat(Ha,X4) ) )
           => ( aa_set_nat_nat(groups708209901874060359at_nat(G),S2) = aa_set_nat_nat(groups708209901874060359at_nat(Ha),T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6845_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,Ha: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_int(Ha,X4) = one_one_int ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_int(G,X4) = aa_nat_int(Ha,X4) ) )
           => ( aa_set_nat_int(groups705719431365010083at_int(G),S2) = aa_set_nat_int(groups705719431365010083at_int(Ha),T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6846_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S2: set_int,Ha: fun_int_int,G: fun_int_int] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_int(Ha,X4) = one_one_int ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_int(G,X4) = aa_int_int(Ha,X4) ) )
           => ( groups1705073143266064639nt_int(G,S2) = groups1705073143266064639nt_int(Ha,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6847_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_nat,Ha: fun_int_nat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_nat(G,X4) = one_one_nat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_nat(G,X4) = aa_int_nat(Ha,X4) ) )
           => ( groups1707563613775114915nt_nat(G,T3) = groups1707563613775114915nt_nat(Ha,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6848_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_Code_integer,Ha: fun_int_Code_integer] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_integer(G,X4) = one_one_Code_integer ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_Code_integer(G,X4) = aa_int_Code_integer(Ha,X4) ) )
           => ( groups3827104343326376752nteger(G,T3) = groups3827104343326376752nteger(Ha,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6849_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_rat,Ha: fun_int_rat] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_rat(G,X4) = one_one_rat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_rat(G,X4) = aa_int_rat(Ha,X4) ) )
           => ( groups1072433553688619179nt_rat(G,T3) = groups1072433553688619179nt_rat(Ha,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6850_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_Code_natural,Ha: fun_int_Code_natural] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_Code_natural(G,X4) = one_one_Code_natural ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_Code_natural(G,X4) = aa_int_Code_natural(Ha,X4) ) )
           => ( groups2650699495083094267atural(G,T3) = groups2650699495083094267atural(Ha,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6851_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_integer(G,X4) = one_one_Code_integer ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_Code_integer(G,X4) = aa_nat_Code_integer(Ha,X4) ) )
           => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(G),T3) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6852_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_rat(G,X4) = one_one_rat ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_rat(G,X4) = aa_nat_rat(Ha,X4) ) )
           => ( aa_set_nat_rat(groups73079841787564623at_rat(G),T3) = aa_set_nat_rat(groups73079841787564623at_rat(Ha),S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6853_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_Code_natural,Ha: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_Code_natural(G,X4) = one_one_Code_natural ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_Code_natural(G,X4) = aa_nat_Code_natural(Ha,X4) ) )
           => ( groups2279045934846249631atural(G,T3) = groups2279045934846249631atural(Ha,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6854_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_nat(G,X4) = one_one_nat ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_nat(G,X4) = aa_nat_nat(Ha,X4) ) )
           => ( aa_set_nat_nat(groups708209901874060359at_nat(G),T3) = aa_set_nat_nat(groups708209901874060359at_nat(Ha),S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6855_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_int,Ha: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),minus_minus_set_nat(T3,S2))
             => ( aa_nat_int(G,X4) = one_one_int ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),S2)
               => ( aa_nat_int(G,X4) = aa_nat_int(Ha,X4) ) )
           => ( aa_set_nat_int(groups705719431365010083at_int(G),T3) = aa_set_nat_int(groups705719431365010083at_int(Ha),S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6856_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_int,Ha: fun_int_int] :
      ( finite_finite_int(T3)
     => ( ord_less_eq_set_int(S2,T3)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T3,S2))
             => ( aa_int_int(G,X4) = one_one_int ) )
         => ( ! [X4: int] :
                ( member_int(X4,S2)
               => ( aa_int_int(G,X4) = aa_int_int(Ha,X4) ) )
           => ( groups1705073143266064639nt_int(G,T3) = groups1705073143266064639nt_int(Ha,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6857_set__bit__eq__or,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),A) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) ).

% set_bit_eq_or
tff(fact_6858_set__bit__eq__or,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer)) ).

% set_bit_eq_or
tff(fact_6859_set__bit__eq__or,axiom,
    ! [N: nat,A: code_natural] : bit_se1617098188084679374atural(N,A) = bit_se9127793120404214118atural(A,bit_se6611745700429515170atural(N,one_one_Code_natural)) ).

% set_bit_eq_or
tff(fact_6860_set__bit__eq__or,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat)) ).

% set_bit_eq_or
tff(fact_6861_sum__eq__Suc0__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( finite_finite_int(A3)
     => ( ( groups4541462559716669496nt_nat(F,A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: int] :
            ( member_int(X3,A3)
            & ( aa_int_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa2: int] :
                ( member_int(Xa2,A3)
               => ( ( X3 != Xa2 )
                 => ( aa_int_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6862_sum__eq__Suc0__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( ( aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A3)
            & ( aa_Pro4243173770044397973at_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa2: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa2),A3)
               => ( ( X3 != Xa2 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6863_sum__eq__Suc0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
            & ( aa_nat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa2),A3)
               => ( ( X3 != Xa2 )
                 => ( aa_nat_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6864_sum__eq__1__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( finite_finite_int(A3)
     => ( ( groups4541462559716669496nt_nat(F,A3) = one_one_nat )
      <=> ? [X3: int] :
            ( member_int(X3,A3)
            & ( aa_int_nat(F,X3) = one_one_nat )
            & ! [Xa2: int] :
                ( member_int(Xa2,A3)
               => ( ( X3 != Xa2 )
                 => ( aa_int_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6865_sum__eq__1__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( ( aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),A3) = one_one_nat )
      <=> ? [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A3)
            & ( aa_Pro4243173770044397973at_nat(F,X3) = one_one_nat )
            & ! [Xa2: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa2),A3)
               => ( ( X3 != Xa2 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6866_sum__eq__1__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = one_one_nat )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
            & ( aa_nat_nat(F,X3) = one_one_nat )
            & ! [Xa2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa2),A3)
               => ( ( X3 != Xa2 )
                 => ( aa_nat_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6867_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_Code_integer,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I2)) )
       => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(F),S) = zero_z3403309356797280102nteger )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S)
           => ( aa_nat_Code_integer(F,I) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6868_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_Code_integer,I: int] :
      ( finite_finite_int(S)
     => ( ! [I2: int] :
            ( member_int(I2,S)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I2)) )
       => ( ( groups7873554091576472773nteger(F,S) = zero_z3403309356797280102nteger )
         => ( member_int(I,S)
           => ( aa_int_Code_integer(F,I) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6869_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_Code_natural,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,I2)) )
       => ( ( groups6325495683096345652atural(F,S) = zero_z2226904508553997617atural )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S)
           => ( aa_nat_Code_natural(F,I) = zero_z2226904508553997617atural ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6870_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_Code_natural,I: int] :
      ( finite_finite_int(S)
     => ( ! [I2: int] :
            ( member_int(I2,S)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,I2)) )
       => ( ( groups6697149243333190288atural(F,S) = zero_z2226904508553997617atural )
         => ( member_int(I,S)
           => ( aa_int_Code_natural(F,I) = zero_z2226904508553997617atural ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6871_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(F),S) = zero_zero_rat )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S)
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6872_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_rat,I: int] :
      ( finite_finite_int(S)
     => ( ! [I2: int] :
            ( member_int(I2,S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,S) = zero_zero_rat )
         => ( member_int(I,S)
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6873_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_nat,I: int] :
      ( finite_finite_int(S)
     => ( ! [I2: int] :
            ( member_int(I2,S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,S) = zero_zero_nat )
         => ( member_int(I,S)
           => ( aa_int_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6874_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_int,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I2)) )
       => ( ( aa_set_nat_int(groups3539618377306564664at_int(F),S) = zero_zero_int )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S)
           => ( aa_nat_int(F,I) = zero_zero_int ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6875_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_nat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,I2)) )
       => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),S) = zero_zero_nat )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S)
           => ( aa_nat_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6876_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_int,I: int] :
      ( finite_finite_int(S)
     => ( ! [I2: int] :
            ( member_int(I2,S)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,I2)) )
       => ( ( groups4538972089207619220nt_int(F,S) = zero_zero_int )
         => ( member_int(I,S)
           => ( aa_int_int(F,I) = zero_zero_int ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6877_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_Code_integer,B3: code_integer,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I2)) )
       => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(F),S) = B3 )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,I)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6878_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_Code_integer,B3: code_integer,I: int] :
      ( finite_finite_int(S)
     => ( ! [I2: int] :
            ( member_int(I2,S)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I2)) )
       => ( ( groups7873554091576472773nteger(F,S) = B3 )
         => ( member_int(I,S)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6879_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_Code_natural,B3: code_natural,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,I2)) )
       => ( ( groups6325495683096345652atural(F,S) = B3 )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(F,I)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6880_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_Code_natural,B3: code_natural,I: int] :
      ( finite_finite_int(S)
     => ( ! [I2: int] :
            ( member_int(I2,S)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,I2)) )
       => ( ( groups6697149243333190288atural(F,S) = B3 )
         => ( member_int(I,S)
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_int_Code_natural(F,I)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6881_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_rat,B3: rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(F),S) = B3 )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6882_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_rat,B3: rat,I: int] :
      ( finite_finite_int(S)
     => ( ! [I2: int] :
            ( member_int(I2,S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,S) = B3 )
         => ( member_int(I,S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6883_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_nat,B3: nat,I: int] :
      ( finite_finite_int(S)
     => ( ! [I2: int] :
            ( member_int(I2,S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,S) = B3 )
         => ( member_int(I,S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6884_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_int,B3: int,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I2)) )
       => ( ( aa_set_nat_int(groups3539618377306564664at_int(F),S) = B3 )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6885_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_nat,B3: nat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,I2)) )
       => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),S) = B3 )
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6886_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_int,B3: int,I: int] :
      ( finite_finite_int(S)
     => ( ! [I2: int] :
            ( member_int(I2,S)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,I2)) )
       => ( ( groups4538972089207619220nt_int(F,S) = B3 )
         => ( member_int(I,S)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,I)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6887_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cx,M))) ) ).

% finite_divisors_nat
tff(fact_6888_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,K),L) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,
        $ite(L = zero_z3403309356797280102nteger,K,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,L),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L))))) ).

% gcd_code_integer
tff(fact_6889_finite__abs__int__segment,axiom,
    ! [A: rat] : finite_finite_rat(collect_rat(aa_rat_fun_rat_o(aTP_Lamm_vv,A))) ).

% finite_abs_int_segment
tff(fact_6890_bit_Ocomplement__unique,axiom,
    ! [A: code_integer,Xa: code_integer,Y3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),Xa) = zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),Xa) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),Y3) = zero_z3403309356797280102nteger )
         => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),Y3) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
           => ( Xa = Y3 ) ) ) ) ) ).

% bit.complement_unique
tff(fact_6891_bit_Ocomplement__unique,axiom,
    ! [A: int,Xa: int,Y3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),Xa) = zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),Xa) = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),Y3) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),Y3) = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( Xa = Y3 ) ) ) ) ) ).

% bit.complement_unique
tff(fact_6892_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),I5)) ) ) ) ) ).

% sum_pos2
tff(fact_6893_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_Code_integer] :
      ( finite_finite_int(I5)
     => ( member_int(I,I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I))
         => ( ! [I2: int] :
                ( member_int(I2,I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups7873554091576472773nteger(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_6894_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I5)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
               => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,I2)) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),groups6325495683096345652atural(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_6895_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_Code_natural] :
      ( finite_finite_int(I5)
     => ( member_int(I,I5)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,I))
         => ( ! [I2: int] :
                ( member_int(I2,I5)
               => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,I2)) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),groups6697149243333190288atural(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_6896_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_set_nat_rat(groups2906978787729119204at_rat(F),I5)) ) ) ) ) ).

% sum_pos2
tff(fact_6897_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( finite_finite_int(I5)
     => ( member_int(I,I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I))
         => ( ! [I2: int] :
                ( member_int(I2,I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_6898_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_nat] :
      ( finite_finite_int(I5)
     => ( member_int(I,I5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I))
         => ( ! [I2: int] :
                ( member_int(I2,I5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_6899_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_set_nat_int(groups3539618377306564664at_int(F),I5)) ) ) ) ) ).

% sum_pos2
tff(fact_6900_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,I2)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups3542108847815614940at_nat(F),I5)) ) ) ) ) ).

% sum_pos2
tff(fact_6901_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_int] :
      ( finite_finite_int(I5)
     => ( member_int(I,I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,I))
         => ( ! [I2: int] :
                ( member_int(I2,I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups4538972089207619220nt_int(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_6902_sum__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_Code_integer] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A3,B3)
       => ( ! [B2: int] :
              ( member_int(B2,minus_minus_set_int(B3,A3))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,B2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(F,A3)),groups7873554091576472773nteger(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6903_sum__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_Code_natural] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A3,B3)
       => ( ! [B2: int] :
              ( member_int(B2,minus_minus_set_int(B3,A3))
             => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,B2)) )
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups6697149243333190288atural(F,A3)),groups6697149243333190288atural(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6904_sum__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_rat] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A3,B3)
       => ( ! [B2: int] :
              ( member_int(B2,minus_minus_set_int(B3,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,B2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6905_sum__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_nat] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A3,B3)
       => ( ! [B2: int] :
              ( member_int(B2,minus_minus_set_int(B3,A3))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6906_sum__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(B3,A3))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,B2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3)),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),B3)) ) ) ) ).

% sum_mono2
tff(fact_6907_sum__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(B3,A3))
             => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,B2)) )
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups6325495683096345652atural(F,A3)),groups6325495683096345652atural(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6908_sum__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(B3,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,B2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),A3)),aa_set_nat_rat(groups2906978787729119204at_rat(F),B3)) ) ) ) ).

% sum_mono2
tff(fact_6909_sum__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(B3,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,B2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),A3)),aa_set_nat_int(groups3539618377306564664at_int(F),B3)) ) ) ) ).

% sum_mono2
tff(fact_6910_sum__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(B3,A3))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B3)) ) ) ) ).

% sum_mono2
tff(fact_6911_sum__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_int] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A3,B3)
       => ( ! [B2: int] :
              ( member_int(B2,minus_minus_set_int(B3,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,B2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4538972089207619220nt_int(F,A3)),groups4538972089207619220nt_int(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6912_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(F,I2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6913_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: fun_int_Code_integer] :
      ( finite_finite_int(I5)
     => ( member_int(I,I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(F,I))
         => ( ! [I2: int] :
                ( member_int(I2,I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F,I2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3827104343326376752nteger(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6914_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_set_nat_rat(groups73079841787564623at_rat(F),I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6915_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( finite_finite_int(I5)
     => ( member_int(I,I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I))
         => ( ! [I2: int] :
                ( member_int(I2,I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6916_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I),I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_set_nat_int(groups705719431365010083at_int(F),I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6917_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: fun_int_int] :
      ( finite_finite_int(I5)
     => ( member_int(I,I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(F,I))
         => ( ! [I2: int] :
                ( member_int(I2,I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups1705073143266064639nt_int(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6918_less__1__prod2,axiom,
    ! [I5: set_set_nat,I: set_nat,F: fun_se7586849273513198688nteger] :
      ( finite1152437895449049373et_nat(I5)
     => ( member_set_nat(I,I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_set871585252387937313nteger(F,I))
         => ( ! [I2: set_nat] :
                ( member_set_nat(I2,I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_set871585252387937313nteger(F,I2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups2761809935038513290nteger(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6919_less__1__prod2,axiom,
    ! [I5: set_set_nat,I: set_nat,F: fun_set_nat_rat] :
      ( finite1152437895449049373et_nat(I5)
     => ( member_set_nat(I,I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_set_nat_rat(F,I))
         => ( ! [I2: set_nat] :
                ( member_set_nat(I2,I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_set_nat_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups3613417700093529605at_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6920_less__1__prod2,axiom,
    ! [I5: set_set_nat,I: set_nat,F: fun_set_nat_int] :
      ( finite1152437895449049373et_nat(I5)
     => ( member_set_nat(I,I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_set_nat_int(F,I))
         => ( ! [I2: set_nat] :
                ( member_set_nat(I2,I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_set_nat_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups4246057289670975065at_int(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6921_less__1__prod2,axiom,
    ! [I5: set_Pr1261947904930325089at_nat,I: product_prod_nat_nat,F: fun_Pr6026251430685297nteger] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,I5)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Pro4764841431850000930nteger(F,I))
         => ( ! [I2: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I2),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Pro4764841431850000930nteger(F,I2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_set5601480140167866242nteger(groups1230400874837758585nteger(F),I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6922_gcd__int__def,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Y3) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,Xa))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Y3)))) ).

% gcd_int_def
tff(fact_6923_sum__strict__mono2,axiom,
    ! [B3: set_int,A3: set_int,B: int,F: fun_int_Code_integer] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A3,B3)
       => ( member_int(B,minus_minus_set_int(B3,A3))
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,B))
           => ( ! [X4: int] :
                  ( member_int(X4,B3)
                 => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X4)) )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups7873554091576472773nteger(F,A3)),groups7873554091576472773nteger(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6924_sum__strict__mono2,axiom,
    ! [B3: set_int,A3: set_int,B: int,F: fun_int_Code_natural] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A3,B3)
       => ( member_int(B,minus_minus_set_int(B3,A3))
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,B))
           => ( ! [X4: int] :
                  ( member_int(X4,B3)
                 => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,X4)) )
             => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,groups6697149243333190288atural(F,A3)),groups6697149243333190288atural(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6925_sum__strict__mono2,axiom,
    ! [B3: set_int,A3: set_int,B: int,F: fun_int_rat] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A3,B3)
       => ( member_int(B,minus_minus_set_int(B3,A3))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,B))
           => ( ! [X4: int] :
                  ( member_int(X4,B3)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6926_sum__strict__mono2,axiom,
    ! [B3: set_int,A3: set_int,B: int,F: fun_int_nat] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A3,B3)
       => ( member_int(B,minus_minus_set_int(B3,A3))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,B))
           => ( ! [X4: int] :
                  ( member_int(X4,B3)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6927_sum__strict__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,B: nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
       => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B),minus_minus_set_nat(B3,A3))
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,B))
           => ( ! [X4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),B3)
                 => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X4)) )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3)),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6928_sum__strict__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,B: nat,F: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
       => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B),minus_minus_set_nat(B3,A3))
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,B))
           => ( ! [X4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),B3)
                 => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,X4)) )
             => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,groups6325495683096345652atural(F,A3)),groups6325495683096345652atural(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6929_sum__strict__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,B: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
       => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B),minus_minus_set_nat(B3,A3))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,B))
           => ( ! [X4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),B3)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),A3)),aa_set_nat_rat(groups2906978787729119204at_rat(F),B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6930_sum__strict__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,B: nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
       => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B),minus_minus_set_nat(B3,A3))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,B))
           => ( ! [X4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),B3)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),A3)),aa_set_nat_int(groups3539618377306564664at_int(F),B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6931_sum__strict__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,B: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
       => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B),minus_minus_set_nat(B3,A3))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,B))
           => ( ! [X4: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),B3)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6932_sum__strict__mono2,axiom,
    ! [B3: set_int,A3: set_int,B: int,F: fun_int_int] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A3,B3)
       => ( member_int(B,minus_minus_set_int(B3,A3))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,B))
           => ( ! [X4: int] :
                  ( member_int(X4,B3)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(F,A3)),groups4538972089207619220nt_int(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6933_prod__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_Code_integer] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A3,B3)
       => ( ! [B2: int] :
              ( member_int(B2,minus_minus_set_int(B3,A3))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F,B2)) )
         => ( ! [A2: int] :
                ( member_int(A2,A3)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,A2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F,A3)),groups3827104343326376752nteger(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_6934_prod__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_rat] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A3,B3)
       => ( ! [B2: int] :
              ( member_int(B2,minus_minus_set_int(B3,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,B2)) )
         => ( ! [A2: int] :
                ( member_int(A2,A3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,A2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),groups1072433553688619179nt_rat(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_6935_prod__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(B3,A3))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(F,B2)) )
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,A2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A3)),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),B3)) ) ) ) ) ).

% prod_mono2
tff(fact_6936_prod__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(B3,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,B2)) )
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,A2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups73079841787564623at_rat(F),A3)),aa_set_nat_rat(groups73079841787564623at_rat(F),B3)) ) ) ) ) ).

% prod_mono2
tff(fact_6937_prod__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),minus_minus_set_nat(B3,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(F,B2)) )
         => ( ! [A2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,A2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups705719431365010083at_int(F),A3)),aa_set_nat_int(groups705719431365010083at_int(F),B3)) ) ) ) ) ).

% prod_mono2
tff(fact_6938_prod__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: fun_int_int] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A3,B3)
       => ( ! [B2: int] :
              ( member_int(B2,minus_minus_set_int(B3,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(F,B2)) )
         => ( ! [A2: int] :
                ( member_int(A2,A3)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,A2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1705073143266064639nt_int(F,A3)),groups1705073143266064639nt_int(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_6939_prod__mono2,axiom,
    ! [B3: set_set_nat,A3: set_set_nat,F: fun_se7586849273513198688nteger] :
      ( finite1152437895449049373et_nat(B3)
     => ( ord_le6893508408891458716et_nat(A3,B3)
       => ( ! [B2: set_nat] :
              ( member_set_nat(B2,minus_2163939370556025621et_nat(B3,A3))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_set871585252387937313nteger(F,B2)) )
         => ( ! [A2: set_nat] :
                ( member_set_nat(A2,A3)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(F,A2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups2761809935038513290nteger(F,A3)),groups2761809935038513290nteger(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_6940_prod__mono2,axiom,
    ! [B3: set_set_nat,A3: set_set_nat,F: fun_set_nat_rat] :
      ( finite1152437895449049373et_nat(B3)
     => ( ord_le6893508408891458716et_nat(A3,B3)
       => ( ! [B2: set_nat] :
              ( member_set_nat(B2,minus_2163939370556025621et_nat(B3,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_set_nat_rat(F,B2)) )
         => ( ! [A2: set_nat] :
                ( member_set_nat(A2,A3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_set_nat_rat(F,A2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3613417700093529605at_rat(F,A3)),groups3613417700093529605at_rat(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_6941_prod__mono2,axiom,
    ! [B3: set_set_nat,A3: set_set_nat,F: fun_set_nat_int] :
      ( finite1152437895449049373et_nat(B3)
     => ( ord_le6893508408891458716et_nat(A3,B3)
       => ( ! [B2: set_nat] :
              ( member_set_nat(B2,minus_2163939370556025621et_nat(B3,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_set_nat_int(F,B2)) )
         => ( ! [A2: set_nat] :
                ( member_set_nat(A2,A3)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_set_nat_int(F,A2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4246057289670975065at_int(F,A3)),groups4246057289670975065at_int(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_6942_prod__mono2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,F: fun_Pr6026251430685297nteger] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
     => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A3),B3)
       => ( ! [B2: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B2),minus_1356011639430497352at_nat(B3,A3))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Pro4764841431850000930nteger(F,B2)) )
         => ( ! [A2: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A2),A3)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Pro4764841431850000930nteger(F,A2)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set5601480140167866242nteger(groups1230400874837758585nteger(F),A3)),aa_set5601480140167866242nteger(groups1230400874837758585nteger(F),B3)) ) ) ) ) ).

% prod_mono2
tff(fact_6943_bezw__aux,axiom,
    ! [Xa: nat,Y3: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Y3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_nat4424723855478970923nt_int(bezw(Xa),Y3))),aa_nat_int(semiri1314217659103216013at_int,Xa))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_nat4424723855478970923nt_int(bezw(Xa),Y3))),aa_nat_int(semiri1314217659103216013at_int,Y3))) ).

% bezw_aux
tff(fact_6944_and__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_option_num(bit_un7362597486090784418nd_num(M),N) = none_num )
    <=> ( bit_se2773287842338716102atural(aa_num_Code_natural(numera5444537566228673987atural,M),aa_num_Code_natural(numera5444537566228673987atural,N)) = zero_z2226904508553997617atural ) ) ).

% and_num_eq_None_iff
tff(fact_6945_and__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_option_num(bit_un7362597486090784418nd_num(M),N) = none_num )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = zero_z3403309356797280102nteger ) ) ).

% and_num_eq_None_iff
tff(fact_6946_and__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_option_num(bit_un7362597486090784418nd_num(M),N) = none_num )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = zero_zero_int ) ) ).

% and_num_eq_None_iff
tff(fact_6947_and__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_option_num(bit_un7362597486090784418nd_num(M),N) = none_num )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = zero_zero_nat ) ) ).

% and_num_eq_None_iff
tff(fact_6948_one__or__eq,axiom,
    ! [A: code_natural] : bit_se9127793120404214118atural(one_one_Code_natural,A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_n8403883297036319079atural(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_6949_one__or__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,one_one_Code_integer),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_6950_one__or__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_6951_one__or__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_6952_or__one__eq,axiom,
    ! [A: code_natural] : bit_se9127793120404214118atural(A,one_one_Code_natural) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_n8403883297036319079atural(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_6953_or__one__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_6954_or__one__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_6955_or__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_6956_mask__Suc__double,axiom,
    ! [N: nat] : aa_nat_Code_natural(bit_se943457434206027407atural,aa_nat_nat(suc,N)) = bit_se9127793120404214118atural(one_one_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_Code_natural(bit_se943457434206027407atural,N))) ).

% mask_Suc_double
tff(fact_6957_mask__Suc__double,axiom,
    ! [N: nat] : aa_nat_Code_integer(bit_se2119862282449309892nteger,aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,N))) ).

% mask_Suc_double
tff(fact_6958_mask__Suc__double,axiom,
    ! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ).

% mask_Suc_double
tff(fact_6959_mask__Suc__double,axiom,
    ! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,N))) ).

% mask_Suc_double
tff(fact_6960_numeral__and__num,axiom,
    ! [M: num,N: num] : bit_se2773287842338716102atural(aa_num_Code_natural(numera5444537566228673987atural,M),aa_num_Code_natural(numera5444537566228673987atural,N)) = case_o5621594795226839503al_num(zero_z2226904508553997617atural,numera5444537566228673987atural,aa_num_option_num(bit_un7362597486090784418nd_num(M),N)) ).

% numeral_and_num
tff(fact_6961_numeral__and__num,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = case_o356765784539232260er_num(zero_z3403309356797280102nteger,numera6620942414471956472nteger,aa_num_option_num(bit_un7362597486090784418nd_num(M),N)) ).

% numeral_and_num
tff(fact_6962_numeral__and__num,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_un7362597486090784418nd_num(M),N)) ).

% numeral_and_num
tff(fact_6963_numeral__and__num,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_option_num_nat(case_option_nat_num(zero_zero_nat,numeral_numeral_nat),aa_num_option_num(bit_un7362597486090784418nd_num(M),N)) ).

% numeral_and_num
tff(fact_6964_Suc__0__or__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% Suc_0_or_eq
tff(fact_6965_or__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% or_Suc_0_eq
tff(fact_6966_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,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_6967_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_hu,K))) ).

% finite_Collect_le_nat
tff(fact_6968_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fl,K))) ).

% finite_Collect_less_nat
tff(fact_6969_finite__Collect__subsets,axiom,
    ! [A3: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(finite2998713641127702882nt_int,A3)
     => finite8744585540193469122nt_int(collec5210948495886036740nt_int(aTP_Lamm_hp(A3))) ) ).

% finite_Collect_subsets
tff(fact_6970_finite__Collect__subsets,axiom,
    ! [A3: set_int] :
      ( finite_finite_int(A3)
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_vw(A3))) ) ).

% finite_Collect_subsets
tff(fact_6971_finite__Collect__subsets,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => finite9047747110432174090at_nat(collec5514110066124741708at_nat(aTP_Lamm_ho(A3))) ) ).

% finite_Collect_subsets
tff(fact_6972_finite__Collect__subsets,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => finite1152437895449049373et_nat(collect_set_nat(aTP_Lamm_hq(A3))) ) ).

% finite_Collect_subsets
tff(fact_6973_gcd__nat_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Xaa) = Y3 )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa))
       => ~ ( ( Y3 = $ite(Xaa = zero_zero_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_6974_finite__Collect__conjI,axiom,
    ! [Pa: fun_se4782206882984618170_nat_o,Q: fun_se4782206882984618170_nat_o] :
      ( ( finite9047747110432174090at_nat(collec5514110066124741708at_nat(Pa))
        | finite9047747110432174090at_nat(collec5514110066124741708at_nat(Q)) )
     => finite9047747110432174090at_nat(collec5514110066124741708at_nat(aa_fun3315923329236548914_nat_o(aTP_Lamm_bx(Pa),Q))) ) ).

% finite_Collect_conjI
tff(fact_6975_finite__Collect__conjI,axiom,
    ! [Pa: fun_se7249947736802449410_int_o,Q: fun_se7249947736802449410_int_o] :
      ( ( finite8744585540193469122nt_int(collec5210948495886036740nt_int(Pa))
        | finite8744585540193469122nt_int(collec5210948495886036740nt_int(Q)) )
     => finite8744585540193469122nt_int(collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_by(Pa),Q))) ) ).

% finite_Collect_conjI
tff(fact_6976_finite__Collect__conjI,axiom,
    ! [Pa: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ( finite1152437895449049373et_nat(collect_set_nat(Pa))
        | finite1152437895449049373et_nat(collect_set_nat(Q)) )
     => finite1152437895449049373et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bz(Pa),Q))) ) ).

% finite_Collect_conjI
tff(fact_6977_finite__Collect__conjI,axiom,
    ! [Pa: fun_nat_o,Q: fun_nat_o] :
      ( ( aa_set_nat_o(finite_finite_nat,collect_nat(Pa))
        | aa_set_nat_o(finite_finite_nat,collect_nat(Q)) )
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cb(Pa),Q))) ) ).

% finite_Collect_conjI
tff(fact_6978_finite__Collect__conjI,axiom,
    ! [Pa: fun_int_o,Q: fun_int_o] :
      ( ( finite_finite_int(collect_int(Pa))
        | finite_finite_int(collect_int(Q)) )
     => finite_finite_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ca(Pa),Q))) ) ).

% finite_Collect_conjI
tff(fact_6979_finite__Collect__conjI,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Pa))
        | aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q)) )
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_vx(Pa),Q))) ) ).

% finite_Collect_conjI
tff(fact_6980_finite__Collect__disjI,axiom,
    ! [Pa: fun_se4782206882984618170_nat_o,Q: fun_se4782206882984618170_nat_o] :
      ( finite9047747110432174090at_nat(collec5514110066124741708at_nat(aa_fun3315923329236548914_nat_o(aTP_Lamm_vy(Pa),Q)))
    <=> ( finite9047747110432174090at_nat(collec5514110066124741708at_nat(Pa))
        & finite9047747110432174090at_nat(collec5514110066124741708at_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6981_finite__Collect__disjI,axiom,
    ! [Pa: fun_se7249947736802449410_int_o,Q: fun_se7249947736802449410_int_o] :
      ( finite8744585540193469122nt_int(collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_vz(Pa),Q)))
    <=> ( finite8744585540193469122nt_int(collec5210948495886036740nt_int(Pa))
        & finite8744585540193469122nt_int(collec5210948495886036740nt_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6982_finite__Collect__disjI,axiom,
    ! [Pa: fun_set_nat_o,Q: fun_set_nat_o] :
      ( finite1152437895449049373et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_wa(Pa),Q)))
    <=> ( finite1152437895449049373et_nat(collect_set_nat(Pa))
        & finite1152437895449049373et_nat(collect_set_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6983_finite__Collect__disjI,axiom,
    ! [Pa: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_wb(Pa),Q)))
    <=> ( aa_set_nat_o(finite_finite_nat,collect_nat(Pa))
        & aa_set_nat_o(finite_finite_nat,collect_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6984_finite__Collect__disjI,axiom,
    ! [Pa: fun_int_o,Q: fun_int_o] :
      ( finite_finite_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_wc(Pa),Q)))
    <=> ( finite_finite_int(collect_int(Pa))
        & finite_finite_int(collect_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6985_finite__Collect__disjI,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_wd(Pa),Q)))
    <=> ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Pa))
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6986_gcd__idem__nat,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Xa) = Xa ).

% gcd_idem_nat
tff(fact_6987_gcd__nat_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ).

% gcd_nat.right_idem
tff(fact_6988_gcd__nat_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ).

% gcd_nat.left_idem
tff(fact_6989_gcd__nat_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),A) = A ).

% gcd_nat.idem
tff(fact_6990_finite__interval__int1,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_we(A),B))) ).

% finite_interval_int1
tff(fact_6991_finite__interval__int4,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_wf(A),B))) ).

% finite_interval_int4
tff(fact_6992_finite__interval__int2,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_wg(A),B))) ).

% finite_interval_int2
tff(fact_6993_finite__interval__int3,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_wh(A),B))) ).

% finite_interval_int3
tff(fact_6994_minus__set__def,axiom,
    ! [A3: set_se7855581050983116737at_nat,B3: set_se7855581050983116737at_nat] : minus_4207664762107033000at_nat(A3,B3) = collec5514110066124741708at_nat(minus_752066780107176211_nat_o(aTP_Lamm_a(A3),aTP_Lamm_a(B3))) ).

% minus_set_def
tff(fact_6995_minus__set__def,axiom,
    ! [A3: set_se6260736226359567993nt_int,B3: set_se6260736226359567993nt_int] : minus_2612819937483484256nt_int(A3,B3) = collec5210948495886036740nt_int(minus_3219807633925007451_int_o(aTP_Lamm_aa(A3),aTP_Lamm_aa(B3))) ).

% minus_set_def
tff(fact_6996_minus__set__def,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] : minus_2163939370556025621et_nat(A3,B3) = collect_set_nat(minus_6087510627544308050_nat_o(aTP_Lamm_ab(A3),aTP_Lamm_ab(B3))) ).

% minus_set_def
tff(fact_6997_minus__set__def,axiom,
    ! [A3: set_int,B3: set_int] : minus_minus_set_int(A3,B3) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_ad(A3),aTP_Lamm_ad(B3))) ).

% minus_set_def
tff(fact_6998_minus__set__def,axiom,
    ! [A3: set_nat,B3: set_nat] : minus_minus_set_nat(A3,B3) = collect_nat(minus_8972599623528345224_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ac,A3),aa_set_nat_fun_nat_o(aTP_Lamm_ac,B3))) ).

% minus_set_def
tff(fact_6999_set__diff__eq,axiom,
    ! [A3: set_se7855581050983116737at_nat,B3: set_se7855581050983116737at_nat] : minus_4207664762107033000at_nat(A3,B3) = collec5514110066124741708at_nat(aa_set7533644503045286079_nat_o(aTP_Lamm_wi(A3),B3)) ).

% set_diff_eq
tff(fact_7000_set__diff__eq,axiom,
    ! [A3: set_se6260736226359567993nt_int,B3: set_se6260736226359567993nt_int] : minus_2612819937483484256nt_int(A3,B3) = collec5210948495886036740nt_int(aa_set4062574822058255039_int_o(aTP_Lamm_wj(A3),B3)) ).

% set_diff_eq
tff(fact_7001_set__diff__eq,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] : minus_2163939370556025621et_nat(A3,B3) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_wk(A3),B3)) ).

% set_diff_eq
tff(fact_7002_set__diff__eq,axiom,
    ! [A3: set_int,B3: set_int] : minus_minus_set_int(A3,B3) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_wl(A3),B3)) ).

% set_diff_eq
tff(fact_7003_set__diff__eq,axiom,
    ! [A3: set_nat,B3: set_nat] : minus_minus_set_nat(A3,B3) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_wm(A3),B3)) ).

% set_diff_eq
tff(fact_7004_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_cw,I))) ) ).

% finite_divisors_int
tff(fact_7005_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_7006_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_nat,R4: fun_nat_fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
             => ? [Xa3: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),B3)
                  & aa_nat_o(aa_nat_fun_nat_o(R4,X4),Xa3) ) )
         => ? [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),B3)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_qv(A3),R4),X4))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7007_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_int,R4: fun_nat_fun_int_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( finite_finite_int(B3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
             => ? [Xa3: int] :
                  ( member_int(Xa3,B3)
                  & aa_int_o(aa_nat_fun_int_o(R4,X4),Xa3) ) )
         => ? [X4: int] :
              ( member_int(X4,B3)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_qo(A3),R4),X4))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7008_pigeonhole__infinite__rel,axiom,
    ! [A3: set_int,B3: set_nat,R4: fun_int_fun_nat_o] :
      ( ~ finite_finite_int(A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X4: int] :
              ( member_int(X4,A3)
             => ? [Xa3: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),B3)
                  & aa_nat_o(aa_int_fun_nat_o(R4,X4),Xa3) ) )
         => ? [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),B3)
              & ~ finite_finite_int(collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_qk(A3),R4),X4))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7009_pigeonhole__infinite__rel,axiom,
    ! [A3: set_int,B3: set_int,R4: fun_int_fun_int_o] :
      ( ~ finite_finite_int(A3)
     => ( finite_finite_int(B3)
       => ( ! [X4: int] :
              ( member_int(X4,A3)
             => ? [Xa3: int] :
                  ( member_int(Xa3,B3)
                  & aa_int_o(aa_int_fun_int_o(R4,X4),Xa3) ) )
         => ? [X4: int] :
              ( member_int(X4,B3)
              & ~ finite_finite_int(collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_rb(A3),R4),X4))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7010_pigeonhole__infinite__rel,axiom,
    ! [A3: set_set_nat,B3: set_nat,R4: fun_se382663106265794036_nat_o] :
      ( ~ finite1152437895449049373et_nat(A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X4: set_nat] :
              ( member_set_nat(X4,A3)
             => ? [Xa3: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),B3)
                  & aa_nat_o(aa_set_nat_fun_nat_o(R4,X4),Xa3) ) )
         => ? [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),B3)
              & ~ finite1152437895449049373et_nat(collect_set_nat(aa_nat_fun_set_nat_o(aa_fun9105140013458882876_nat_o(aTP_Lamm_rg(A3),R4),X4))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7011_pigeonhole__infinite__rel,axiom,
    ! [A3: set_set_nat,B3: set_int,R4: fun_se7360890175450123928_int_o] :
      ( ~ finite1152437895449049373et_nat(A3)
     => ( finite_finite_int(B3)
       => ( ! [X4: set_nat] :
              ( member_set_nat(X4,A3)
             => ? [Xa3: int] :
                  ( member_int(Xa3,B3)
                  & aa_int_o(aa_set_nat_fun_int_o(R4,X4),Xa3) ) )
         => ? [X4: int] :
              ( member_int(X4,B3)
              & ~ finite1152437895449049373et_nat(collect_set_nat(aa_int_fun_set_nat_o(aa_fun9086678093774146108_nat_o(aTP_Lamm_rl(A3),R4),X4))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7012_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_Pr1261947904930325089at_nat,R4: fun_na3970208099681795815_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),A3)
             => ? [Xa3: product_prod_nat_nat] :
                  ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa3),B3)
                  & aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(R4,X4),Xa3) ) )
         => ? [X4: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),B3)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_Pro4733806262894120092_nat_o(aa_fun5389978208584994248_nat_o(aTP_Lamm_wn(A3),R4),X4))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7013_pigeonhole__infinite__rel,axiom,
    ! [A3: set_int,B3: set_Pr1261947904930325089at_nat,R4: fun_in2375804543815556931_nat_o] :
      ( ~ finite_finite_int(A3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3)
       => ( ! [X4: int] :
              ( member_int(X4,A3)
             => ? [Xa3: product_prod_nat_nat] :
                  ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa3),B3)
                  & aa_Pro6150637604445964947_nat_o(aa_int6101145382336329604_nat_o(R4,X4),Xa3) ) )
         => ? [X4: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),B3)
              & ~ finite_finite_int(collect_int(aa_Pro2912763917461432128_int_o(aa_fun8623471109766342856_int_o(aTP_Lamm_wo(A3),R4),X4))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7014_pigeonhole__infinite__rel,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_nat,R4: fun_Pr6984107991879650211_nat_o] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X4: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A3)
             => ? [Xa3: nat] :
                  ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa3),B3)
                  & aa_nat_o(aa_Pro4733806262894120092_nat_o(R4,X4),Xa3) ) )
         => ? [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),B3)
              & ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_nat7695548938202568488_nat_o(aa_fun2489310642977492936_nat_o(aTP_Lamm_wp(A3),R4),X4))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7015_pigeonhole__infinite__rel,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_int,R4: fun_Pr4738963024209204295_int_o] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( finite_finite_int(B3)
       => ( ! [X4: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A3)
             => ? [Xa3: int] :
                  ( member_int(Xa3,B3)
                  & aa_int_o(aa_Pro2912763917461432128_int_o(R4,X4),Xa3) ) )
         => ? [X4: int] :
              ( member_int(X4,B3)
              & ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_int6101145382336329604_nat_o(aa_fun1144309370076579528_nat_o(aTP_Lamm_wq(A3),R4),X4))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7016_not__finite__existsD,axiom,
    ! [Pa: fun_se4782206882984618170_nat_o] :
      ( ~ finite9047747110432174090at_nat(collec5514110066124741708at_nat(Pa))
     => ? [X_13: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(Pa,X_13) ) ).

% not_finite_existsD
tff(fact_7017_not__finite__existsD,axiom,
    ! [Pa: fun_se7249947736802449410_int_o] :
      ( ~ finite8744585540193469122nt_int(collec5210948495886036740nt_int(Pa))
     => ? [X_13: set_Pr958786334691620121nt_int] : aa_set6170482117830564987_int_o(Pa,X_13) ) ).

% not_finite_existsD
tff(fact_7018_not__finite__existsD,axiom,
    ! [Pa: fun_set_nat_o] :
      ( ~ finite1152437895449049373et_nat(collect_set_nat(Pa))
     => ? [X_13: set_nat] : aa_set_nat_o(Pa,X_13) ) ).

% not_finite_existsD
tff(fact_7019_not__finite__existsD,axiom,
    ! [Pa: fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,collect_nat(Pa))
     => ? [X_13: nat] : aa_nat_o(Pa,X_13) ) ).

% not_finite_existsD
tff(fact_7020_not__finite__existsD,axiom,
    ! [Pa: fun_int_o] :
      ( ~ finite_finite_int(collect_int(Pa))
     => ? [X_13: int] : aa_int_o(Pa,X_13) ) ).

% not_finite_existsD
tff(fact_7021_not__finite__existsD,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Pa))
     => ? [X_13: product_prod_nat_nat] : aa_Pro6150637604445964947_nat_o(Pa,X_13) ) ).

% not_finite_existsD
tff(fact_7022_in__finite__psubset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A3,B3),finite_psubset_nat)
    <=> ( aa_set_nat_o(ord_less_set_nat(A3),B3)
        & aa_set_nat_o(finite_finite_nat,B3) ) ) ).

% in_finite_psubset
tff(fact_7023_in__finite__psubset,axiom,
    ! [A3: set_int,B3: set_int] :
      ( member2572552093476627150et_int(produc6363374080413544029et_int(A3,B3),finite_psubset_int)
    <=> ( ord_less_set_int(A3,B3)
        & finite_finite_int(B3) ) ) ).

% in_finite_psubset
tff(fact_7024_in__finite__psubset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A3,B3),finite469560695537375940at_nat)
    <=> ( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(A3),B3)
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B3) ) ) ).

% in_finite_psubset
tff(fact_7025_even__set__encode__iff,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A3))
      <=> ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),A3) ) ) ).

% even_set_encode_iff
tff(fact_7026_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = $ite(
        M = zero_zero_nat,
        N,
        $ite(N = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_7027_even__sum__iff,axiom,
    ! [A3: set_Product_unit,F: fun_Product_unit_int] :
      ( finite4290736615968046902t_unit(A3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups8764306707065906943it_int(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite410649719033368117t_unit(collect_Product_unit(aa_fun693351410520992030unit_o(aTP_Lamm_wr(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_7028_even__sum__iff,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(F),A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(finite_card_nat,collect_nat(aa_fun2180201864171524266_nat_o(aTP_Lamm_ws(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_7029_even__sum__iff,axiom,
    ! [A3: set_Product_unit,F: fun_Product_unit_nat] :
      ( finite4290736615968046902t_unit(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups8766797177574957219it_nat(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite410649719033368117t_unit(collect_Product_unit(aa_fun9021389585855579330unit_o(aTP_Lamm_wt(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_7030_even__sum__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( finite_finite_int(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups4541462559716669496nt_nat(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_int(collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_wu(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_7031_even__sum__iff,axiom,
    ! [A3: set_Product_unit,F: fun_Pr1820225556744648189nteger] :
      ( finite4290736615968046902t_unit(A3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups5393439102872367408nteger(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite410649719033368117t_unit(collect_Product_unit(aa_fun7797430150634904527unit_o(aTP_Lamm_wv(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_7032_even__sum__iff,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(finite_card_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_ww(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_7033_even__sum__iff,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( finite_finite_int(A3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7873554091576472773nteger(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_int(collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_wx(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_7034_even__sum__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(finite_card_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_wy(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_7035_even__sum__iff,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( finite_finite_int(A3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups4538972089207619220nt_int(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_int(collect_int(aa_fun8155382727227203058_int_o(aTP_Lamm_wz(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_7036_even__sum__iff,axiom,
    ! [A3: set_set_nat,F: fun_set_nat_int] :
      ( finite1152437895449049373et_nat(A3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups8292507037921071086at_int(F,A3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_set_nat(collect_set_nat(aa_fun1141455960422277694_nat_o(aTP_Lamm_xa(A3),F)))) ) ) ).

% even_sum_iff
tff(fact_7037_drop__bit__rec,axiom,
    ! [N: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),A) = $ite(N = zero_zero_nat,A,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% drop_bit_rec
tff(fact_7038_drop__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(N),A) = $ite(N = zero_zero_nat,A,aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% drop_bit_rec
tff(fact_7039_drop__bit__rec,axiom,
    ! [N: nat,A: code_natural] :
      bit_se2751692689150723149atural(N,A) = $ite(N = zero_zero_nat,A,bit_se2751692689150723149atural(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))))) ).

% drop_bit_rec
tff(fact_7040_drop__bit__rec,axiom,
    ! [N: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),A) = $ite(N = zero_zero_nat,A,aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% drop_bit_rec
tff(fact_7041_finite__psubset__def,axiom,
    finite_psubset_nat = collec6662362479098859352et_nat(produc6247414631856714078_nat_o(aTP_Lamm_xb)) ).

% finite_psubset_def
tff(fact_7042_finite__psubset__def,axiom,
    finite_psubset_int = collec957716948307931664et_int(produc4109468873575309990_int_o(aTP_Lamm_xc)) ).

% finite_psubset_def
tff(fact_7043_finite__psubset__def,axiom,
    finite469560695537375940at_nat = collec6321179662152712658at_nat(produc410239310623530412_nat_o(aTP_Lamm_xd)) ).

% finite_psubset_def
tff(fact_7044_card__Collect__less__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fl,N))) = N ).

% card_Collect_less_nat
tff(fact_7045_drop__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),zero_zero_int) = zero_zero_int ).

% drop_bit_of_0
tff(fact_7046_drop__bit__of__0,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(N),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% drop_bit_of_0
tff(fact_7047_drop__bit__of__0,axiom,
    ! [N: nat] : bit_se2751692689150723149atural(N,zero_z2226904508553997617atural) = zero_z2226904508553997617atural ).

% drop_bit_of_0
tff(fact_7048_drop__bit__of__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),zero_zero_nat) = zero_zero_nat ).

% drop_bit_of_0
tff(fact_7049_drop__bit__drop__bit,axiom,
    ! [M: nat,N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),A) ).

% drop_bit_drop_bit
tff(fact_7050_of__bool__or__iff,axiom,
    ! [Pa: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (Pa)
        | (Q) )) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_o_int(zero_n2684676970156552555ol_int,(Pa))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_or_iff
tff(fact_7051_of__bool__or__iff,axiom,
    ! [Pa: $o,Q: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (Pa)
        | (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ).

% of_bool_or_iff
tff(fact_7052_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_7053_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_7054_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_7055_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_7056_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_7057_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_7058_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_7059_card__Collect__le__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_hu,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_7060_max__0__1_I4_J,axiom,
    ! [Xa: num] : ord_max_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,Xa),zero_z2226904508553997617atural) = aa_num_Code_natural(numera5444537566228673987atural,Xa) ).

% max_0_1(4)
tff(fact_7061_max__0__1_I4_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,Xa)),zero_zero_int) = aa_num_int(numeral_numeral_int,Xa) ).

% max_0_1(4)
tff(fact_7062_max__0__1_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xa)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,Xa) ).

% max_0_1(4)
tff(fact_7063_max__0__1_I4_J,axiom,
    ! [Xa: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,Xa)),zero_z3403309356797280102nteger) = aa_num_Code_integer(numera6620942414471956472nteger,Xa) ).

% max_0_1(4)
tff(fact_7064_max__0__1_I4_J,axiom,
    ! [Xa: num] : ord_max_rat(numeral_numeral_rat(Xa),zero_zero_rat) = numeral_numeral_rat(Xa) ).

% max_0_1(4)
tff(fact_7065_max__0__1_I3_J,axiom,
    ! [Xa: num] : ord_max_Code_natural(zero_z2226904508553997617atural,aa_num_Code_natural(numera5444537566228673987atural,Xa)) = aa_num_Code_natural(numera5444537566228673987atural,Xa) ).

% max_0_1(3)
tff(fact_7066_max__0__1_I3_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,Xa)) = aa_num_int(numeral_numeral_int,Xa) ).

% max_0_1(3)
tff(fact_7067_max__0__1_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Xa)) = aa_num_nat(numeral_numeral_nat,Xa) ).

% max_0_1(3)
tff(fact_7068_max__0__1_I3_J,axiom,
    ! [Xa: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,Xa)) = aa_num_Code_integer(numera6620942414471956472nteger,Xa) ).

% max_0_1(3)
tff(fact_7069_max__0__1_I3_J,axiom,
    ! [Xa: num] : ord_max_rat(zero_zero_rat,numeral_numeral_rat(Xa)) = numeral_numeral_rat(Xa) ).

% max_0_1(3)
tff(fact_7070_max__0__1_I1_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) = one_one_Code_integer ).

% max_0_1(1)
tff(fact_7071_max__0__1_I1_J,axiom,
    ord_max_rat(zero_zero_rat,one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_7072_max__0__1_I1_J,axiom,
    ord_max_Code_natural(zero_z2226904508553997617atural,one_one_Code_natural) = one_one_Code_natural ).

% max_0_1(1)
tff(fact_7073_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_7074_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_7075_max__0__1_I2_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,one_one_Code_integer),zero_z3403309356797280102nteger) = one_one_Code_integer ).

% max_0_1(2)
tff(fact_7076_max__0__1_I2_J,axiom,
    ord_max_rat(one_one_rat,zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_7077_max__0__1_I2_J,axiom,
    ord_max_Code_natural(one_one_Code_natural,zero_z2226904508553997617atural) = one_one_Code_natural ).

% max_0_1(2)
tff(fact_7078_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_7079_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_7080_max__0__1_I6_J,axiom,
    ! [Xa: num] : ord_max_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,Xa),one_one_Code_natural) = aa_num_Code_natural(numera5444537566228673987atural,Xa) ).

% max_0_1(6)
tff(fact_7081_max__0__1_I6_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,Xa)),one_one_int) = aa_num_int(numeral_numeral_int,Xa) ).

% max_0_1(6)
tff(fact_7082_max__0__1_I6_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xa)),one_one_nat) = aa_num_nat(numeral_numeral_nat,Xa) ).

% max_0_1(6)
tff(fact_7083_max__0__1_I6_J,axiom,
    ! [Xa: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,Xa)),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,Xa) ).

% max_0_1(6)
tff(fact_7084_max__0__1_I6_J,axiom,
    ! [Xa: num] : ord_max_rat(numeral_numeral_rat(Xa),one_one_rat) = numeral_numeral_rat(Xa) ).

% max_0_1(6)
tff(fact_7085_max__0__1_I5_J,axiom,
    ! [Xa: num] : ord_max_Code_natural(one_one_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,Xa)) = aa_num_Code_natural(numera5444537566228673987atural,Xa) ).

% max_0_1(5)
tff(fact_7086_max__0__1_I5_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),aa_num_int(numeral_numeral_int,Xa)) = aa_num_int(numeral_numeral_int,Xa) ).

% max_0_1(5)
tff(fact_7087_max__0__1_I5_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xa)) = aa_num_nat(numeral_numeral_nat,Xa) ).

% max_0_1(5)
tff(fact_7088_max__0__1_I5_J,axiom,
    ! [Xa: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,Xa)) = aa_num_Code_integer(numera6620942414471956472nteger,Xa) ).

% max_0_1(5)
tff(fact_7089_max__0__1_I5_J,axiom,
    ! [Xa: num] : ord_max_rat(one_one_rat,numeral_numeral_rat(Xa)) = numeral_numeral_rat(Xa) ).

% max_0_1(5)
tff(fact_7090_card_Oinfinite,axiom,
    ! [A3: set_Product_unit] :
      ( ~ finite4290736615968046902t_unit(A3)
     => ( finite410649719033368117t_unit(A3) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_7091_card_Oinfinite,axiom,
    ! [A3: set_set_nat] :
      ( ~ finite1152437895449049373et_nat(A3)
     => ( finite_card_set_nat(A3) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_7092_card_Oinfinite,axiom,
    ! [A3: set_list_nat] :
      ( ~ finite8100373058378681591st_nat(A3)
     => ( finite_card_list_nat(A3) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_7093_card_Oinfinite,axiom,
    ! [A3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_nat(finite_card_nat,A3) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_7094_card_Oinfinite,axiom,
    ! [A3: set_int] :
      ( ~ finite_finite_int(A3)
     => ( finite_card_int(A3) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_7095_card_Oinfinite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_7096_drop__bit__of__bool,axiom,
    ! [N: nat,B: $o] :
      aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_o_int(zero_n2684676970156552555ol_int,(B))) = aa_o_int(zero_n2684676970156552555ol_int,
        ( ( N = zero_zero_nat )
        & (B) )) ).

% drop_bit_of_bool
tff(fact_7097_drop__bit__of__bool,axiom,
    ! [N: nat,B: $o] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_o_nat(zero_n2687167440665602831ol_nat,(B))) = aa_o_nat(zero_n2687167440665602831ol_nat,
        ( ( N = zero_zero_nat )
        & (B) )) ).

% drop_bit_of_bool
tff(fact_7098_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,N = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_7099_prod__constant,axiom,
    ! [Y3: nat,A3: set_Product_unit] : groups396421249026171384it_nat(aTP_Lamm_xe(Y3),A3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),finite410649719033368117t_unit(A3)) ).

% prod_constant
tff(fact_7100_prod__constant,axiom,
    ! [Y3: nat,A3: set_set_nat] : groups4248547760180025341at_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_xf,Y3),A3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),finite_card_set_nat(A3)) ).

% prod_constant
tff(fact_7101_prod__constant,axiom,
    ! [Y3: nat,A3: set_list_nat] : groups2907647131375434839at_nat(aTP_Lamm_xg(Y3),A3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),finite_card_list_nat(A3)) ).

% prod_constant
tff(fact_7102_prod__constant,axiom,
    ! [Y3: int,A3: set_Product_unit] : groups393930778517121108it_int(aTP_Lamm_xh(Y3),A3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),finite410649719033368117t_unit(A3)) ).

% prod_constant
tff(fact_7103_prod__constant,axiom,
    ! [Y3: int,A3: set_set_nat] : groups4246057289670975065at_int(aa_int2448287375063835929at_int(aTP_Lamm_xi,Y3),A3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),finite_card_set_nat(A3)) ).

% prod_constant
tff(fact_7104_prod__constant,axiom,
    ! [Y3: int,A3: set_list_nat] : groups2905156660866384563at_int(aTP_Lamm_xj(Y3),A3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),finite_card_list_nat(A3)) ).

% prod_constant
tff(fact_7105_prod__constant,axiom,
    ! [Y3: nat,A3: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ar,Y3)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),aa_set_nat_nat(finite_card_nat,A3)) ).

% prod_constant
tff(fact_7106_prod__constant,axiom,
    ! [Y3: int,A3: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_xk,Y3)),A3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_set_nat_nat(finite_card_nat,A3)) ).

% prod_constant
tff(fact_7107_prod__constant,axiom,
    ! [Y3: int,A3: set_int] : groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_ft,Y3),A3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),finite_card_int(A3)) ).

% prod_constant
tff(fact_7108_drop__bit__of__1,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(N),one_one_Code_integer) = zero_n356916108424825756nteger(N = zero_zero_nat) ).

% drop_bit_of_1
tff(fact_7109_drop__bit__of__1,axiom,
    ! [N: nat] : bit_se2751692689150723149atural(N,one_one_Code_natural) = zero_n8403883297036319079atural(N = zero_zero_nat) ).

% drop_bit_of_1
tff(fact_7110_drop__bit__of__1,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,N = zero_zero_nat) ).

% drop_bit_of_1
tff(fact_7111_drop__bit__of__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,N = zero_zero_nat) ).

% drop_bit_of_1
tff(fact_7112_sum__constant,axiom,
    ! [Y3: code_integer,A3: set_nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_xl(Y3)),A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(finite_card_nat,A3))),Y3) ).

% sum_constant
tff(fact_7113_sum__constant,axiom,
    ! [Y3: code_integer,A3: set_Product_unit] : groups5393439102872367408nteger(aTP_Lamm_xm(Y3),A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite410649719033368117t_unit(A3))),Y3) ).

% sum_constant
tff(fact_7114_sum__constant,axiom,
    ! [Y3: rat,A3: set_nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_xn(Y3)),A3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A3))),Y3) ).

% sum_constant
tff(fact_7115_sum__constant,axiom,
    ! [Y3: rat,A3: set_Product_unit] : groups8131667117488461483it_rat(aTP_Lamm_xo(Y3),A3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite410649719033368117t_unit(A3))),Y3) ).

% sum_constant
tff(fact_7116_sum__constant,axiom,
    ! [Y3: int,A3: set_nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_xk,Y3)),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,A3))),Y3) ).

% sum_constant
tff(fact_7117_sum__constant,axiom,
    ! [Y3: int,A3: set_Product_unit] : groups8764306707065906943it_int(aTP_Lamm_xh(Y3),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite410649719033368117t_unit(A3))),Y3) ).

% sum_constant
tff(fact_7118_sum__constant,axiom,
    ! [Y3: nat,A3: set_Product_unit] : groups8766797177574957219it_nat(aTP_Lamm_xe(Y3),A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite410649719033368117t_unit(A3))),Y3) ).

% sum_constant
tff(fact_7119_sum__constant,axiom,
    ! [Y3: nat,A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ar,Y3)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(finite_card_nat,A3))),Y3) ).

% sum_constant
tff(fact_7120_sum__constant,axiom,
    ! [Y3: int,A3: set_int] : groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_ft,Y3),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_int(A3))),Y3) ).

% sum_constant
tff(fact_7121_sum__constant,axiom,
    ! [Y3: code_integer,A3: set_set_nat] : groups9190459664516455967nteger(aTP_Lamm_xp(Y3),A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_set_nat(A3))),Y3) ).

% sum_constant
tff(fact_7122_max__add__distrib__left,axiom,
    ! [Xa: rat,Y3: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,ord_max_rat(Xa,Y3)),Z) = ord_max_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Z),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y3),Z)) ).

% max_add_distrib_left
tff(fact_7123_max__add__distrib__left,axiom,
    ! [Xa: nat,Y3: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y3)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y3),Z)) ).

% max_add_distrib_left
tff(fact_7124_max__add__distrib__left,axiom,
    ! [Xa: int,Y3: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xa),Y3)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),Z)) ).

% max_add_distrib_left
tff(fact_7125_max__add__distrib__right,axiom,
    ! [Xa: rat,Y3: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),ord_max_rat(Y3,Z)) = ord_max_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Z)) ).

% max_add_distrib_right
tff(fact_7126_max__add__distrib__right,axiom,
    ! [Xa: nat,Y3: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y3),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Z)) ).

% max_add_distrib_right
tff(fact_7127_max__add__distrib__right,axiom,
    ! [Xa: int,Y3: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),aa_int_int(aa_int_fun_int_int(ord_max_int,Y3),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Z)) ).

% max_add_distrib_right
tff(fact_7128_max__diff__distrib__left,axiom,
    ! [Xa: int,Y3: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xa),Y3)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y3),Z)) ).

% max_diff_distrib_left
tff(fact_7129_max__diff__distrib__left,axiom,
    ! [Xa: code_integer,Y3: code_integer,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,Xa),Y3)),Z) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xa),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Y3),Z)) ).

% max_diff_distrib_left
tff(fact_7130_max__diff__distrib__left,axiom,
    ! [Xa: rat,Y3: rat,Z: rat] : minus_minus_rat(ord_max_rat(Xa,Y3),Z) = ord_max_rat(minus_minus_rat(Xa,Z),minus_minus_rat(Y3,Z)) ).

% max_diff_distrib_left
tff(fact_7131_of__int__max,axiom,
    ! [Xa: int,Y3: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(ord_max_int,Xa),Y3)) = ord_max_rat(aa_int_rat(ring_1_of_int_rat,Xa),aa_int_rat(ring_1_of_int_rat,Y3)) ).

% of_int_max
tff(fact_7132_of__int__max,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xa),Y3)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(ring_1_of_int_int,Xa)),aa_int_int(ring_1_of_int_int,Y3)) ).

% of_int_max
tff(fact_7133_of__nat__max,axiom,
    ! [Xa: nat,Y3: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y3)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(semiri1314217659103216013at_int,Y3)) ).

% of_nat_max
tff(fact_7134_of__nat__max,axiom,
    ! [Xa: nat,Y3: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),aa_nat_nat(semiri1316708129612266289at_nat,Y3)) ).

% of_nat_max
tff(fact_7135_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)) ).

% nat_add_max_right
tff(fact_7136_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q2)) ).

% nat_add_max_left
tff(fact_7137_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_max_right
tff(fact_7138_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_max_left
tff(fact_7139_max__def__raw,axiom,
    ! [X: set_nat,Xa3: set_nat] :
      ord_max_set_nat(X,Xa3) = $ite(aa_set_nat_o(ord_less_eq_set_nat(X),Xa3),Xa3,X) ).

% max_def_raw
tff(fact_7140_max__def__raw,axiom,
    ! [X: rat,Xa3: rat] :
      ord_max_rat(X,Xa3) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa3),Xa3,X) ).

% max_def_raw
tff(fact_7141_max__def__raw,axiom,
    ! [X: num,Xa3: num] :
      aa_num_num(ord_max_num(X),Xa3) = $ite(aa_num_o(ord_less_eq_num(X),Xa3),Xa3,X) ).

% max_def_raw
tff(fact_7142_max__def__raw,axiom,
    ! [X: nat,Xa3: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Xa3) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa3),Xa3,X) ).

% max_def_raw
tff(fact_7143_max__def__raw,axiom,
    ! [X: int,Xa3: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,X),Xa3) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa3),Xa3,X) ).

% max_def_raw
tff(fact_7144_n__subsets,axiom,
    ! [A3: set_Pr958786334691620121nt_int,K: nat] :
      ( aa_set6170482117830564987_int_o(finite2998713641127702882nt_int,A3)
     => ( finite4053189226111446337nt_int(collec5210948495886036740nt_int(aa_nat4194932087953772816_int_o(aTP_Lamm_xq(A3),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_set4753215920113914285nt_nat(finite6756421564338198497nt_int,A3)),K) ) ) ).

% n_subsets
tff(fact_7145_n__subsets,axiom,
    ! [A3: set_Product_unit,K: nat] :
      ( finite4290736615968046902t_unit(A3)
     => ( finite22775419188893461t_unit(collec7787489603430924120t_unit(aa_nat5385523842031975612unit_o(aTP_Lamm_xr(A3),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite410649719033368117t_unit(A3)),K) ) ) ).

% n_subsets
tff(fact_7146_n__subsets,axiom,
    ! [A3: set_set_nat,K: nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( finite1149291290879098388et_nat(collect_set_set_nat(aa_nat5160323483289836711_nat_o(aTP_Lamm_xs(A3),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_set_nat(A3)),K) ) ) ).

% n_subsets
tff(fact_7147_n__subsets,axiom,
    ! [A3: set_list_nat,K: nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( finite2364142230527598318st_nat(collect_set_list_nat(aa_nat7721107121239158477_nat_o(aTP_Lamm_xt(A3),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_list_nat(A3)),K) ) ) ).

% n_subsets
tff(fact_7148_n__subsets,axiom,
    ! [A3: set_int,K: nat] :
      ( finite_finite_int(A3)
     => ( finite_card_set_int(collect_set_int(aa_nat_fun_set_int_o(aTP_Lamm_xu(A3),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_int(A3)),K) ) ) ).

% n_subsets
tff(fact_7149_n__subsets,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( finite4356350796350151305at_nat(collec5514110066124741708at_nat(aa_nat1727191234135941576_nat_o(aTP_Lamm_xv(A3),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),K) ) ) ).

% n_subsets
tff(fact_7150_n__subsets,axiom,
    ! [A3: set_nat,K: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_xw(A3),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_set_nat_nat(finite_card_nat,A3)),K) ) ) ).

% n_subsets
tff(fact_7151_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_7152_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [N: nat,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),A) = A )
    <=> ( aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(N),A) = zero_z3403309356797280102nteger ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_7153_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se569199155075624693atural(N,A) = A )
    <=> ( bit_se2751692689150723149atural(N,A) = zero_z2226904508553997617atural ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_7154_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [N: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),A) = zero_zero_nat ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_7155_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [N: nat,A: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = A )
    <=> ( aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),A) = zero_zero_int ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_7156_take__bit__drop__bit,axiom,
    ! [M: nat,N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),A)) ).

% take_bit_drop_bit
tff(fact_7157_take__bit__drop__bit,axiom,
    ! [M: nat,N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),A)) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),A)) ).

% take_bit_drop_bit
tff(fact_7158_sum__multicount__gen,axiom,
    ! [S: set_Product_unit,T: set_Product_unit,R4: fun_Pr1520749766685617582unit_o,K: fun_Product_unit_nat] :
      ( finite4290736615968046902t_unit(S)
     => ( finite4290736615968046902t_unit(T)
       => ( ! [X4: product_unit] :
              ( member_Product_unit(X4,T)
             => ( finite410649719033368117t_unit(collect_Product_unit(aa_Pro4895447120693452255unit_o(aa_fun7194487344118582332unit_o(aTP_Lamm_xx(S),R4),X4))) = aa_Product_unit_nat(K,X4) ) )
         => ( groups8766797177574957219it_nat(aa_fun2167693721382921342it_nat(aTP_Lamm_xz(T),R4),S) = groups8766797177574957219it_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_7159_sum__multicount__gen,axiom,
    ! [S: set_Product_unit,T: set_int,R4: fun_Pr2157672562078508475_int_o,K: fun_int_nat] :
      ( finite4290736615968046902t_unit(S)
     => ( finite_finite_int(T)
       => ( ! [X4: int] :
              ( member_int(X4,T)
             => ( finite410649719033368117t_unit(collect_Product_unit(aa_int8968792772080532472unit_o(aa_fun249790601762366384unit_o(aTP_Lamm_ya(S),R4),X4))) = aa_int_nat(K,X4) ) )
         => ( groups8766797177574957219it_nat(aa_fun4006079477318474295it_nat(aTP_Lamm_yc(T),R4),S) = groups4541462559716669496nt_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_7160_sum__multicount__gen,axiom,
    ! [S: set_int,T: set_Product_unit,R4: fun_in8984369272479023031unit_o,K: fun_Product_unit_nat] :
      ( finite_finite_int(S)
     => ( finite4290736615968046902t_unit(T)
       => ( ! [X4: product_unit] :
              ( member_Product_unit(X4,T)
             => ( finite_card_int(collect_int(aa_Pro8249700398669690036_int_o(aa_fun7970191300016449456_int_o(aTP_Lamm_yd(S),R4),X4))) = aa_Product_unit_nat(K,X4) ) )
         => ( groups4541462559716669496nt_nat(aa_fun5797050646745590254nt_nat(aTP_Lamm_yf(T),R4),S) = groups8766797177574957219it_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_7161_sum__multicount__gen,axiom,
    ! [S: set_int,T: set_int,R4: fun_int_fun_int_o,K: fun_int_nat] :
      ( finite_finite_int(S)
     => ( finite_finite_int(T)
       => ( ! [X4: int] :
              ( member_int(X4,T)
             => ( finite_card_int(collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_rb(S),R4),X4))) = aa_int_nat(K,X4) ) )
         => ( groups4541462559716669496nt_nat(aa_fun4460091444687325999nt_nat(aTP_Lamm_yg(T),R4),S) = groups4541462559716669496nt_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_7162_sum__multicount__gen,axiom,
    ! [S: set_Product_unit,T: set_nat,R4: fun_Pr4402817529748954391_nat_o,K: fun_nat_nat] :
      ( finite4290736615968046902t_unit(S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),T)
             => ( finite410649719033368117t_unit(collect_Product_unit(aa_nat5955485856412068252unit_o(aa_fun3452303748553993392unit_o(aTP_Lamm_yh(S),R4),X4))) = aa_nat_nat(K,X4) ) )
         => ( groups8766797177574957219it_nat(aa_fun1580523165164119515it_nat(aTP_Lamm_yj(T),R4),S) = aa_set_nat_nat(groups3542108847815614940at_nat(K),T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_7163_sum__multicount__gen,axiom,
    ! [S: set_int,T: set_nat,R4: fun_int_fun_nat_o,K: fun_nat_nat] :
      ( finite_finite_int(S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),T)
             => ( finite_card_int(collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_qk(S),R4),X4))) = aa_nat_nat(K,X4) ) )
         => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aTP_Lamm_yk(T),R4),S) = aa_set_nat_nat(groups3542108847815614940at_nat(K),T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_7164_sum__multicount__gen,axiom,
    ! [S: set_nat,T: set_Product_unit,R4: fun_na5971062356810558811unit_o,K: fun_Product_unit_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( finite4290736615968046902t_unit(T)
       => ( ! [X4: product_unit] :
              ( member_Product_unit(X4,T)
             => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_Pro847370707247602192_nat_o(aa_fun6915127332797442224_nat_o(aTP_Lamm_yl(S),R4),X4))) = aa_Product_unit_nat(K,X4) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8994848686300240182at_nat(aTP_Lamm_yn(T),R4)),S) = groups8766797177574957219it_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_7165_sum__multicount__gen,axiom,
    ! [S: set_nat,T: set_int,R4: fun_nat_fun_int_o,K: fun_int_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( finite_finite_int(T)
       => ( ! [X4: int] :
              ( member_int(X4,T)
             => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_qo(S),R4),X4))) = aa_int_nat(K,X4) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8666973529875159599at_nat(aTP_Lamm_yo(T),R4)),S) = groups4541462559716669496nt_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_7166_sum__multicount__gen,axiom,
    ! [S: set_nat,T: set_nat,R4: fun_nat_fun_nat_o,K: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),T)
             => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_qv(S),R4),X4))) = aa_nat_nat(K,X4) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8857679445107423627at_nat(aTP_Lamm_yp(T),R4)),S) = aa_set_nat_nat(groups3542108847815614940at_nat(K),T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_7167_sum__multicount__gen,axiom,
    ! [S: set_Product_unit,T: set_set_nat,R4: fun_Pr1646868699942875873_nat_o,K: fun_set_nat_nat] :
      ( finite4290736615968046902t_unit(S)
     => ( finite1152437895449049373et_nat(T)
       => ( ! [X4: set_nat] :
              ( member_set_nat(X4,T)
             => ( finite410649719033368117t_unit(collect_Product_unit(aa_set3064922119574097746unit_o(aa_fun6872337739698859568unit_o(aTP_Lamm_yq(S),R4),X4))) = aa_set_nat_nat(K,X4) ) )
         => ( groups8766797177574957219it_nat(aa_fun6562703090490685585it_nat(aTP_Lamm_ys(T),R4),S) = groups8294997508430121362at_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_7168_card__eq__sum,axiom,
    ! [A3: set_Product_unit] : finite410649719033368117t_unit(A3) = groups8766797177574957219it_nat(aTP_Lamm_yt,A3) ).

% card_eq_sum
tff(fact_7169_card__eq__sum,axiom,
    ! [A3: set_set_nat] : finite_card_set_nat(A3) = groups8294997508430121362at_nat(aTP_Lamm_yu,A3) ).

% card_eq_sum
tff(fact_7170_card__eq__sum,axiom,
    ! [A3: set_list_nat] : finite_card_list_nat(A3) = groups4396056296759096172at_nat(aTP_Lamm_yv,A3) ).

% card_eq_sum
tff(fact_7171_card__eq__sum,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(finite_card_nat,A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kq),A3) ).

% card_eq_sum
tff(fact_7172_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_yw,N)),M) ).

% max_Suc2
tff(fact_7173_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_yx,N)),M) ).

% max_Suc1
tff(fact_7174_card__ge__0__finite,axiom,
    ! [A3: set_Product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite410649719033368117t_unit(A3))
     => finite4290736615968046902t_unit(A3) ) ).

% card_ge_0_finite
tff(fact_7175_card__ge__0__finite,axiom,
    ! [A3: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_set_nat(A3))
     => finite1152437895449049373et_nat(A3) ) ).

% card_ge_0_finite
tff(fact_7176_card__ge__0__finite,axiom,
    ! [A3: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_list_nat(A3))
     => finite8100373058378681591st_nat(A3) ) ).

% card_ge_0_finite
tff(fact_7177_card__ge__0__finite,axiom,
    ! [A3: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A3))
     => aa_set_nat_o(finite_finite_nat,A3) ) ).

% card_ge_0_finite
tff(fact_7178_card__ge__0__finite,axiom,
    ! [A3: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A3))
     => finite_finite_int(A3) ) ).

% card_ge_0_finite
tff(fact_7179_card__ge__0__finite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3))
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3) ) ).

% card_ge_0_finite
tff(fact_7180_card__less__Suc2,axiom,
    ! [M2: set_nat,I: nat] :
      ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),M2)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_yy(M2),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_yz(M2),I))) ) ) ).

% card_less_Suc2
tff(fact_7181_card__less__Suc,axiom,
    ! [M2: set_nat,I: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),M2)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_yy(M2),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_yz(M2),I))) ) ) ).

% card_less_Suc
tff(fact_7182_card__less,axiom,
    ! [M2: set_nat,I: nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),M2)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_yz(M2),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_7183_sum__Suc,axiom,
    ! [F: fun_Product_unit_nat,A3: set_Product_unit] : groups8766797177574957219it_nat(aTP_Lamm_za(F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8766797177574957219it_nat(F,A3)),finite410649719033368117t_unit(A3)) ).

% sum_Suc
tff(fact_7184_sum__Suc,axiom,
    ! [F: fun_set_nat_nat,A3: set_set_nat] : groups8294997508430121362at_nat(aTP_Lamm_zb(F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8294997508430121362at_nat(F,A3)),finite_card_set_nat(A3)) ).

% sum_Suc
tff(fact_7185_sum__Suc,axiom,
    ! [F: fun_list_nat_nat,A3: set_list_nat] : groups4396056296759096172at_nat(aTP_Lamm_zc(F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4396056296759096172at_nat(F,A3)),finite_card_list_nat(A3)) ).

% sum_Suc
tff(fact_7186_sum__Suc,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_zd(F)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(finite_card_nat,A3)) ).

% sum_Suc
tff(fact_7187_sum__multicount,axiom,
    ! [S2: set_Product_unit,T3: set_Product_unit,R4: fun_Pr1520749766685617582unit_o,K: nat] :
      ( finite4290736615968046902t_unit(S2)
     => ( finite4290736615968046902t_unit(T3)
       => ( ! [X4: product_unit] :
              ( member_Product_unit(X4,T3)
             => ( finite410649719033368117t_unit(collect_Product_unit(aa_Pro4895447120693452255unit_o(aa_fun7194487344118582332unit_o(aTP_Lamm_xx(S2),R4),X4))) = K ) )
         => ( groups8766797177574957219it_nat(aa_fun2167693721382921342it_nat(aTP_Lamm_xz(T3),R4),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite410649719033368117t_unit(T3)) ) ) ) ) ).

% sum_multicount
tff(fact_7188_sum__multicount,axiom,
    ! [S2: set_Product_unit,T3: set_nat,R4: fun_Pr4402817529748954391_nat_o,K: nat] :
      ( finite4290736615968046902t_unit(S2)
     => ( aa_set_nat_o(finite_finite_nat,T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),T3)
             => ( finite410649719033368117t_unit(collect_Product_unit(aa_nat5955485856412068252unit_o(aa_fun3452303748553993392unit_o(aTP_Lamm_yh(S2),R4),X4))) = K ) )
         => ( groups8766797177574957219it_nat(aa_fun1580523165164119515it_nat(aTP_Lamm_yj(T3),R4),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_set_nat_nat(finite_card_nat,T3)) ) ) ) ) ).

% sum_multicount
tff(fact_7189_sum__multicount,axiom,
    ! [S2: set_Product_unit,T3: set_int,R4: fun_Pr2157672562078508475_int_o,K: nat] :
      ( finite4290736615968046902t_unit(S2)
     => ( finite_finite_int(T3)
       => ( ! [X4: int] :
              ( member_int(X4,T3)
             => ( finite410649719033368117t_unit(collect_Product_unit(aa_int8968792772080532472unit_o(aa_fun249790601762366384unit_o(aTP_Lamm_ya(S2),R4),X4))) = K ) )
         => ( groups8766797177574957219it_nat(aa_fun4006079477318474295it_nat(aTP_Lamm_yc(T3),R4),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_int(T3)) ) ) ) ) ).

% sum_multicount
tff(fact_7190_sum__multicount,axiom,
    ! [S2: set_int,T3: set_Product_unit,R4: fun_in8984369272479023031unit_o,K: nat] :
      ( finite_finite_int(S2)
     => ( finite4290736615968046902t_unit(T3)
       => ( ! [X4: product_unit] :
              ( member_Product_unit(X4,T3)
             => ( finite_card_int(collect_int(aa_Pro8249700398669690036_int_o(aa_fun7970191300016449456_int_o(aTP_Lamm_yd(S2),R4),X4))) = K ) )
         => ( groups4541462559716669496nt_nat(aa_fun5797050646745590254nt_nat(aTP_Lamm_yf(T3),R4),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite410649719033368117t_unit(T3)) ) ) ) ) ).

% sum_multicount
tff(fact_7191_sum__multicount,axiom,
    ! [S2: set_int,T3: set_nat,R4: fun_int_fun_nat_o,K: nat] :
      ( finite_finite_int(S2)
     => ( aa_set_nat_o(finite_finite_nat,T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),T3)
             => ( finite_card_int(collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_qk(S2),R4),X4))) = K ) )
         => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aTP_Lamm_yk(T3),R4),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_set_nat_nat(finite_card_nat,T3)) ) ) ) ) ).

% sum_multicount
tff(fact_7192_sum__multicount,axiom,
    ! [S2: set_int,T3: set_int,R4: fun_int_fun_int_o,K: nat] :
      ( finite_finite_int(S2)
     => ( finite_finite_int(T3)
       => ( ! [X4: int] :
              ( member_int(X4,T3)
             => ( finite_card_int(collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_rb(S2),R4),X4))) = K ) )
         => ( groups4541462559716669496nt_nat(aa_fun4460091444687325999nt_nat(aTP_Lamm_yg(T3),R4),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_int(T3)) ) ) ) ) ).

% sum_multicount
tff(fact_7193_sum__multicount,axiom,
    ! [S2: set_nat,T3: set_Product_unit,R4: fun_na5971062356810558811unit_o,K: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( finite4290736615968046902t_unit(T3)
       => ( ! [X4: product_unit] :
              ( member_Product_unit(X4,T3)
             => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_Pro847370707247602192_nat_o(aa_fun6915127332797442224_nat_o(aTP_Lamm_yl(S2),R4),X4))) = K ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8994848686300240182at_nat(aTP_Lamm_yn(T3),R4)),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite410649719033368117t_unit(T3)) ) ) ) ) ).

% sum_multicount
tff(fact_7194_sum__multicount,axiom,
    ! [S2: set_nat,T3: set_nat,R4: fun_nat_fun_nat_o,K: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(finite_finite_nat,T3)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),T3)
             => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_qv(S2),R4),X4))) = K ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8857679445107423627at_nat(aTP_Lamm_yp(T3),R4)),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_set_nat_nat(finite_card_nat,T3)) ) ) ) ) ).

% sum_multicount
tff(fact_7195_sum__multicount,axiom,
    ! [S2: set_nat,T3: set_int,R4: fun_nat_fun_int_o,K: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( finite_finite_int(T3)
       => ( ! [X4: int] :
              ( member_int(X4,T3)
             => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_qo(S2),R4),X4))) = K ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8666973529875159599at_nat(aTP_Lamm_yo(T3),R4)),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_int(T3)) ) ) ) ) ).

% sum_multicount
tff(fact_7196_sum__multicount,axiom,
    ! [S2: set_Product_unit,T3: set_set_nat,R4: fun_Pr1646868699942875873_nat_o,K: nat] :
      ( finite4290736615968046902t_unit(S2)
     => ( finite1152437895449049373et_nat(T3)
       => ( ! [X4: set_nat] :
              ( member_set_nat(X4,T3)
             => ( finite410649719033368117t_unit(collect_Product_unit(aa_set3064922119574097746unit_o(aa_fun6872337739698859568unit_o(aTP_Lamm_yq(S2),R4),X4))) = K ) )
         => ( groups8766797177574957219it_nat(aa_fun6562703090490685585it_nat(aTP_Lamm_ys(T3),R4),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_set_nat(T3)) ) ) ) ) ).

% sum_multicount
tff(fact_7197_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: int,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),A) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_7198_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer)) = aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(N),A) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_7199_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: code_natural,N: nat] : divide5121882707175180666atural(A,bit_se6611745700429515170atural(N,one_one_Code_natural)) = bit_se2751692689150723149atural(N,A) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_7200_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),A) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_7201_bits__ident,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),A))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) = A ).

% bits_ident
tff(fact_7202_bits__ident,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),A))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)) = A ).

% bits_ident
tff(fact_7203_subset__card__intvl__is__intvl,axiom,
    ! [A3: set_nat,K: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),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,A3))))
     => ( A3 = 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,A3))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_7204_set__encode__inf,axiom,
    ! [A3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_nat(nat_set_encode,A3) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_7205_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_Product_unit] :
      ( finite4290736615968046902t_unit(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(A3)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X3: product_unit] :
            ( member_Product_unit(X3,A3)
           => ! [Xa2: product_unit] :
                ( member_Product_unit(Xa2,A3)
               => ( X3 = Xa2 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_7206_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_set_nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A3)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X3: set_nat] :
            ( member_set_nat(X3,A3)
           => ! [Xa2: set_nat] :
                ( member_set_nat(Xa2,A3)
               => ( X3 = Xa2 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_7207_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A3)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X3: list_nat] :
            ( member_list_nat(X3,A3)
           => ! [Xa2: list_nat] :
                ( member_list_nat(Xa2,A3)
               => ( X3 = Xa2 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_7208_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X3: nat] :
            ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),A3)
           => ! [Xa2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa2),A3)
               => ( X3 = Xa2 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_7209_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_int] :
      ( finite_finite_int(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A3)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X3: int] :
            ( member_int(X3,A3)
           => ! [Xa2: int] :
                ( member_int(Xa2,A3)
               => ( X3 = Xa2 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_7210_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A3)
           => ! [Xa2: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa2),A3)
               => ( X3 = Xa2 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_7211_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set_nat,N: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N5)),N) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_7212_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_av),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_av),S2)) ).

% card_sum_le_nat_sum
tff(fact_7213_prod__le__power,axiom,
    ! [A3: set_int,F: fun_int_Code_integer,N: code_integer,K: nat] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I2))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I2)),N) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A3)),K)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),N)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F,A3)),aa_nat_Code_integer(power_8256067586552552935nteger(N),K)) ) ) ) ).

% prod_le_power
tff(fact_7214_prod__le__power,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer,N: code_integer,K: nat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A3)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I2))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,I2)),N) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),K)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),N)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A3)),aa_nat_Code_integer(power_8256067586552552935nteger(N),K)) ) ) ) ).

% prod_le_power
tff(fact_7215_prod__le__power,axiom,
    ! [A3: set_Product_unit,F: fun_Pr1820225556744648189nteger,N: code_integer,K: nat] :
      ( ! [I2: product_unit] :
          ( member_Product_unit(I2,A3)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Pro1705297278780387758nteger(F,I2))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Pro1705297278780387758nteger(F,I2)),N) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(A3)),K)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),N)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups8185359811944078981nteger(F,A3)),aa_nat_Code_integer(power_8256067586552552935nteger(N),K)) ) ) ) ).

% prod_le_power
tff(fact_7216_prod__le__power,axiom,
    ! [A3: set_int,F: fun_int_Code_natural,N: code_natural,K: nat] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,I2))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_int_Code_natural(F,I2)),N) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A3)),K)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),N)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups2650699495083094267atural(F,A3)),aa_nat_Code_natural(power_7079662738309270450atural(N),K)) ) ) ) ).

% prod_le_power
tff(fact_7217_prod__le__power,axiom,
    ! [A3: set_nat,F: fun_nat_Code_natural,N: code_natural,K: nat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A3)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,I2))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(F,I2)),N) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),K)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),N)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups2279045934846249631atural(F,A3)),aa_nat_Code_natural(power_7079662738309270450atural(N),K)) ) ) ) ).

% prod_le_power
tff(fact_7218_prod__le__power,axiom,
    ! [A3: set_Product_unit,F: fun_Pr6278675377255837256atural,N: code_natural,K: nat] :
      ( ! [I2: product_unit] :
          ( member_Product_unit(I2,A3)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_Pro528892430537105273atural(F,I2))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Pro528892430537105273atural(F,I2)),N) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(A3)),K)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),N)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups7008954963700796496atural(F,A3)),aa_nat_Code_natural(power_7079662738309270450atural(N),K)) ) ) ) ).

% prod_le_power
tff(fact_7219_prod__le__power,axiom,
    ! [A3: set_int,F: fun_int_rat,N: rat,K: nat] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),N) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A3)),K)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),aa_nat_rat(power_power_rat(N),K)) ) ) ) ).

% prod_le_power
tff(fact_7220_prod__le__power,axiom,
    ! [A3: set_nat,F: fun_nat_rat,N: rat,K: nat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),N) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),K)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups73079841787564623at_rat(F),A3)),aa_nat_rat(power_power_rat(N),K)) ) ) ) ).

% prod_le_power
tff(fact_7221_prod__le__power,axiom,
    ! [A3: set_Product_unit,F: fun_Product_unit_rat,N: rat,K: nat] :
      ( ! [I2: product_unit] :
          ( member_Product_unit(I2,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Product_unit_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Product_unit_rat(F,I2)),N) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(A3)),K)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups8984663225794451456it_rat(F,A3)),aa_nat_rat(power_power_rat(N),K)) ) ) ) ).

% prod_le_power
tff(fact_7222_prod__le__power,axiom,
    ! [A3: set_int,F: fun_int_nat,N: nat,K: nat] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),N) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A3)),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K)) ) ) ) ).

% prod_le_power
tff(fact_7223_sum__bounded__above__strict,axiom,
    ! [A3: set_int,F: fun_int_Code_integer,K5: code_integer] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,I2)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A3))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups7873554091576472773nteger(F,A3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_int(A3))),K5)) ) ) ).

% sum_bounded_above_strict
tff(fact_7224_sum__bounded__above__strict,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer,K5: code_integer] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,I2)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A3))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(finite_card_nat,A3))),K5)) ) ) ).

% sum_bounded_above_strict
tff(fact_7225_sum__bounded__above__strict,axiom,
    ! [A3: set_Product_unit,F: fun_Pr1820225556744648189nteger,K5: code_integer] :
      ( ! [I2: product_unit] :
          ( member_Product_unit(I2,A3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Pro1705297278780387758nteger(F,I2)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite410649719033368117t_unit(A3))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups5393439102872367408nteger(F,A3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite410649719033368117t_unit(A3))),K5)) ) ) ).

% sum_bounded_above_strict
tff(fact_7226_sum__bounded__above__strict,axiom,
    ! [A3: set_int,F: fun_int_rat,K5: rat] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,I2)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A3))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A3))),K5)) ) ) ).

% sum_bounded_above_strict
tff(fact_7227_sum__bounded__above__strict,axiom,
    ! [A3: set_nat,F: fun_nat_rat,K5: rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,I2)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A3))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_set_nat_rat(groups2906978787729119204at_rat(F),A3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A3))),K5)) ) ) ).

% sum_bounded_above_strict
tff(fact_7228_sum__bounded__above__strict,axiom,
    ! [A3: set_Product_unit,F: fun_Product_unit_rat,K5: rat] :
      ( ! [I2: product_unit] :
          ( member_Product_unit(I2,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Product_unit_rat(F,I2)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite410649719033368117t_unit(A3))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups8131667117488461483it_rat(F,A3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite410649719033368117t_unit(A3))),K5)) ) ) ).

% sum_bounded_above_strict
tff(fact_7229_sum__bounded__above__strict,axiom,
    ! [A3: set_nat,F: fun_nat_int,K5: int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,I2),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,I2)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A3))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),A3)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,A3))),K5)) ) ) ).

% sum_bounded_above_strict
tff(fact_7230_sum__bounded__above__strict,axiom,
    ! [A3: set_Product_unit,F: fun_Product_unit_int,K5: int] :
      ( ! [I2: product_unit] :
          ( member_Product_unit(I2,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Product_unit_int(F,I2)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite410649719033368117t_unit(A3))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups8764306707065906943it_int(F,A3)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite410649719033368117t_unit(A3))),K5)) ) ) ).

% sum_bounded_above_strict
tff(fact_7231_sum__bounded__above__strict,axiom,
    ! [A3: set_int,F: fun_int_nat,K5: nat] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,I2)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A3))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(A3))),K5)) ) ) ).

% sum_bounded_above_strict
tff(fact_7232_sum__bounded__above__strict,axiom,
    ! [A3: set_Product_unit,F: fun_Product_unit_nat,K5: nat] :
      ( ! [I2: product_unit] :
          ( member_Product_unit(I2,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Product_unit_nat(F,I2)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite410649719033368117t_unit(A3))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups8766797177574957219it_nat(F,A3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite410649719033368117t_unit(A3))),K5)) ) ) ).

% sum_bounded_above_strict
tff(fact_7233_prod__gen__delta,axiom,
    ! [S2: set_Product_unit,A: product_unit,B: fun_Product_unit_nat,C: nat] :
      ( finite4290736615968046902t_unit(S2)
     => ( groups396421249026171384it_nat(aa_nat7626694396248770914it_nat(aa_fun277559562889681605it_nat(aTP_Lamm_ze(A),B),C),S2) = $ite(member_Product_unit(A,S2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Product_unit_nat(B,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite410649719033368117t_unit(S2)),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),finite410649719033368117t_unit(S2))) ) ) ).

% prod_gen_delta
tff(fact_7234_prod__gen__delta,axiom,
    ! [S2: set_int,A: int,B: fun_int_nat,C: nat] :
      ( finite_finite_int(S2)
     => ( groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aa_fun1022895345097303677nt_nat(aTP_Lamm_zf(A),B),C),S2) = $ite(member_int(A,S2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(B,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(S2)),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),finite_card_int(S2))) ) ) ).

% prod_gen_delta
tff(fact_7235_prod__gen__delta,axiom,
    ! [S2: set_Product_unit,A: product_unit,B: fun_Product_unit_int,C: int] :
      ( finite4290736615968046902t_unit(S2)
     => ( groups393930778517121108it_int(aa_int1856962249220300386it_int(aa_fun4982840383720920169it_int(aTP_Lamm_zg(A),B),C),S2) = $ite(member_Product_unit(A,S2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Product_unit_int(B,A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite410649719033368117t_unit(S2)),one_one_nat))),aa_nat_int(aa_int_fun_nat_int(power_power_int,C),finite410649719033368117t_unit(S2))) ) ) ).

% prod_gen_delta
tff(fact_7236_prod__gen__delta,axiom,
    ! [S2: set_Product_unit,A: product_unit,B: fun_Pr1820225556744648189nteger,C: code_integer] :
      ( finite4290736615968046902t_unit(S2)
     => ( groups8185359811944078981nteger(aa_Cod3669976074529508130nteger(aa_fun3412420425980457528nteger(aTP_Lamm_zh(A),B),C),S2) = $ite(member_Product_unit(A,S2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Pro1705297278780387758nteger(B,A)),aa_nat_Code_integer(power_8256067586552552935nteger(C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite410649719033368117t_unit(S2)),one_one_nat))),aa_nat_Code_integer(power_8256067586552552935nteger(C),finite410649719033368117t_unit(S2))) ) ) ).

% prod_gen_delta
tff(fact_7237_prod__gen__delta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_Code_integer,C: code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_Cod1578151519121547077nteger(aa_fun8469111845059130348nteger(aTP_Lamm_zi(A),B),C)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(B,A)),aa_nat_Code_integer(power_8256067586552552935nteger(C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,S2)),one_one_nat))),aa_nat_Code_integer(power_8256067586552552935nteger(C),aa_set_nat_nat(finite_card_nat,S2))) ) ) ).

% prod_gen_delta
tff(fact_7238_prod__gen__delta,axiom,
    ! [S2: set_int,A: int,B: fun_int_Code_integer,C: code_integer] :
      ( finite_finite_int(S2)
     => ( groups3827104343326376752nteger(aa_Cod1949805079358391713nteger(aa_fun2884842898402423460nteger(aTP_Lamm_zj(A),B),C),S2) = $ite(member_int(A,S2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(B,A)),aa_nat_Code_integer(power_8256067586552552935nteger(C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(S2)),one_one_nat))),aa_nat_Code_integer(power_8256067586552552935nteger(C),finite_card_int(S2))) ) ) ).

% prod_gen_delta
tff(fact_7239_prod__gen__delta,axiom,
    ! [S2: set_Product_unit,A: product_unit,B: fun_Product_unit_rat,C: rat] :
      ( finite4290736615968046902t_unit(S2)
     => ( groups8984663225794451456it_rat(aa_rat6507435564262174562it_rat(aa_fun5217320890768598205it_rat(aTP_Lamm_zk(A),B),C),S2) = $ite(member_Product_unit(A,S2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Product_unit_rat(B,A)),aa_nat_rat(power_power_rat(C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite410649719033368117t_unit(S2)),one_one_nat))),aa_nat_rat(power_power_rat(C),finite410649719033368117t_unit(S2))) ) ) ).

% prod_gen_delta
tff(fact_7240_prod__gen__delta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat,C: rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_zl(A),B),C)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(B,A)),aa_nat_rat(power_power_rat(C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,S2)),one_one_nat))),aa_nat_rat(power_power_rat(C),aa_set_nat_nat(finite_card_nat,S2))) ) ) ).

% prod_gen_delta
tff(fact_7241_prod__gen__delta,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat,C: rat] :
      ( finite_finite_int(S2)
     => ( groups1072433553688619179nt_rat(aa_rat_fun_int_rat(aa_fun9155288305517341845nt_rat(aTP_Lamm_zm(A),B),C),S2) = $ite(member_int(A,S2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(B,A)),aa_nat_rat(power_power_rat(C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(S2)),one_one_nat))),aa_nat_rat(power_power_rat(C),finite_card_int(S2))) ) ) ).

% prod_gen_delta
tff(fact_7242_prod__gen__delta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_nat,C: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_zn(A),B),C)),S2) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),S2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(B,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,S2)),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_set_nat_nat(finite_card_nat,S2))) ) ) ).

% prod_gen_delta
tff(fact_7243_set__encode__insert,axiom,
    ! [A3: set_nat,N: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,N),A3)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_set_nat_nat(nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_7244_card__lists__length__le,axiom,
    ! [A3: set_Product_unit,N: nat] :
      ( finite4290736615968046902t_unit(A3)
     => ( finite3744894192721311803t_unit(collec4736972532740828926t_unit(aa_nat382646585846990998unit_o(aTP_Lamm_zo(A3),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,finite410649719033368117t_unit(A3))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7245_card__lists__length__le,axiom,
    ! [A3: set_set_nat,N: nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( finite5631907774883551598et_nat(collect_list_set_nat(aa_nat2247512698059773005_nat_o(aTP_Lamm_zp(A3),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_set_nat(A3))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7246_card__lists__length__le,axiom,
    ! [A3: set_list_nat,N: nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( finite7325466520557071688st_nat(collec5989764272469232197st_nat(aa_nat158606821896287091_nat_o(aTP_Lamm_zq(A3),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_list_nat(A3))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7247_card__lists__length__le,axiom,
    ! [A3: set_int,N: nat] :
      ( finite_finite_int(A3)
     => ( finite_card_list_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_zr(A3),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_int(A3))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7248_card__lists__length__le,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( finite249151656366948015at_nat(collec3343600615725829874at_nat(aa_nat1970911183850433186_nat_o(aTP_Lamm_zs(A3),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7249_card__lists__length__le,axiom,
    ! [A3: set_char,N: nat] :
      ( finite_finite_char(A3)
     => ( finite1726956870759998265t_char(collect_list_char(aa_nat3000828340120075458char_o(aTP_Lamm_zt(A3),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_char(A3))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7250_card__lists__length__le,axiom,
    ! [A3: set_nat,N: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_zu(A3),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(finite_card_nat,A3))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7251_bit__rec,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),N)
    <=> $ite(N = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% bit_rec
tff(fact_7252_bit__rec,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_nat_o(bit_se8040316288895769887atural(A),N)
    <=> $ite(N = zero_zero_nat,~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A),aa_nat_o(bit_se8040316288895769887atural(divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% bit_rec
tff(fact_7253_bit__rec,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
    <=> $ite(N = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% bit_rec
tff(fact_7254_bit__rec,axiom,
    ! [A: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N)
    <=> $ite(N = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% bit_rec
tff(fact_7255_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se9216721137139052372nteger(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),N)
      <=> $ite(N = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)),N)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_7256_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),N)
      <=> $ite(N = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)),N)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_7257_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,N: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),N)
      <=> $ite(N = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),N)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_7258_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_hu,aTP_Lamm_fl) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_7259_apsnd__apfst,axiom,
    ! [F: fun_in4744367027289715400nteger,G: fun_in4421688504851875865e_term,Xa: product_prod_int_int] : produc7159730696577248097e_term(F,produc6151042875033135238rm_int(G,Xa)) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,aa_int5322781614173852506e_term(G,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa))),aa_int77912060771590321nteger(F,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa))) ).

% apsnd_apfst
tff(fact_7260_apsnd__apfst,axiom,
    ! [F: fun_in7496697025202066910atural,G: fun_in1403664741593307672e_term,Xa: product_prod_int_int] : produc4003887134565203958e_term(F,produc7982337911579641349rm_int(G,Xa)) = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,aa_int5547325084924761305e_term(G,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa))),aa_int8200833983109115591atural(F,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa))) ).

% apsnd_apfst
tff(fact_7261_apsnd__apfst,axiom,
    ! [F: fun_in4744367027289715400nteger,G: fun_in7158797356850311459e_term,Xa: product_prod_int_int] : produc5655117854151192939e_term(F,produc3213193131064493712rm_int(G,Xa)) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,aa_int1137397288593391972e_term(G,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa))),aa_int77912060771590321nteger(F,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa))) ).

% apsnd_apfst
tff(fact_7262_apsnd__apfst,axiom,
    ! [F: fun_in4211310926729683942nt_int,G: fun_in237891044757944084e_term,Xa: product_prod_int_int] : produc5266752770087298298e_term(F,produc1502976015219776769rm_int(G,Xa)) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,aa_int8219552307768696789e_term(G,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa))),aa_int7462565939943104719nt_int(F,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa))) ).

% apsnd_apfst
tff(fact_7263_apsnd__apfst,axiom,
    ! [F: fun_in4211310926729683942nt_int,G: fun_in94812177438132456e_term,Xa: product_prod_int_int] : produc4251495004083222734e_term(F,produc6429934853825515733rm_int(G,Xa)) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,aa_int408404503994301353e_term(G,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa))),aa_int7462565939943104719nt_int(F,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa))) ).

% apsnd_apfst
tff(fact_7264_apsnd__apfst,axiom,
    ! [F: fun_na3822476419092417572nteger,G: fun_na1605103330871484349e_term,Xa: product_prod_nat_nat] : produc8601001262878435005e_term(F,produc9061699220754655694rm_nat(G,Xa)) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,aa_nat2506196440193460990e_term(G,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa))),aa_nat8379393489429068301nteger(F,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa))) ).

% apsnd_apfst
tff(fact_7265_apsnd__apfst,axiom,
    ! [F: fun_na6574806417004769082atural,G: fun_na6030372222970483132e_term,Xa: product_prod_nat_nat] : produc1378413197345364818e_term(F,produc2369655474374764365rm_nat(G,Xa)) = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,aa_nat950660529447160957e_term(G,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa))),aa_nat7278943374911817763atural(F,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa))) ).

% apsnd_apfst
tff(fact_7266_apsnd__apfst,axiom,
    ! [F: fun_na3822476419092417572nteger,G: fun_na2562132801372711111e_term,Xa: product_prod_nat_nat] : produc3029643916931353799e_term(F,produc6823882730714392536rm_nat(G,Xa)) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,aa_nat5764104769970567432e_term(G,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa))),aa_nat8379393489429068301nteger(F,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa))) ).

% apsnd_apfst
tff(fact_7267_apsnd__apfst,axiom,
    ! [F: fun_na1173468842265550146nt_int,G: fun_na6101041082579861176e_term,Xa: product_prod_nat_nat] : produc357814514533863510e_term(F,produc6132173967106002505rm_nat(G,Xa)) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,aa_nat4859330308735838073e_term(G,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa))),aa_nat4424723855478970923nt_int(F,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa))) ).

% apsnd_apfst
tff(fact_7268_apsnd__apfst,axiom,
    ! [F: fun_na1173468842265550146nt_int,G: fun_na3603975746771071116e_term,Xa: product_prod_nat_nat] : produc4148695890800444458e_term(F,produc3615840150354174493rm_nat(G,Xa)) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,aa_nat3917568073327240013e_term(G,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa))),aa_nat4424723855478970923nt_int(F,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa))) ).

% apsnd_apfst
tff(fact_7269_apfst__conv,axiom,
    ! [F: fun_fu9036519777977489245e_term,Xa: fun_Co3460408610320954864e_term,Y3: produc8923325533196201883nteger] : produc5151272107987982333nteger(F,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Xa),Y3)) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,aa_fun8764196472072707742e_term(F,Xa)),Y3) ).

% apfst_conv
tff(fact_7270_apfst__conv,axiom,
    ! [F: fun_fu3093988009420988519e_term,Xa: fun_Co3460408610320954864e_term,Y3: produc8923325533196201883nteger] : produc2260500861290198387nteger(F,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Xa),Y3)) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,aa_fun1525513021908036776e_term(F,Xa)),Y3) ).

% apfst_conv
tff(fact_7271_apfst__conv,axiom,
    ! [F: fun_fu7248590935147425627e_term,Xa: fun_Pr285225112257423215e_term,Y3: produc7822875418678951345atural] : produc1921288387873046805atural(F,aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,Xa),Y3)) = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,aa_fun6933112156808570396e_term(F,Xa)),Y3) ).

% apfst_conv
tff(fact_7272_apfst__conv,axiom,
    ! [F: fun_fu1456393871774969959e_term,Xa: fun_Pr5098669352780829690e_term,Y3: produc8923325533196201883nteger] : produc177741965484698227nteger(F,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Xa),Y3)) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,aa_fun1536964767670826920e_term(F,Xa)),Y3) ).

% apfst_conv
tff(fact_7273_apfst__conv,axiom,
    ! [F: fun_fu2505754153367914865e_term,Xa: fun_Pr5098669352780829690e_term,Y3: produc8923325533196201883nteger] : produc183566614307960809nteger(F,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Xa),Y3)) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,aa_fun1248587000295462322e_term(F,Xa)),Y3) ).

% apfst_conv
tff(fact_7274_apfst__conv,axiom,
    ! [F: fun_fu836511555612062035e_term,Xa: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int] : produc7857123342865048869nt_int(F,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Xa),Y3)) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,aa_fun49903357033182740e_term(F,Xa)),Y3) ).

% apfst_conv
tff(fact_7275_apfst__conv,axiom,
    ! [F: fun_fu4835546211937200935e_term,Xa: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int] : produc7965835600623382609nt_int(F,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Xa),Y3)) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,aa_fun3544567364070095336e_term(F,Xa)),Y3) ).

% apfst_conv
tff(fact_7276_apfst__conv,axiom,
    ! [F: fun_fu1176768540439120679e_term,Xa: fun_in5730379987311621695e_term,Y3: product_prod_int_int] : produc479425830829021777nt_int(F,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Xa),Y3)) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,aa_fun1932935823081541608e_term(F,Xa)),Y3) ).

% apfst_conv
tff(fact_7277_apfst__conv,axiom,
    ! [F: fun_fu4810317505167811835e_term,Xa: fun_in5730379987311621695e_term,Y3: product_prod_int_int] : produc5901819666759714173nt_int(F,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Xa),Y3)) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,aa_fun5947755341433680828e_term(F,Xa)),Y3) ).

% apfst_conv
tff(fact_7278_fst__apfst,axiom,
    ! [F: fun_nat_nat,Xa: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_Pro262675715343573244at_nat(produc986720760941809198at_nat(F),Xa)) = aa_nat_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa)) ).

% fst_apfst
tff(fact_7279_fst__apfst,axiom,
    ! [F: fun_int_int,Xa: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_Pro6970504127699319548nt_int(produc2355237562570658114nt_int(F),Xa)) = aa_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)) ).

% fst_apfst
tff(fact_7280_snd__apfst,axiom,
    ! [F: fun_int_int,Xa: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_Pro6970504127699319548nt_int(produc2355237562570658114nt_int(F),Xa)) = aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa) ).

% snd_apfst
tff(fact_7281_snd__apfst,axiom,
    ! [F: fun_nat_nat,Xa: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_Pro262675715343573244at_nat(produc986720760941809198at_nat(F),Xa)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa) ).

% snd_apfst
tff(fact_7282_snd__comp__apfst,axiom,
    ! [F: fun_int_int] : comp_P5356232565778667509nt_int(product_snd_int_int,produc2355237562570658114nt_int(F)) = product_snd_int_int ).

% snd_comp_apfst
tff(fact_7283_snd__comp__apfst,axiom,
    ! [F: fun_nat_nat] : comp_P1567445206330693457at_nat(product_snd_nat_nat,produc986720760941809198at_nat(F)) = product_snd_nat_nat ).

% snd_comp_apfst
tff(fact_7284_sum_Oinsert,axiom,
    ! [A3: set_Product_unit,Xa: product_unit,G: fun_Product_unit_rat] :
      ( finite4290736615968046902t_unit(A3)
     => ( ~ member_Product_unit(Xa,A3)
       => ( groups8131667117488461483it_rat(G,insert_Product_unit(Xa,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Product_unit_rat(G,Xa)),groups8131667117488461483it_rat(G,A3)) ) ) ) ).

% sum.insert
tff(fact_7285_sum_Oinsert,axiom,
    ! [A3: set_nat,Xa: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa),A3)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),insert_nat(Xa,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,Xa)),aa_set_nat_rat(groups2906978787729119204at_rat(G),A3)) ) ) ) ).

% sum.insert
tff(fact_7286_sum_Oinsert,axiom,
    ! [A3: set_int,Xa: int,G: fun_int_rat] :
      ( finite_finite_int(A3)
     => ( ~ member_int(Xa,A3)
       => ( groups3906332499630173760nt_rat(G,insert_int(Xa,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G,Xa)),groups3906332499630173760nt_rat(G,A3)) ) ) ) ).

% sum.insert
tff(fact_7287_sum_Oinsert,axiom,
    ! [A3: set_Product_unit,Xa: product_unit,G: fun_Product_unit_nat] :
      ( finite4290736615968046902t_unit(A3)
     => ( ~ member_Product_unit(Xa,A3)
       => ( groups8766797177574957219it_nat(G,insert_Product_unit(Xa,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Product_unit_nat(G,Xa)),groups8766797177574957219it_nat(G,A3)) ) ) ) ).

% sum.insert
tff(fact_7288_sum_Oinsert,axiom,
    ! [A3: set_int,Xa: int,G: fun_int_nat] :
      ( finite_finite_int(A3)
     => ( ~ member_int(Xa,A3)
       => ( groups4541462559716669496nt_nat(G,insert_int(Xa,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,Xa)),groups4541462559716669496nt_nat(G,A3)) ) ) ) ).

% sum.insert
tff(fact_7289_sum_Oinsert,axiom,
    ! [A3: set_Product_unit,Xa: product_unit,G: fun_Product_unit_int] :
      ( finite4290736615968046902t_unit(A3)
     => ( ~ member_Product_unit(Xa,A3)
       => ( groups8764306707065906943it_int(G,insert_Product_unit(Xa,A3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Product_unit_int(G,Xa)),groups8764306707065906943it_int(G,A3)) ) ) ) ).

% sum.insert
tff(fact_7290_sum_Oinsert,axiom,
    ! [A3: set_nat,Xa: nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa),A3)
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),insert_nat(Xa,A3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Xa)),aa_set_nat_int(groups3539618377306564664at_int(G),A3)) ) ) ) ).

% sum.insert
tff(fact_7291_sum_Oinsert,axiom,
    ! [A3: set_nat,Xa: nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa),A3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),insert_nat(Xa,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Xa)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)) ) ) ) ).

% sum.insert
tff(fact_7292_sum_Oinsert,axiom,
    ! [A3: set_int,Xa: int,G: fun_int_int] :
      ( finite_finite_int(A3)
     => ( ~ member_int(Xa,A3)
       => ( groups4538972089207619220nt_int(G,insert_int(Xa,A3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(G,Xa)),groups4538972089207619220nt_int(G,A3)) ) ) ) ).

% sum.insert
tff(fact_7293_sum_Oinsert,axiom,
    ! [A3: set_set_nat,Xa: set_nat,G: fun_set_nat_rat] :
      ( finite1152437895449049373et_nat(A3)
     => ( ~ member_set_nat(Xa,A3)
       => ( groups7659867448343625626at_rat(G,insert_set_nat(Xa,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(G,Xa)),groups7659867448343625626at_rat(G,A3)) ) ) ) ).

% sum.insert
tff(fact_7294_fst__comp__apfst,axiom,
    ! [F: fun_nat_nat] : comp_P1567445206330693457at_nat(product_fst_nat_nat,produc986720760941809198at_nat(F)) = comp_n2711240574518091068at_nat(F,product_fst_nat_nat) ).

% fst_comp_apfst
tff(fact_7295_fst__comp__apfst,axiom,
    ! [F: fun_int_int] : comp_P5356232565778667509nt_int(product_fst_int_int,produc2355237562570658114nt_int(F)) = comp_i4330622015127193916nt_int(F,product_fst_int_int) ).

% fst_comp_apfst
tff(fact_7296_card__Diff__insert,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A3)
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B3)
       => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,minus_1356011639430497352at_nat(A3,insert8211810215607154385at_nat(A,B3))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,minus_1356011639430497352at_nat(A3,B3))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7297_card__Diff__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(A,A3)
     => ( ~ member8757157785044589968at_nat(A,B3)
       => ( finite3771342082235030671at_nat(minus_3314409938677909166at_nat(A3,insert9069300056098147895at_nat(A,B3))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite3771342082235030671at_nat(minus_3314409938677909166at_nat(A3,B3))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7298_card__Diff__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B3: set_se7855581050983116737at_nat] :
      ( member2643936169264416010at_nat(A,A3)
     => ( ~ member2643936169264416010at_nat(A,B3)
       => ( finite4356350796350151305at_nat(minus_4207664762107033000at_nat(A3,insert9200635055090092081at_nat(A,B3))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite4356350796350151305at_nat(minus_4207664762107033000at_nat(A3,B3))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7299_card__Diff__insert,axiom,
    ! [A: set_Pr958786334691620121nt_int,A3: set_se6260736226359567993nt_int,B3: set_se6260736226359567993nt_int] :
      ( member2340774599025711042nt_int(A,A3)
     => ( ~ member2340774599025711042nt_int(A,B3)
       => ( finite4053189226111446337nt_int(minus_2612819937483484256nt_int(A3,insert8897473484851387113nt_int(A,B3))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite4053189226111446337nt_int(minus_2612819937483484256nt_int(A3,B3))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7300_card__Diff__insert,axiom,
    ! [A: int,A3: set_int,B3: set_int] :
      ( member_int(A,A3)
     => ( ~ member_int(A,B3)
       => ( finite_card_int(minus_minus_set_int(A3,insert_int(A,B3))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(minus_minus_set_int(A3,B3))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7301_card__Diff__insert,axiom,
    ! [A: product_unit,A3: set_Product_unit,B3: set_Product_unit] :
      ( member_Product_unit(A,A3)
     => ( ~ member_Product_unit(A,B3)
       => ( finite410649719033368117t_unit(minus_6452836326544984404t_unit(A3,insert_Product_unit(A,B3))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A3,B3))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7302_card__Diff__insert,axiom,
    ! [A: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( member_set_nat(A,A3)
     => ( ~ member_set_nat(A,B3)
       => ( finite_card_set_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(A,B3))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A3,B3))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7303_card__Diff__insert,axiom,
    ! [A: list_nat,A3: set_list_nat,B3: set_list_nat] :
      ( member_list_nat(A,A3)
     => ( ~ member_list_nat(A,B3)
       => ( finite_card_list_nat(minus_7954133019191499631st_nat(A3,insert_list_nat(A,B3))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A3,B3))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7304_card__Diff__insert,axiom,
    ! [A: nat,A3: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A3)
     => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),B3)
       => ( aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A3,insert_nat(A,B3))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A3,B3))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7305_apfst__apsnd,axiom,
    ! [F: fun_in4421688504851875865e_term,G: fun_in4744367027289715400nteger,Xa: product_prod_int_int] : produc3354640145386789057nteger(F,produc2935421404040714269er_int(G,Xa)) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,aa_int5322781614173852506e_term(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa))),aa_int77912060771590321nteger(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa))) ).

% apfst_apsnd
tff(fact_7306_apfst__apsnd,axiom,
    ! [F: fun_in1403664741593307672e_term,G: fun_in7496697025202066910atural,Xa: product_prod_int_int] : produc2162821685372896920atural(F,produc1633769443012300339al_int(G,Xa)) = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,aa_int5547325084924761305e_term(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa))),aa_int8200833983109115591atural(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa))) ).

% apfst_apsnd
tff(fact_7307_apfst__apsnd,axiom,
    ! [F: fun_in7158797356850311459e_term,G: fun_in4744367027289715400nteger,Xa: product_prod_int_int] : produc8552504676914255415nteger(F,produc2935421404040714269er_int(G,Xa)) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,aa_int1137397288593391972e_term(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa))),aa_int77912060771590321nteger(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa))) ).

% apfst_apsnd
tff(fact_7308_apfst__apsnd,axiom,
    ! [F: fun_in237891044757944084e_term,G: fun_in4211310926729683942nt_int,Xa: product_prod_int_int] : produc7648640027116863140nt_int(F,produc6100211334659168315nt_int(G,Xa)) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,aa_int8219552307768696789e_term(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa))),aa_int7462565939943104719nt_int(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa))) ).

% apfst_apsnd
tff(fact_7309_apfst__apsnd,axiom,
    ! [F: fun_in94812177438132456e_term,G: fun_in4211310926729683942nt_int,Xa: product_prod_int_int] : produc1852316473262731728nt_int(F,produc6100211334659168315nt_int(G,Xa)) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,aa_int408404503994301353e_term(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa))),aa_int7462565939943104719nt_int(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa))) ).

% apfst_apsnd
tff(fact_7310_apfst__apsnd,axiom,
    ! [F: fun_na1605103330871484349e_term,G: fun_na3822476419092417572nteger,Xa: product_prod_nat_nat] : produc4795910711687975965nteger(F,produc2688509591584439069er_nat(G,Xa)) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,aa_nat2506196440193460990e_term(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa))),aa_nat8379393489429068301nteger(G,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa))) ).

% apfst_apsnd
tff(fact_7311_apfst__apsnd,axiom,
    ! [F: fun_na6030372222970483132e_term,G: fun_na6574806417004769082atural,Xa: product_prod_nat_nat] : produc8760719785007833588atural(F,produc1386857630556025139al_nat(G,Xa)) = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,aa_nat950660529447160957e_term(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa))),aa_nat7278943374911817763atural(G,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa))) ).

% apfst_apsnd
tff(fact_7312_apfst__apsnd,axiom,
    ! [F: fun_na2562132801372711111e_term,G: fun_na3822476419092417572nteger,Xa: product_prod_nat_nat] : produc5927030739694416275nteger(F,produc2688509591584439069er_nat(G,Xa)) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,aa_nat5764104769970567432e_term(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa))),aa_nat8379393489429068301nteger(G,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa))) ).

% apfst_apsnd
tff(fact_7313_apfst__apsnd,axiom,
    ! [F: fun_na6101041082579861176e_term,G: fun_na1173468842265550146nt_int,Xa: product_prod_nat_nat] : produc2739701771563428352nt_int(F,produc4342665141415107387nt_nat(G,Xa)) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,aa_nat4859330308735838073e_term(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa))),aa_nat4424723855478970923nt_int(G,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa))) ).

% apfst_apsnd
tff(fact_7314_apfst__apsnd,axiom,
    ! [F: fun_na3603975746771071116e_term,G: fun_na1173468842265550146nt_int,Xa: product_prod_nat_nat] : produc1749517359979953452nt_int(F,produc4342665141415107387nt_nat(G,Xa)) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,aa_nat3917568073327240013e_term(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa))),aa_nat4424723855478970923nt_int(G,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa))) ).

% apfst_apsnd
tff(fact_7315_bit__0,axiom,
    ! [A: code_integer] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),zero_zero_nat)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_7316_bit__0,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),zero_zero_nat)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_7317_bit__0,axiom,
    ! [A: int] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),zero_zero_nat)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_7318_bit__mod__2__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),N)
    <=> ( ( N = zero_zero_nat )
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_7319_bit__mod__2__iff,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N)
    <=> ( ( N = zero_zero_nat )
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_7320_bit__mod__2__iff,axiom,
    ! [A: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),N)
    <=> ( ( N = zero_zero_nat )
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_7321_insert__compr,axiom,
    ! [A: product_unit,B3: set_Product_unit] : insert_Product_unit(A,B3) = collect_Product_unit(aa_set2953008695252327103unit_o(aTP_Lamm_zv(A),B3)) ).

% insert_compr
tff(fact_7322_insert__compr,axiom,
    ! [A: product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,B3) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_zw(A),B3)) ).

% insert_compr
tff(fact_7323_insert__compr,axiom,
    ! [A: produc3843707927480180839at_nat,B3: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,B3) = collec6321179662152712658at_nat(aa_set6353399069419443519_nat_o(aTP_Lamm_zx(A),B3)) ).

% insert_compr
tff(fact_7324_insert__compr,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B3: set_se7855581050983116737at_nat] : insert9200635055090092081at_nat(A,B3) = collec5514110066124741708at_nat(aa_set7533644503045286079_nat_o(aTP_Lamm_zy(A),B3)) ).

% insert_compr
tff(fact_7325_insert__compr,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_se6260736226359567993nt_int] : insert8897473484851387113nt_int(A,B3) = collec5210948495886036740nt_int(aa_set4062574822058255039_int_o(aTP_Lamm_zz(A),B3)) ).

% insert_compr
tff(fact_7326_insert__compr,axiom,
    ! [A: set_nat,B3: set_set_nat] : insert_set_nat(A,B3) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_aaa(A),B3)) ).

% insert_compr
tff(fact_7327_insert__compr,axiom,
    ! [A: nat,B3: set_nat] : insert_nat(A,B3) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_aab(A),B3)) ).

% insert_compr
tff(fact_7328_insert__compr,axiom,
    ! [A: int,B3: set_int] : insert_int(A,B3) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_aac(A),B3)) ).

% insert_compr
tff(fact_7329_insert__Collect,axiom,
    ! [A: product_unit,Pa: fun_Product_unit_o] : insert_Product_unit(A,collect_Product_unit(Pa)) = collect_Product_unit(aa_fun644178081534259290unit_o(aTP_Lamm_aad(A),Pa)) ).

% insert_Collect
tff(fact_7330_insert__Collect,axiom,
    ! [A: product_prod_nat_nat,Pa: fun_Pr4796943543331071514_nat_o] : insert8211810215607154385at_nat(A,collec3392354462482085612at_nat(Pa)) = collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_aae(A),Pa)) ).

% insert_Collect
tff(fact_7331_insert__Collect,axiom,
    ! [A: produc3843707927480180839at_nat,Pa: fun_Pr3059359313727237556_nat_o] : insert9069300056098147895at_nat(A,collec6321179662152712658at_nat(Pa)) = collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_aaf(A),Pa)) ).

% insert_Collect
tff(fact_7332_insert__Collect,axiom,
    ! [A: set_Pr1261947904930325089at_nat,Pa: fun_se4782206882984618170_nat_o] : insert9200635055090092081at_nat(A,collec5514110066124741708at_nat(Pa)) = collec5514110066124741708at_nat(aa_fun3315923329236548914_nat_o(aTP_Lamm_aag(A),Pa)) ).

% insert_Collect
tff(fact_7333_insert__Collect,axiom,
    ! [A: set_Pr958786334691620121nt_int,Pa: fun_se7249947736802449410_int_o] : insert8897473484851387113nt_int(A,collec5210948495886036740nt_int(Pa)) = collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_aah(A),Pa)) ).

% insert_Collect
tff(fact_7334_insert__Collect,axiom,
    ! [A: set_nat,Pa: fun_set_nat_o] : insert_set_nat(A,collect_set_nat(Pa)) = collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_aai(A),Pa)) ).

% insert_Collect
tff(fact_7335_insert__Collect,axiom,
    ! [A: nat,Pa: fun_nat_o] : insert_nat(A,collect_nat(Pa)) = collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aaj(A),Pa)) ).

% insert_Collect
tff(fact_7336_insert__Collect,axiom,
    ! [A: int,Pa: fun_int_o] : insert_int(A,collect_int(Pa)) = collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_aak(A),Pa)) ).

% insert_Collect
tff(fact_7337_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ! [N2: nat] :
          ( ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N2)
          | ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),N2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),N)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
          | aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),N) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_7338_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( ! [N2: nat] :
          ( ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N2)
          | ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,B),N2) )
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N)
      <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N)
          | aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,B),N) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_7339_apsnd__apfst__commute,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger,P: produc8923325533196201883nteger] : aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc3195571438000033749nteger,G),P)) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc3195571438000033749nteger,G),aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),P)) ).

% apsnd_apfst_commute
tff(fact_7340_not__bit__1__Suc,axiom,
    ! [N: nat] : ~ aa_nat_o(bit_se9216721137139052372nteger(one_one_Code_integer),aa_nat_nat(suc,N)) ).

% not_bit_1_Suc
tff(fact_7341_not__bit__1__Suc,axiom,
    ! [N: nat] : ~ aa_nat_o(bit_se8040316288895769887atural(one_one_Code_natural),aa_nat_nat(suc,N)) ).

% not_bit_1_Suc
tff(fact_7342_not__bit__1__Suc,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_nat_nat(suc,N)) ).

% not_bit_1_Suc
tff(fact_7343_not__bit__1__Suc,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,one_one_int),aa_nat_nat(suc,N)) ).

% not_bit_1_Suc
tff(fact_7344_bit__1__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(one_one_Code_integer),N)
    <=> ( N = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_7345_bit__1__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(bit_se8040316288895769887atural(one_one_Code_natural),N)
    <=> ( N = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_7346_bit__1__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),N)
    <=> ( N = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_7347_bit__1__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,one_one_int),N)
    <=> ( N = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_7348_bit__numeral__simps_I1_J,axiom,
    ! [N: num] : ~ aa_nat_o(bit_se9216721137139052372nteger(one_one_Code_integer),aa_num_nat(numeral_numeral_nat,N)) ).

% bit_numeral_simps(1)
tff(fact_7349_bit__numeral__simps_I1_J,axiom,
    ! [N: num] : ~ aa_nat_o(bit_se8040316288895769887atural(one_one_Code_natural),aa_num_nat(numeral_numeral_nat,N)) ).

% bit_numeral_simps(1)
tff(fact_7350_bit__numeral__simps_I1_J,axiom,
    ! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) ).

% bit_numeral_simps(1)
tff(fact_7351_bit__numeral__simps_I1_J,axiom,
    ! [N: num] : ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,one_one_int),aa_num_nat(numeral_numeral_nat,N)) ).

% bit_numeral_simps(1)
tff(fact_7352_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,N)) ).

% not_bit_Suc_0_Suc
tff(fact_7353_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),N)
    <=> ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_7354_disjunctive__add,axiom,
    ! [A: int,B: int] :
      ( ! [N2: nat] :
          ( ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N2)
          | ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,B),N2) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B) ) ) ).

% disjunctive_add
tff(fact_7355_disjunctive__add,axiom,
    ! [A: nat,B: nat] :
      ( ! [N2: nat] :
          ( ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N2)
          | ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),N2) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) ) ) ).

% disjunctive_add
tff(fact_7356_bit__of__bool__iff,axiom,
    ! [B: $o,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B))),N)
    <=> ( (B)
        & ( N = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_7357_bit__of__bool__iff,axiom,
    ! [B: $o,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_o_int(zero_n2684676970156552555ol_int,(B))),N)
    <=> ( (B)
        & ( N = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_7358_bit__drop__bit__eq,axiom,
    ! [N: nat,A: int] : aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),A)) = comp_nat_o_nat(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),aa_nat_fun_nat_nat(plus_plus_nat,N)) ).

% bit_drop_bit_eq
tff(fact_7359_bit__drop__bit__eq,axiom,
    ! [N: nat,A: nat] : aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),A)) = comp_nat_o_nat(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_fun_nat_nat(plus_plus_nat,N)) ).

% bit_drop_bit_eq
tff(fact_7360_length__pos__if__in__set,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat] :
      ( member2643936169264416010at_nat(Xa,set_se5049602875457034614at_nat(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s8736152011456118867at_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_7361_length__pos__if__in__set,axiom,
    ! [Xa: set_Pr958786334691620121nt_int,Xs: list_s8839374986383574687nt_int] :
      ( member2340774599025711042nt_int(Xa,set_se4746441305218329646nt_int(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s7141307186832570123nt_int(Xs)) ) ).

% length_pos_if_in_set
tff(fact_7362_length__pos__if__in__set,axiom,
    ! [Xa: set_nat,Xs: list_set_nat] :
      ( member_set_nat(Xa,set_set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_7363_length__pos__if__in__set,axiom,
    ! [Xa: int,Xs: list_int] :
      ( member_int(Xa,set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)) ) ).

% length_pos_if_in_set
tff(fact_7364_length__pos__if__in__set,axiom,
    ! [Xa: char,Xs: list_char] :
      ( member_char(Xa,set_char2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_list_char_nat(size_size_list_char,Xs)) ) ).

% length_pos_if_in_set
tff(fact_7365_length__pos__if__in__set,axiom,
    ! [Xa: nat,Xs: list_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa),set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs)) ) ).

% length_pos_if_in_set
tff(fact_7366_finite__lists__length__eq,axiom,
    ! [A3: set_int,N: nat] :
      ( finite_finite_int(A3)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_aal(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_7367_finite__lists__length__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1970911183850433186_nat_o(aTP_Lamm_aam(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_7368_finite__lists__length__eq,axiom,
    ! [A3: set_char,N: nat] :
      ( finite_finite_char(A3)
     => finite3619747586194699384t_char(collect_list_char(aa_nat3000828340120075458char_o(aTP_Lamm_aan(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_7369_finite__lists__length__eq,axiom,
    ! [A3: set_nat,N: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aao(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_7370_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,N)) ).

% not_bit_Suc_0_numeral
tff(fact_7371_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),N)
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(N),A)),one_one_Code_integer) = one_one_Code_integer ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_7372_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_nat_o(bit_se8040316288895769887atural(A),N)
    <=> ( bit_se2773287842338716102atural(bit_se2751692689150723149atural(N,A),one_one_Code_natural) = one_one_Code_natural ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_7373_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N)
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),A)),one_one_int) = one_one_int ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_7374_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),A)),one_one_nat) = one_one_nat ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_7375_sum_Oinsert__if,axiom,
    ! [A3: set_Product_unit,G: fun_Product_unit_rat,Xa: product_unit] :
      ( finite4290736615968046902t_unit(A3)
     => ( groups8131667117488461483it_rat(G,insert_Product_unit(Xa,A3)) = $ite(member_Product_unit(Xa,A3),groups8131667117488461483it_rat(G,A3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Product_unit_rat(G,Xa)),groups8131667117488461483it_rat(G,A3))) ) ) ).

% sum.insert_if
tff(fact_7376_sum_Oinsert__if,axiom,
    ! [A3: set_nat,G: fun_nat_rat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(G),insert_nat(Xa,A3)) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa),A3),aa_set_nat_rat(groups2906978787729119204at_rat(G),A3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,Xa)),aa_set_nat_rat(groups2906978787729119204at_rat(G),A3))) ) ) ).

% sum.insert_if
tff(fact_7377_sum_Oinsert__if,axiom,
    ! [A3: set_int,G: fun_int_rat,Xa: int] :
      ( finite_finite_int(A3)
     => ( groups3906332499630173760nt_rat(G,insert_int(Xa,A3)) = $ite(member_int(Xa,A3),groups3906332499630173760nt_rat(G,A3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G,Xa)),groups3906332499630173760nt_rat(G,A3))) ) ) ).

% sum.insert_if
tff(fact_7378_sum_Oinsert__if,axiom,
    ! [A3: set_Product_unit,G: fun_Product_unit_nat,Xa: product_unit] :
      ( finite4290736615968046902t_unit(A3)
     => ( groups8766797177574957219it_nat(G,insert_Product_unit(Xa,A3)) = $ite(member_Product_unit(Xa,A3),groups8766797177574957219it_nat(G,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Product_unit_nat(G,Xa)),groups8766797177574957219it_nat(G,A3))) ) ) ).

% sum.insert_if
tff(fact_7379_sum_Oinsert__if,axiom,
    ! [A3: set_int,G: fun_int_nat,Xa: int] :
      ( finite_finite_int(A3)
     => ( groups4541462559716669496nt_nat(G,insert_int(Xa,A3)) = $ite(member_int(Xa,A3),groups4541462559716669496nt_nat(G,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,Xa)),groups4541462559716669496nt_nat(G,A3))) ) ) ).

% sum.insert_if
tff(fact_7380_sum_Oinsert__if,axiom,
    ! [A3: set_Product_unit,G: fun_Product_unit_int,Xa: product_unit] :
      ( finite4290736615968046902t_unit(A3)
     => ( groups8764306707065906943it_int(G,insert_Product_unit(Xa,A3)) = $ite(member_Product_unit(Xa,A3),groups8764306707065906943it_int(G,A3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Product_unit_int(G,Xa)),groups8764306707065906943it_int(G,A3))) ) ) ).

% sum.insert_if
tff(fact_7381_sum_Oinsert__if,axiom,
    ! [A3: set_nat,G: fun_nat_int,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),insert_nat(Xa,A3)) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa),A3),aa_set_nat_int(groups3539618377306564664at_int(G),A3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Xa)),aa_set_nat_int(groups3539618377306564664at_int(G),A3))) ) ) ).

% sum.insert_if
tff(fact_7382_sum_Oinsert__if,axiom,
    ! [A3: set_nat,G: fun_nat_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),insert_nat(Xa,A3)) = $ite(aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Xa),A3),aa_set_nat_nat(groups3542108847815614940at_nat(G),A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Xa)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A3))) ) ) ).

% sum.insert_if
tff(fact_7383_sum_Oinsert__if,axiom,
    ! [A3: set_int,G: fun_int_int,Xa: int] :
      ( finite_finite_int(A3)
     => ( groups4538972089207619220nt_int(G,insert_int(Xa,A3)) = $ite(member_int(Xa,A3),groups4538972089207619220nt_int(G,A3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(G,Xa)),groups4538972089207619220nt_int(G,A3))) ) ) ).

% sum.insert_if
tff(fact_7384_sum_Oinsert__if,axiom,
    ! [A3: set_set_nat,G: fun_set_nat_rat,Xa: set_nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( groups7659867448343625626at_rat(G,insert_set_nat(Xa,A3)) = $ite(member_set_nat(Xa,A3),groups7659867448343625626at_rat(G,A3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(G,Xa)),groups7659867448343625626at_rat(G,A3))) ) ) ).

% sum.insert_if
tff(fact_7385_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_7386_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_7387_finite__lists__length__le,axiom,
    ! [A3: set_int,N: nat] :
      ( finite_finite_int(A3)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_zr(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7388_finite__lists__length__le,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1970911183850433186_nat_o(aTP_Lamm_zs(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7389_finite__lists__length__le,axiom,
    ! [A3: set_char,N: nat] :
      ( finite_finite_char(A3)
     => finite3619747586194699384t_char(collect_list_char(aa_nat3000828340120075458char_o(aTP_Lamm_zt(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7390_finite__lists__length__le,axiom,
    ! [A3: set_nat,N: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_zu(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7391_bit__numeral__rec_I1_J,axiom,
    ! [W: num,N: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,W))),N)
    <=> aa_nat_o(case_nat_o($false,bit_se9216721137139052372nteger(aa_num_Code_integer(numera6620942414471956472nteger,W))),N) ) ).

% bit_numeral_rec(1)
tff(fact_7392_bit__numeral__rec_I1_J,axiom,
    ! [W: num,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))),N)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W))),N) ) ).

% bit_numeral_rec(1)
tff(fact_7393_bit__numeral__rec_I1_J,axiom,
    ! [W: num,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))),N)
    <=> aa_nat_o(case_nat_o($false,aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W))),N) ) ).

% bit_numeral_rec(1)
tff(fact_7394_bit__numeral__rec_I2_J,axiom,
    ! [W: num,N: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,W))),N)
    <=> aa_nat_o(case_nat_o($true,bit_se9216721137139052372nteger(aa_num_Code_integer(numera6620942414471956472nteger,W))),N) ) ).

% bit_numeral_rec(2)
tff(fact_7395_bit__numeral__rec_I2_J,axiom,
    ! [W: num,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))),N)
    <=> aa_nat_o(case_nat_o($true,aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W))),N) ) ).

% bit_numeral_rec(2)
tff(fact_7396_bit__numeral__rec_I2_J,axiom,
    ! [W: num,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))),N)
    <=> aa_nat_o(case_nat_o($true,aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W))),N) ) ).

% bit_numeral_rec(2)
tff(fact_7397_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),N)
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer)) != zero_z3403309356797280102nteger ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_7398_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_nat_o(bit_se8040316288895769887atural(A),N)
    <=> ( bit_se2773287842338716102atural(A,bit_se6611745700429515170atural(N,one_one_Code_natural)) != zero_z2226904508553997617atural ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_7399_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N)
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) != zero_zero_int ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_7400_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat)) != zero_zero_nat ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_7401_card__lists__length__eq,axiom,
    ! [A3: set_Product_unit,N: nat] :
      ( finite4290736615968046902t_unit(A3)
     => ( finite3744894192721311803t_unit(collec4736972532740828926t_unit(aa_nat382646585846990998unit_o(aTP_Lamm_aap(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite410649719033368117t_unit(A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7402_card__lists__length__eq,axiom,
    ! [A3: set_set_nat,N: nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( finite5631907774883551598et_nat(collect_list_set_nat(aa_nat2247512698059773005_nat_o(aTP_Lamm_aaq(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_set_nat(A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7403_card__lists__length__eq,axiom,
    ! [A3: set_list_nat,N: nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( finite7325466520557071688st_nat(collec5989764272469232197st_nat(aa_nat158606821896287091_nat_o(aTP_Lamm_aar(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_list_nat(A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7404_card__lists__length__eq,axiom,
    ! [A3: set_int,N: nat] :
      ( finite_finite_int(A3)
     => ( finite_card_list_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_aal(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_int(A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7405_card__lists__length__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( finite249151656366948015at_nat(collec3343600615725829874at_nat(aa_nat1970911183850433186_nat_o(aTP_Lamm_aam(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7406_card__lists__length__eq,axiom,
    ! [A3: set_char,N: nat] :
      ( finite_finite_char(A3)
     => ( finite1726956870759998265t_char(collect_list_char(aa_nat3000828340120075458char_o(aTP_Lamm_aan(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_char(A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7407_card__lists__length__eq,axiom,
    ! [A3: set_nat,N: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aao(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(finite_card_nat,A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7408_exp__eq__0__imp__not__bit,axiom,
    ! [N: nat,A: code_natural] :
      ( ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N) = zero_z2226904508553997617atural )
     => ~ aa_nat_o(bit_se8040316288895769887atural(A),N) ) ).

% exp_eq_0_imp_not_bit
tff(fact_7409_exp__eq__0__imp__not__bit,axiom,
    ! [N: nat,A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
     => ~ aa_nat_o(bit_se9216721137139052372nteger(A),N) ) ).

% exp_eq_0_imp_not_bit
tff(fact_7410_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,aa_num_num(bit0,one))),N) = zero_zero_nat )
     => ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N) ) ).

% exp_eq_0_imp_not_bit
tff(fact_7411_exp__eq__0__imp__not__bit,axiom,
    ! [N: nat,A: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
     => ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N) ) ).

% exp_eq_0_imp_not_bit
tff(fact_7412_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M3)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),M3)
              <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N2) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N2) ) ) ) ).

% int_bit_bound
tff(fact_7413_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: code_natural,N: nat] :
      ( ( bit_se2773287842338716102atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N)) = zero_z2226904508553997617atural )
    <=> ~ aa_nat_o(bit_se8040316288895769887atural(A),N) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_7414_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = zero_z3403309356797280102nteger )
    <=> ~ aa_nat_o(bit_se9216721137139052372nteger(A),N) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_7415_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = zero_zero_int )
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_7416_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,aa_num_num(bit0,one))),N)) = zero_zero_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_7417_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_aas) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_7418_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(Y3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(insert_int(Xa,Y3))),N) ) ) ).

% card_insert_le_m1
tff(fact_7419_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_Pr1261947904930325089at_nat,Xa: product_prod_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,Y3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,insert8211810215607154385at_nat(Xa,Y3))),N) ) ) ).

% card_insert_le_m1
tff(fact_7420_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_Pr4329608150637261639at_nat,Xa: produc3843707927480180839at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite3771342082235030671at_nat(Y3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite3771342082235030671at_nat(insert9069300056098147895at_nat(Xa,Y3))),N) ) ) ).

% card_insert_le_m1
tff(fact_7421_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,Y3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,insert_nat(Xa,Y3))),N) ) ) ).

% card_insert_le_m1
tff(fact_7422_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_Product_unit,Xa: product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(Y3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(insert_Product_unit(Xa,Y3))),N) ) ) ).

% card_insert_le_m1
tff(fact_7423_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_set_nat,Xa: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(Y3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(insert_set_nat(Xa,Y3))),N) ) ) ).

% card_insert_le_m1
tff(fact_7424_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_list_nat,Xa: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(Y3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(insert_list_nat(Xa,Y3))),N) ) ) ).

% card_insert_le_m1
tff(fact_7425_take__bit__sum,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aat,A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% take_bit_sum
tff(fact_7426_take__bit__sum,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_aau,A)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% take_bit_sum
tff(fact_7427_even__bit__succ__iff,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se8040316288895769887atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),A)),N)
      <=> ( aa_nat_o(bit_se8040316288895769887atural(A),N)
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_7428_even__bit__succ__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),N)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(A),N)
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_7429_even__bit__succ__iff,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),N)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_7430_even__bit__succ__iff,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),N)
      <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N)
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_7431_odd__bit__iff__bit__pred,axiom,
    ! [A: code_natural,N: nat] :
      ( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se8040316288895769887atural(A),N)
      <=> ( aa_nat_o(bit_se8040316288895769887atural(minus_7197305767214868737atural(A,one_one_Code_natural)),N)
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_7432_odd__bit__iff__bit__pred,axiom,
    ! [A: code_integer,N: nat] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se9216721137139052372nteger(A),N)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),one_one_Code_integer)),N)
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_7433_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),one_one_nat)),N)
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_7434_odd__bit__iff__bit__pred,axiom,
    ! [A: int,N: nat] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N)
      <=> ( aa_nat_o(aa_int_fun_nat_o(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_7435_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,N),aa_nat_set_nat(nat_set_decode,Z))
     => ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),Z)) = insert_nat(N,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_7436_signed__take__bit__code,axiom,
    ! [N: nat,A: code_integer] :
      aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),A) = $let(
        l: code_integer,
        l:= aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),A),
        $ite(aa_nat_o(bit_se9216721137139052372nteger(l),N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,l),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_nat_nat(suc,N)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))),l) ) ).

% signed_take_bit_code
tff(fact_7437_signed__take__bit__code,axiom,
    ! [N: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),A) = $let(
        l: int,
        l:= aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),A),
        $ite(aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,l),N),aa_int_int(aa_int_fun_int_int(plus_plus_int,l),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,one_one_int))),l) ) ).

% signed_take_bit_code
tff(fact_7438_card__lists__distinct__length__eq,axiom,
    ! [A3: set_Product_unit,K: nat] :
      ( finite4290736615968046902t_unit(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),finite410649719033368117t_unit(A3))
       => ( finite3744894192721311803t_unit(collec4736972532740828926t_unit(aa_nat382646585846990998unit_o(aTP_Lamm_aav(A3),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite410649719033368117t_unit(A3)),K)),one_one_nat)),finite410649719033368117t_unit(A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_7439_card__lists__distinct__length__eq,axiom,
    ! [A3: set_set_nat,K: nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),finite_card_set_nat(A3))
       => ( finite5631907774883551598et_nat(collect_list_set_nat(aa_nat2247512698059773005_nat_o(aTP_Lamm_aaw(A3),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A3)),K)),one_one_nat)),finite_card_set_nat(A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_7440_card__lists__distinct__length__eq,axiom,
    ! [A3: set_list_nat,K: nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),finite_card_list_nat(A3))
       => ( finite7325466520557071688st_nat(collec5989764272469232197st_nat(aa_nat158606821896287091_nat_o(aTP_Lamm_aax(A3),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A3)),K)),one_one_nat)),finite_card_list_nat(A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_7441_card__lists__distinct__length__eq,axiom,
    ! [A3: set_int,K: nat] :
      ( finite_finite_int(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),finite_card_int(A3))
       => ( finite_card_list_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_aay(A3),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A3)),K)),one_one_nat)),finite_card_int(A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_7442_card__lists__distinct__length__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3))
       => ( finite249151656366948015at_nat(collec3343600615725829874at_nat(aa_nat1970911183850433186_nat_o(aTP_Lamm_aaz(A3),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),K)),one_one_nat)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_7443_card__lists__distinct__length__eq,axiom,
    ! [A3: set_char,K: nat] :
      ( finite_finite_char(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),finite_card_char(A3))
       => ( finite1726956870759998265t_char(collect_list_char(aa_nat3000828340120075458char_o(aTP_Lamm_aba(A3),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_char(A3)),K)),one_one_nat)),finite_card_char(A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_7444_card__lists__distinct__length__eq,axiom,
    ! [A3: set_nat,K: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_set_nat_nat(finite_card_nat,A3))
       => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_abb(A3),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A3)),K)),one_one_nat)),aa_set_nat_nat(finite_card_nat,A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_7445_card__lists__distinct__length__eq_H,axiom,
    ! [K: nat,A3: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),finite_card_int(A3))
     => ( finite_card_list_int(collect_list_int(aa_set1055383481605741461_int_o(aTP_Lamm_abc(K),A3))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A3)),K)),one_one_nat)),finite_card_int(A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7446_card__lists__distinct__length__eq_H,axiom,
    ! [K: nat,A3: set_Product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),finite410649719033368117t_unit(A3))
     => ( finite3744894192721311803t_unit(collec4736972532740828926t_unit(aa_set1739257761580076089unit_o(aTP_Lamm_abd(K),A3))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite410649719033368117t_unit(A3)),K)),one_one_nat)),finite410649719033368117t_unit(A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7447_card__lists__distinct__length__eq_H,axiom,
    ! [K: nat,A3: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),finite_card_set_nat(A3))
     => ( finite5631907774883551598et_nat(collect_list_set_nat(aa_set474910819808044769_nat_o(aTP_Lamm_abe(K),A3))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A3)),K)),one_one_nat)),finite_card_set_nat(A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7448_card__lists__distinct__length__eq_H,axiom,
    ! [K: nat,A3: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),finite_card_list_nat(A3))
     => ( finite7325466520557071688st_nat(collec5989764272469232197st_nat(aa_set5589568368246272301_nat_o(aTP_Lamm_abf(K),A3))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A3)),K)),one_one_nat)),finite_card_list_nat(A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7449_card__lists__distinct__length__eq_H,axiom,
    ! [K: nat,A3: set_char] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),finite_card_char(A3))
     => ( finite1726956870759998265t_char(collect_list_char(aa_set3791436276075643467char_o(aTP_Lamm_abg(K),A3))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_char(A3)),K)),one_one_nat)),finite_card_char(A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7450_card__lists__distinct__length__eq_H,axiom,
    ! [K: nat,A3: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_set_nat_nat(finite_card_nat,A3))
     => ( finite_card_list_nat(collect_list_nat(aa_set3062951307625612365_nat_o(aTP_Lamm_abh(K),A3))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A3)),K)),one_one_nat)),aa_set_nat_nat(finite_card_nat,A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7451_apfst__convE,axiom,
    ! [Q2: produc7509114972854644512nteger,F: fun_fu9036519777977489245e_term,P: produc7509114972854644512nteger] :
      ( ( Q2 = produc5151272107987982333nteger(F,P) )
     => ~ ! [X4: fun_Co3460408610320954864e_term,Y4: produc8923325533196201883nteger] :
            ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X4),Y4) )
           => ( Q2 != aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,aa_fun8764196472072707742e_term(F,X4)),Y4) ) ) ) ).

% apfst_convE
tff(fact_7452_apfst__convE,axiom,
    ! [Q2: produc2453742332425399446nteger,F: fun_fu3093988009420988519e_term,P: produc7509114972854644512nteger] :
      ( ( Q2 = produc2260500861290198387nteger(F,P) )
     => ~ ! [X4: fun_Co3460408610320954864e_term,Y4: produc8923325533196201883nteger] :
            ( ( P = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X4),Y4) )
           => ( Q2 != aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,aa_fun1525513021908036776e_term(F,X4)),Y4) ) ) ) ).

% apfst_convE
tff(fact_7453_apfst__convE,axiom,
    ! [Q2: produc9140211490168418807atural,F: fun_fu7248590935147425627e_term,P: produc9140211490168418807atural] :
      ( ( Q2 = produc1921288387873046805atural(F,P) )
     => ~ ! [X4: fun_Pr285225112257423215e_term,Y4: produc7822875418678951345atural] :
            ( ( P = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,X4),Y4) )
           => ( Q2 != aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,aa_fun6933112156808570396e_term(F,X4)),Y4) ) ) ) ).

% apfst_convE
tff(fact_7454_apfst__convE,axiom,
    ! [Q2: produc7509114972854644512nteger,F: fun_fu1456393871774969959e_term,P: produc2453742332425399446nteger] :
      ( ( Q2 = produc177741965484698227nteger(F,P) )
     => ~ ! [X4: fun_Pr5098669352780829690e_term,Y4: produc8923325533196201883nteger] :
            ( ( P = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X4),Y4) )
           => ( Q2 != aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,aa_fun1536964767670826920e_term(F,X4)),Y4) ) ) ) ).

% apfst_convE
tff(fact_7455_apfst__convE,axiom,
    ! [Q2: produc2453742332425399446nteger,F: fun_fu2505754153367914865e_term,P: produc2453742332425399446nteger] :
      ( ( Q2 = produc183566614307960809nteger(F,P) )
     => ~ ! [X4: fun_Pr5098669352780829690e_term,Y4: produc8923325533196201883nteger] :
            ( ( P = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X4),Y4) )
           => ( Q2 != aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,aa_fun1248587000295462322e_term(F,X4)),Y4) ) ) ) ).

% apfst_convE
tff(fact_7456_apfst__convE,axiom,
    ! [Q2: produc3043004577601930755nt_int,F: fun_fu836511555612062035e_term,P: produc3043004577601930755nt_int] :
      ( ( Q2 = produc7857123342865048869nt_int(F,P) )
     => ~ ! [X4: fun_Pr7763680713840462443e_term,Y4: product_prod_int_int] :
            ( ( P = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Y4) )
           => ( Q2 != aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,aa_fun49903357033182740e_term(F,X4)),Y4) ) ) ) ).

% apfst_convE
tff(fact_7457_apfst__convE,axiom,
    ! [Q2: produc4788131091073775919nt_int,F: fun_fu4835546211937200935e_term,P: produc3043004577601930755nt_int] :
      ( ( Q2 = produc7965835600623382609nt_int(F,P) )
     => ~ ! [X4: fun_Pr7763680713840462443e_term,Y4: product_prod_int_int] :
            ( ( P = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Y4) )
           => ( Q2 != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,aa_fun3544567364070095336e_term(F,X4)),Y4) ) ) ) ).

% apfst_convE
tff(fact_7458_apfst__convE,axiom,
    ! [Q2: produc3043004577601930755nt_int,F: fun_fu1176768540439120679e_term,P: produc4788131091073775919nt_int] :
      ( ( Q2 = produc479425830829021777nt_int(F,P) )
     => ~ ! [X4: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
            ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y4) )
           => ( Q2 != aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,aa_fun1932935823081541608e_term(F,X4)),Y4) ) ) ) ).

% apfst_convE
tff(fact_7459_apfst__convE,axiom,
    ! [Q2: produc4788131091073775919nt_int,F: fun_fu4810317505167811835e_term,P: produc4788131091073775919nt_int] :
      ( ( Q2 = produc5901819666759714173nt_int(F,P) )
     => ~ ! [X4: fun_in5730379987311621695e_term,Y4: product_prod_int_int] :
            ( ( P = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Y4) )
           => ( Q2 != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,aa_fun5947755341433680828e_term(F,X4)),Y4) ) ) ) ).

% apfst_convE
tff(fact_7460_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_abj),Xa),Xb)) ).

% times_int.abs_eq
tff(fact_7461_rat__minus__code,axiom,
    ! [P: rat,Q2: rat] : quotient_of(minus_minus_rat(P,Q2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_abl(Q2)),quotient_of(P)) ).

% rat_minus_code
tff(fact_7462_rat__one__code,axiom,
    quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int) ).

% rat_one_code
tff(fact_7463_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% rat_zero_code
tff(fact_7464_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(numeral_numeral_rat(K)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(3)
tff(fact_7465_quotient__of__number_I4_J,axiom,
    quotient_of(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% quotient_of_number(4)
tff(fact_7466_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(K))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).

% quotient_of_number(5)
tff(fact_7467_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_int,N: nat] :
      ( finite_finite_int(A3)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_aay(A3),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_7468_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A3)
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1970911183850433186_nat_o(aTP_Lamm_aaz(A3),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_7469_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_char,N: nat] :
      ( finite_finite_char(A3)
     => finite3619747586194699384t_char(collect_list_char(aa_nat3000828340120075458char_o(aTP_Lamm_aba(A3),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_7470_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_nat,N: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_abb(A3),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_7471_int_Oabs__induct,axiom,
    ! [Pa: fun_int_o,Xa: int] :
      ( ! [Y4: product_prod_nat_nat] : aa_int_o(Pa,aa_Pro4240683299535347697at_int(abs_Integ,Y4))
     => aa_int_o(Pa,Xa) ) ).

% int.abs_induct
tff(fact_7472_quotient__of__div,axiom,
    ! [R: rat,N: int,D: int] :
      ( ( quotient_of(R) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,N),D) )
     => ( R = divide_divide_rat(aa_int_rat(ring_1_of_int_rat,N),aa_int_rat(ring_1_of_int_rat,D)) ) ) ).

% quotient_of_div
tff(fact_7473_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X4: nat,Y4: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4)) ).

% eq_Abs_Integ
tff(fact_7474_nat_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,Xa)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),Xa) ).

% nat.abs_eq
tff(fact_7475_quotient__of__denom__pos,axiom,
    ! [R: rat,P: int,Q2: int] :
      ( ( quotient_of(R) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% quotient_of_denom_pos
tff(fact_7476_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_7477_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_7478_rat__floor__code,axiom,
    ! [P: rat] : archim3151403230148437115or_rat(P) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,divide_divide_int),quotient_of(P)) ).

% rat_floor_code
tff(fact_7479_rat__uminus__code,axiom,
    ! [P: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_abm),quotient_of(P)) ).

% rat_uminus_code
tff(fact_7480_rat__abs__code,axiom,
    ! [P: rat] : quotient_of(abs_abs_rat(P)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_abn),quotient_of(P)) ).

% rat_abs_code
tff(fact_7481_rat__less__eq__code,axiom,
    ! [P: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P),Q2)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_abp(Q2)),quotient_of(P)) ) ).

% rat_less_eq_code
tff(fact_7482_rat__less__code,axiom,
    ! [P: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P),Q2)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_abr(Q2)),quotient_of(P)) ) ).

% rat_less_code
tff(fact_7483_uminus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_mh),Xa)) ).

% uminus_int.abs_eq
tff(fact_7484_rat__sgn__code,axiom,
    ! [P: rat] : quotient_of(sgn_sgn_rat(P)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(sgn_sgn_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,quotient_of(P)))),one_one_int) ).

% rat_sgn_code
tff(fact_7485_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_7486_of__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_Pro4240683299535347697at_int(abs_Integ,Xa)) = aa_Pro4764841431850000930nteger(produc1830744345554046123nteger(aTP_Lamm_abs),Xa) ).

% of_int.abs_eq
tff(fact_7487_of__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat] : aa_int_rat(ring_1_of_int_rat,aa_Pro4240683299535347697at_int(abs_Integ,Xa)) = aa_Pro3608043709957902237at_rat(produc6207742614233964070at_rat(aTP_Lamm_abt),Xa) ).

% of_int.abs_eq
tff(fact_7488_of__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat] : aa_int_int(ring_1_of_int_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)) = aa_Pro4240683299535347697at_int(produc6840382203811409530at_int(aTP_Lamm_abu),Xa) ).

% of_int.abs_eq
tff(fact_7489_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_abw),Xa),Xb) ) ).

% less_int.abs_eq
tff(fact_7490_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_aby),Xa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_7491_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_aca),Xa),Xb)) ).

% plus_int.abs_eq
tff(fact_7492_rat__times__code,axiom,
    ! [P: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P),Q2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_acc(Q2)),quotient_of(P)) ).

% rat_times_code
tff(fact_7493_rat__divide__code,axiom,
    ! [P: rat,Q2: rat] : quotient_of(divide_divide_rat(P,Q2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_ace(Q2)),quotient_of(P)) ).

% rat_divide_code
tff(fact_7494_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_acg),Xa),Xb)) ).

% minus_int.abs_eq
tff(fact_7495_rat__plus__code,axiom,
    ! [P: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P),Q2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aci(Q2)),quotient_of(P)) ).

% rat_plus_code
tff(fact_7496_quotient__of__int,axiom,
    ! [A: int] : quotient_of(aa_int_rat(of_int,A)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),one_one_int) ).

% quotient_of_int
tff(fact_7497_rat__inverse__code,axiom,
    ! [P: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_acj),quotient_of(P)) ).

% rat_inverse_code
tff(fact_7498_numeral__xor__num,axiom,
    ! [M: num,N: num] : bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,M),aa_num_Code_natural(numera5444537566228673987atural,N)) = case_o5621594795226839503al_num(zero_z2226904508553997617atural,numera5444537566228673987atural,aa_num_option_num(bit_un2480387367778600638or_num(M),N)) ).

% numeral_xor_num
tff(fact_7499_numeral__xor__num,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = case_o356765784539232260er_num(zero_z3403309356797280102nteger,numera6620942414471956472nteger,aa_num_option_num(bit_un2480387367778600638or_num(M),N)) ).

% numeral_xor_num
tff(fact_7500_numeral__xor__num,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_un2480387367778600638or_num(M),N)) ).

% numeral_xor_num
tff(fact_7501_numeral__xor__num,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_option_num_nat(case_option_nat_num(zero_zero_nat,numeral_numeral_nat),aa_num_option_num(bit_un2480387367778600638or_num(M),N)) ).

% numeral_xor_num
tff(fact_7502_diff__numeral__special_I8_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(neg_numeral_sub_int(one),M) ).

% diff_numeral_special(8)
tff(fact_7503_diff__numeral__special_I8_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = neg_nu5755505904847501662nteger(one,M) ).

% diff_numeral_special(8)
tff(fact_7504_diff__numeral__special_I8_J,axiom,
    ! [M: num] : minus_minus_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = neg_numeral_sub_rat(one,M) ).

% diff_numeral_special(8)
tff(fact_7505_diff__numeral__special_I7_J,axiom,
    ! [N: num] : 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,aa_num_int(numeral_numeral_int,N))) = aa_num_int(neg_numeral_sub_int(N),one) ).

% diff_numeral_special(7)
tff(fact_7506_diff__numeral__special_I7_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = neg_nu5755505904847501662nteger(N,one) ).

% diff_numeral_special(7)
tff(fact_7507_diff__numeral__special_I7_J,axiom,
    ! [N: num] : minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = neg_numeral_sub_rat(N,one) ).

% diff_numeral_special(7)
tff(fact_7508_inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% inverse_zero
tff(fact_7509_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_7510_inverse__eq__1__iff,axiom,
    ! [Xa: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,Xa) = one_one_rat )
    <=> ( Xa = one_one_rat ) ) ).

% inverse_eq_1_iff
tff(fact_7511_inverse__1,axiom,
    aa_rat_rat(inverse_inverse_rat,one_one_rat) = one_one_rat ).

% inverse_1
tff(fact_7512_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_7513_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_7514_inverse__less__iff__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_7515_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_7516_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% inverse_negative_iff_negative
tff(fact_7517_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% inverse_positive_iff_positive
tff(fact_7518_sub__num__simps_I1_J,axiom,
    neg_nu5755505904847501662nteger(one,one) = zero_z3403309356797280102nteger ).

% sub_num_simps(1)
tff(fact_7519_sub__num__simps_I1_J,axiom,
    neg_numeral_sub_rat(one,one) = zero_zero_rat ).

% sub_num_simps(1)
tff(fact_7520_sub__num__simps_I1_J,axiom,
    aa_num_int(neg_numeral_sub_int(one),one) = zero_zero_int ).

% sub_num_simps(1)
tff(fact_7521_inverse__le__iff__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_7522_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_7523_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% left_inverse
tff(fact_7524_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(inverse_inverse_rat,A)) = one_one_rat ) ) ).

% right_inverse
tff(fact_7525_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,numeral_numeral_rat(W)) = divide_divide_rat(one_one_rat,numeral_numeral_rat(W)) ).

% inverse_eq_divide_numeral
tff(fact_7526_add__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(neg_numeral_sub_int(M),N) ).

% add_neg_numeral_simps(1)
tff(fact_7527_add__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = neg_nu5755505904847501662nteger(M,N) ).

% add_neg_numeral_simps(1)
tff(fact_7528_add__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = neg_numeral_sub_rat(M,N) ).

% add_neg_numeral_simps(1)
tff(fact_7529_add__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_num_int(neg_numeral_sub_int(N),M) ).

% add_neg_numeral_simps(2)
tff(fact_7530_add__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N)) = neg_nu5755505904847501662nteger(N,M) ).

% add_neg_numeral_simps(2)
tff(fact_7531_add__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),numeral_numeral_rat(N)) = neg_numeral_sub_rat(N,M) ).

% add_neg_numeral_simps(2)
tff(fact_7532_semiring__norm_I167_J,axiom,
    ! [V: num,W: num,Y3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W)),Y3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(neg_numeral_sub_int(W),V)),Y3) ).

% semiring_norm(167)
tff(fact_7533_semiring__norm_I167_J,axiom,
    ! [V: num,W: num,Y3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Y3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,neg_nu5755505904847501662nteger(W,V)),Y3) ).

% semiring_norm(167)
tff(fact_7534_semiring__norm_I167_J,axiom,
    ! [V: num,W: num,Y3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(W)),Y3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,neg_numeral_sub_rat(W,V)),Y3) ).

% semiring_norm(167)
tff(fact_7535_semiring__norm_I166_J,axiom,
    ! [V: num,W: num,Y3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(neg_numeral_sub_int(V),W)),Y3) ).

% semiring_norm(166)
tff(fact_7536_semiring__norm_I166_J,axiom,
    ! [V: num,W: num,Y3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,neg_nu5755505904847501662nteger(V,W)),Y3) ).

% semiring_norm(166)
tff(fact_7537_semiring__norm_I166_J,axiom,
    ! [V: num,W: num,Y3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),Y3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,neg_numeral_sub_rat(V,W)),Y3) ).

% semiring_norm(166)
tff(fact_7538_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) = divide_divide_rat(one_one_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_7539_diff__numeral__special_I2_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int) = aa_num_int(neg_numeral_sub_int(M),one) ).

% diff_numeral_special(2)
tff(fact_7540_diff__numeral__special_I2_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),one_one_Code_integer) = neg_nu5755505904847501662nteger(M,one) ).

% diff_numeral_special(2)
tff(fact_7541_diff__numeral__special_I2_J,axiom,
    ! [M: num] : minus_minus_rat(numeral_numeral_rat(M),one_one_rat) = neg_numeral_sub_rat(M,one) ).

% diff_numeral_special(2)
tff(fact_7542_diff__numeral__special_I1_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_num_int(neg_numeral_sub_int(one),N) ).

% diff_numeral_special(1)
tff(fact_7543_diff__numeral__special_I1_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N)) = neg_nu5755505904847501662nteger(one,N) ).

% diff_numeral_special(1)
tff(fact_7544_diff__numeral__special_I1_J,axiom,
    ! [N: num] : minus_minus_rat(one_one_rat,numeral_numeral_rat(N)) = neg_numeral_sub_rat(one,N) ).

% diff_numeral_special(1)
tff(fact_7545_add__neg__numeral__special_I1_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) = aa_num_int(neg_numeral_sub_int(one),M) ).

% add_neg_numeral_special(1)
tff(fact_7546_add__neg__numeral__special_I1_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))) = neg_nu5755505904847501662nteger(one,M) ).

% add_neg_numeral_special(1)
tff(fact_7547_add__neg__numeral__special_I1_J,axiom,
    ! [M: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))) = neg_numeral_sub_rat(one,M) ).

% add_neg_numeral_special(1)
tff(fact_7548_add__neg__numeral__special_I2_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) = aa_num_int(neg_numeral_sub_int(one),M) ).

% add_neg_numeral_special(2)
tff(fact_7549_add__neg__numeral__special_I2_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) = neg_nu5755505904847501662nteger(one,M) ).

% add_neg_numeral_special(2)
tff(fact_7550_add__neg__numeral__special_I2_J,axiom,
    ! [M: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M))),one_one_rat) = neg_numeral_sub_rat(one,M) ).

% add_neg_numeral_special(2)
tff(fact_7551_add__neg__numeral__special_I3_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(neg_numeral_sub_int(M),one) ).

% add_neg_numeral_special(3)
tff(fact_7552_add__neg__numeral__special_I3_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = neg_nu5755505904847501662nteger(M,one) ).

% add_neg_numeral_special(3)
tff(fact_7553_add__neg__numeral__special_I3_J,axiom,
    ! [M: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = neg_numeral_sub_rat(M,one) ).

% add_neg_numeral_special(3)
tff(fact_7554_add__neg__numeral__special_I4_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_num_int(numeral_numeral_int,N)) = aa_num_int(neg_numeral_sub_int(N),one) ).

% add_neg_numeral_special(4)
tff(fact_7555_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = neg_nu5755505904847501662nteger(N,one) ).

% add_neg_numeral_special(4)
tff(fact_7556_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),numeral_numeral_rat(N)) = neg_numeral_sub_rat(N,one) ).

% add_neg_numeral_special(4)
tff(fact_7557_minus__sub__one__diff__one,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(neg_numeral_sub_int(M),one))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) ).

% minus_sub_one_diff_one
tff(fact_7558_minus__sub__one__diff__one,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5755505904847501662nteger(M,one))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) ).

% minus_sub_one_diff_one
tff(fact_7559_minus__sub__one__diff__one,axiom,
    ! [M: num] : minus_minus_rat(aa_rat_rat(uminus_uminus_rat,neg_numeral_sub_rat(M,one)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M)) ).

% minus_sub_one_diff_one
tff(fact_7560_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_7561_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_7562_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
     => ( ( A != zero_zero_rat )
       => ( ( B != zero_zero_rat )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_7563_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
tff(fact_7564_field__class_Ofield__inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% field_class.field_inverse_zero
tff(fact_7565_inverse__less__imp__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% inverse_less_imp_less
tff(fact_7566_less__imp__inverse__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less
tff(fact_7567_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_7568_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_7569_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% inverse_negative_imp_negative
tff(fact_7570_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_7571_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat) ) ).

% negative_imp_inverse_negative
tff(fact_7572_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ).

% positive_imp_inverse_positive
tff(fact_7573_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_7574_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_7575_inverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = one_one_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = B ) ) ).

% inverse_unique
tff(fact_7576_inverse__eq__divide,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,A) = divide_divide_rat(one_one_rat,A) ).

% inverse_eq_divide
tff(fact_7577_power__mult__power__inverse__commute,axiom,
    ! [Xa: rat,M: nat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xa),M)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xa)),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xa)),N)),aa_nat_rat(power_power_rat(Xa),M)) ).

% power_mult_power_inverse_commute
tff(fact_7578_power__mult__inverse__distrib,axiom,
    ! [Xa: rat,M: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xa),M)),aa_rat_rat(inverse_inverse_rat,Xa)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,Xa)),aa_nat_rat(power_power_rat(Xa),M)) ).

% power_mult_inverse_distrib
tff(fact_7579_nonzero__abs__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( abs_abs_rat(aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,abs_abs_rat(A)) ) ) ).

% nonzero_abs_inverse
tff(fact_7580_mult__inverse__of__int__commute,axiom,
    ! [Xa: int,Xb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xa))),Xb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xa))) ).

% mult_inverse_of_int_commute
tff(fact_7581_inverse__le__imp__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% inverse_le_imp_le
tff(fact_7582_le__imp__inverse__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le
tff(fact_7583_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_7584_le__imp__inverse__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_7585_inverse__le__1__iff,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,Xa)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xa) ) ) ).

% inverse_le_1_iff
tff(fact_7586_one__less__inverse,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% one_less_inverse
tff(fact_7587_one__less__inverse__iff,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,Xa))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),one_one_rat) ) ) ).

% one_less_inverse_iff
tff(fact_7588_inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(inverse_inverse_rat,A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% inverse_add
tff(fact_7589_division__ring__inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_7590_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% field_class.field_inverse
tff(fact_7591_division__ring__inverse__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),minus_minus_rat(B,A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_7592_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = divide_divide_rat(one_one_rat,A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_7593_inverse__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ).

% inverse_le_iff
tff(fact_7594_inverse__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ) ).

% inverse_less_iff
tff(fact_7595_sub__BitM__One__eq,axiom,
    ! [N: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,N)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(N),one)) ).

% sub_BitM_One_eq
tff(fact_7596_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),aa_num_int(numeral_numeral_int,K))) = divide_divide_rat(one_one_rat,numeral_numeral_rat(K)) ).

% Frct_code_post(5)
tff(fact_7597_floor__rat__def,axiom,
    ! [Xa: rat] : archim3151403230148437115or_rat(Xa) = the_int(aTP_Lamm_ack(Xa)) ).

% floor_rat_def
tff(fact_7598_less__eq__int_Orep__eq,axiom,
    ! [Xa: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_aby),aa_int1417691210696471063at_nat(rep_Integ,Xa)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_7599_less__int_Orep__eq,axiom,
    ! [Xa: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_abw),aa_int1417691210696471063at_nat(rep_Integ,Xa)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_int.rep_eq
tff(fact_7600_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L))) = divide_divide_rat(numeral_numeral_rat(K),numeral_numeral_rat(L)) ).

% Frct_code_post(6)
tff(fact_7601_nat_Orep__eq,axiom,
    ! [Xa: int] : aa_int_nat(nat2,Xa) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,Xa)) ).

% nat.rep_eq
tff(fact_7602_Frct__code__post_I1_J,axiom,
    ! [A: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A)) = zero_zero_rat ).

% Frct_code_post(1)
tff(fact_7603_Frct__code__post_I2_J,axiom,
    ! [A: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),zero_zero_int)) = zero_zero_rat ).

% Frct_code_post(2)
tff(fact_7604_Frct__code__post_I3_J,axiom,
    frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_7605_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,A)),B)) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ).

% Frct_code_post(7)
tff(fact_7606_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),aa_int_int(uminus_uminus_int,B))) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ).

% Frct_code_post(8)
tff(fact_7607_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int)) = numeral_numeral_rat(K) ).

% Frct_code_post(4)
tff(fact_7608_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_mh)) ).

% uminus_int_def
tff(fact_7609_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_acl) ).

% prod_encode_def
tff(fact_7610_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ).

% le_prod_encode_1
tff(fact_7611_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ).

% le_prod_encode_2
tff(fact_7612_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_7613_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_abj)) ).

% times_int_def
tff(fact_7614_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_acg)) ).

% minus_int_def
tff(fact_7615_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_aca)) ).

% plus_int_def
tff(fact_7616_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_acm)) ).

% pred_nat_def
tff(fact_7617_gcd__integer__def,axiom,
    gcd_gcd_Code_integer = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),gcd_gcd_int) ).

% gcd_integer_def
tff(fact_7618_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_aby)) ).

% less_eq_int_def
tff(fact_7619_less__int__def,axiom,
    ord_less_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_abw)) ).

% less_int_def
tff(fact_7620_Code__Numeral_Odup__def,axiom,
    code_dup = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),aTP_Lamm_acn) ).

% Code_Numeral.dup_def
tff(fact_7621_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_7622_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_7623_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_7624_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_7625_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_7626_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_7627_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_7628_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_7629_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_7630_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_min_right
tff(fact_7631_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_min_left
tff(fact_7632_atLeastLessThan0,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_7633_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_7634_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_aco,N)),M) ).

% min_Suc1
tff(fact_7635_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_acp,N)),M) ).

% min_Suc2
tff(fact_7636_num__of__integer__def,axiom,
    code_num_of_integer = map_fu1227494855608507351um_num(code_int_of_integer,id_num,comp_nat_num_int(num_of_nat,nat2)) ).

% num_of_integer_def
tff(fact_7637_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_7638_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_7639_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_7640_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L))
     => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
            ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(L,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.psimps
tff(fact_7641_and__int_Opelims,axiom,
    ! [Xa: int,Xaa: int,Y3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Xaa) = Y3 )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Xaa))
       => ~ ( ( Y3 = $ite(
                  ( member_int(Xa,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & member_int(Xaa,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
                  aa_int_int(aa_int_fun_int_int(plus_plus_int,
                      aa_o_int(zero_n2684676970156552555ol_int,
                        ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_7642_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,Pa: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [K2: int,L3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K2),L3))
           => ( ( ~ ( member_int(K2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & member_int(L3,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(Pa,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(Pa,K2),L3) ) )
       => aa_int_o(aa_int_fun_int_o(Pa,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_7643_upto_Opinduct,axiom,
    ! [A0: int,A1: int,Pa: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [I2: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I2),J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
               => aa_int_o(aa_int_fun_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(Pa,I2),J2) ) )
       => aa_int_o(aa_int_fun_int_o(Pa,A0),A1) ) ) ).

% upto.pinduct
tff(fact_7644_Code__Numeral_Osub__def,axiom,
    code_sub = map_fu8540202848259802501nteger(id_num,map_fu8638147718074629079nteger(id_num,code_integer_of_int),aTP_Lamm_acq) ).

% Code_Numeral.sub_def
tff(fact_7645_Gcd__remove0__nat,axiom,
    ! [M2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M2)
     => ( aa_set_nat_nat(gcd_Gcd_nat,M2) = aa_set_nat_nat(gcd_Gcd_nat,minus_minus_set_nat(M2,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_7646_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xa: nat,Y3: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acr,C),aa_nat_set_nat(set_or4665077453230672383an_nat(Xa),Y3)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y3),
        aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y3),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y3),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_7647_Gcd__greatest__nat,axiom,
    ! [A3: set_nat,A: nat] :
      ( ! [B2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),A3)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_set_nat_nat(gcd_Gcd_nat,A3)) ) ).

% Gcd_greatest_nat
tff(fact_7648_Gcd__dvd__nat,axiom,
    ! [A: nat,A3: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A),A3)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(gcd_Gcd_nat,A3)),A) ) ).

% Gcd_dvd_nat
tff(fact_7649_Gcd__nat__eq__one,axiom,
    ! [N5: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,one_one_nat),N5)
     => ( aa_set_nat_nat(gcd_Gcd_nat,N5) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_7650_zero__notin__Suc__image,axiom,
    ! [A3: set_nat] : ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),image_nat_nat(suc,A3)) ).

% zero_notin_Suc_image
tff(fact_7651_Gcd__in,axiom,
    ! [A3: set_nat] :
      ( ! [A2: nat,B2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,A2),A3)
         => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,B2),A3)
           => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B2)),A3) ) )
     => ( ( A3 != bot_bot_set_nat )
       => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_set_nat_nat(gcd_Gcd_nat,A3)),A3) ) ) ).

% Gcd_in
tff(fact_7652_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_7653_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_7654_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_7655_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_7656_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_7657_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_7658_abs__Gcd__eq,axiom,
    ! [K5: set_int] : aa_int_int(abs_abs_int,gcd_Gcd_int(K5)) = gcd_Gcd_int(K5) ).

% abs_Gcd_eq
tff(fact_7659_Gcd__abs__eq,axiom,
    ! [K5: set_int] : gcd_Gcd_int(image_int_int(abs_abs_int,K5)) = gcd_Gcd_int(K5) ).

% Gcd_abs_eq
tff(fact_7660_Gcd__int__eq,axiom,
    ! [N5: set_nat] : gcd_Gcd_int(image_nat_int(semiri1314217659103216013at_int,N5)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,N5)) ).

% Gcd_int_eq
tff(fact_7661_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] : aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(aTP_Lamm_acs,K5)) = aa_int_nat(nat2,gcd_Gcd_int(K5)) ).

% Gcd_nat_abs_eq
tff(fact_7662_Gcd__dvd__int,axiom,
    ! [A: int,A3: set_int] :
      ( member_int(A,A3)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,gcd_Gcd_int(A3)),A) ) ).

% Gcd_dvd_int
tff(fact_7663_Gcd__greatest__int,axiom,
    ! [A3: set_int,A: int] :
      ( ! [B2: int] :
          ( member_int(B2,A3)
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),gcd_Gcd_int(A3)) ) ).

% Gcd_greatest_int
tff(fact_7664_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_act,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_7665_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K5)) ).

% Gcd_int_greater_eq_0
tff(fact_7666_Gcd__int__def,axiom,
    ! [K5: set_int] : gcd_Gcd_int(K5) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),K5))) ).

% Gcd_int_def
tff(fact_7667_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),N)) ).

% int_numeral_and_not_num
tff(fact_7668_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(N),M)) ).

% int_numeral_not_and_num
tff(fact_7669_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_7670_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_7671_Inf__unit__def,axiom,
    ! [Uu: set_Product_unit] : comple2584293577114468500t_unit(Uu) = product_Unity ).

% Inf_unit_def
tff(fact_7672_bot__unit__def,axiom,
    bot_bot_Product_unit = product_Unity ).

% bot_unit_def
tff(fact_7673_Sup__unit__def,axiom,
    ! [Uu: set_Product_unit] : comple4687483117567863418t_unit(Uu) = product_Unity ).

% Sup_unit_def
tff(fact_7674_inf__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : inf_inf_Product_unit(Uu,Uv) = product_Unity ).

% inf_unit_def
tff(fact_7675_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_7676_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_7677_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_7678_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
tff(fact_7679_range__mod,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acu,N),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ) ) ).

% range_mod
tff(fact_7680_measure__function__int,axiom,
    fun_is_measure_int(comp_int_nat_int(nat2,abs_abs_int)) ).

% measure_function_int
tff(fact_7681_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_7682_range__abs__Nats,axiom,
    image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_7683_top__unit__def,axiom,
    top_top_Product_unit = product_Unity ).

% top_unit_def
tff(fact_7684_UNIV__unit,axiom,
    top_to1996260823553986621t_unit = insert_Product_unit(product_Unity,bot_bo3957492148770167129t_unit) ).

% UNIV_unit
tff(fact_7685_infinite__UNIV__int,axiom,
    ~ finite_finite_int(top_top_set_int) ).

% infinite_UNIV_int
tff(fact_7686_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_7687_int__in__range__abs,axiom,
    ! [N: nat] : member_int(aa_nat_int(semiri1314217659103216013at_int,N),image_int_int(abs_abs_int,top_top_set_int)) ).

% int_in_range_abs
tff(fact_7688_Gcd__eq__Max,axiom,
    ! [M2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M2)
     => ( ( M2 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),M2)
         => ( aa_set_nat_nat(gcd_Gcd_nat,M2) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_acv,M2))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_7689_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cx,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_7690_sup__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : sup_sup_Product_unit(Uu,Uv) = product_Unity ).

% sup_unit_def
tff(fact_7691_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_7692_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_7693_Sup__nat__def,axiom,
    ! [X6: set_nat] :
      aa_set_nat_nat(complete_Sup_Sup_nat,X6) = $ite(X6 = bot_bot_set_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,X6)) ).

% Sup_nat_def
tff(fact_7694_divide__nat__def,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = $ite(N = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_acw(M),N)))) ).

% divide_nat_def
tff(fact_7695_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_acx(N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_7696_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_7697_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_cw,N))) = aa_int_int(abs_abs_int,N) ) ) ).

% Max_divisors_self_int
tff(fact_7698_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_acy(N),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_7699_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_7700_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : aa_set_nat_nat(gcd_Gcd_nat,set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_gcd_nat,Xs),zero_zero_nat) ).

% Gcd_nat_set_eq_fold
tff(fact_7701_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list_int] : gcd_Gcd_int(set_int2(Xs)) = aa_int_int(fold_int_int(gcd_gcd_int,Xs),zero_zero_int) ).

% Gcd_int_set_eq_fold
tff(fact_7702_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_nat_fun_set_nat_o(aTP_Lamm_acz(N),K))) ).

% binomial_def
tff(fact_7703_Field__natLeq__on,axiom,
    ! [N: nat] : field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ada(N)))) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fl,N)) ).

% Field_natLeq_on
tff(fact_7704_full__exhaustive__int_H_Ocases,axiom,
    ! [Xa: produc3043004577601930755nt_int] :
      ~ ! [F5: fun_Pr7763680713840462443e_term,D3: int,I2: int] : Xa != aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,F5),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D3),I2)) ).

% full_exhaustive_int'.cases
tff(fact_7705_exhaustive__int_H_Ocases,axiom,
    ! [Xa: produc4788131091073775919nt_int] :
      ~ ! [F5: fun_in5730379987311621695e_term,D3: int,I2: int] : Xa != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,F5),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D3),I2)) ).

% exhaustive_int'.cases
tff(fact_7706_full__exhaustive__integer_H_Ocases,axiom,
    ! [Xa: produc2453742332425399446nteger] :
      ~ ! [F5: fun_Pr5098669352780829690e_term,D3: code_integer,I2: code_integer] : Xa != aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,F5),produc1086072967326762835nteger(D3,I2)) ).

% full_exhaustive_integer'.cases
tff(fact_7707_exhaustive__integer_H_Ocases,axiom,
    ! [Xa: produc7509114972854644512nteger] :
      ~ ! [F5: fun_Co3460408610320954864e_term,D3: code_integer,I2: code_integer] : Xa != aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,F5),produc1086072967326762835nteger(D3,I2)) ).

% exhaustive_integer'.cases
tff(fact_7708_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat)) ).

% natLess_def
tff(fact_7709_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_adb,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_7710_sub_Orsp,axiom,
    bNF_re6378542964492772123um_int(fequal_num,bNF_re1822329894187522285nt_int(fequal_num,fequal_int),aTP_Lamm_acq,aTP_Lamm_acq) ).

% sub.rsp
tff(fact_7711_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_7712_gcd__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),gcd_gcd_int,gcd_gcd_int) ).

% gcd_integer.rsp
tff(fact_7713_plus__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),plus_plus_nat,plus_plus_nat) ).

% plus_natural.rsp
tff(fact_7714_dup_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),aTP_Lamm_acn),aTP_Lamm_acn) ).

% dup.rsp
tff(fact_7715_mono__times__nat,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_7716_num__of__integer_Orsp,axiom,
    bNF_re7626690874201225453um_num(fequal_int,fequal_num,comp_nat_num_int(num_of_nat,nat2),comp_nat_num_int(num_of_nat,nat2)) ).

% num_of_integer.rsp
tff(fact_7717_plus__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_adc,plus_plus_rat) ).

% plus_rat.transfer
tff(fact_7718_inverse__rat_Otransfer,axiom,
    aa_fun_rat_rat_o(aa_fun382765352565706371_rat_o(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_add),inverse_inverse_rat) ).

% inverse_rat.transfer
tff(fact_7719_one__rat_Otransfer,axiom,
    aa_rat_o(aa_Pro2298929076809959388_rat_o(pcr_rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)),one_one_rat) ).

% one_rat.transfer
tff(fact_7720_zero__rat_Otransfer,axiom,
    aa_rat_o(aa_Pro2298929076809959388_rat_o(pcr_rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)),zero_zero_rat) ).

% zero_rat.transfer
tff(fact_7721_uminus__rat_Otransfer,axiom,
    aa_fun_rat_rat_o(aa_fun382765352565706371_rat_o(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_ade),uminus_uminus_rat) ).

% uminus_rat.transfer
tff(fact_7722_times__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_adf,times_times_rat) ).

% times_rat.transfer
tff(fact_7723_positive_Otransfer,axiom,
    bNF_re1494630372529172596at_o_o(pcr_rat,fequal_o,aTP_Lamm_adg,positive) ).

% positive.transfer
tff(fact_7724_positive__def,axiom,
    positive = map_fu898904425404107465nt_o_o(rep_Rat,id_o,aTP_Lamm_adg) ).

% positive_def
tff(fact_7725_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_abj),times_times_int) ).

% times_int.transfer
tff(fact_7726_zero__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int) ).

% zero_int.transfer
tff(fact_7727_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_adh,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_7728_uminus__int_Otransfer,axiom,
    aa_fun_int_int_o(aa_fun856177901615811547_int_o(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_mh)),uminus_uminus_int) ).

% uminus_int.transfer
tff(fact_7729_nat_Otransfer,axiom,
    bNF_re4555766996558763186at_nat(pcr_int,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),nat2) ).

% nat.transfer
tff(fact_7730_one__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int) ).

% one_int.transfer
tff(fact_7731_less__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_abw),ord_less_int) ).

% less_int.transfer
tff(fact_7732_less__eq__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_aby),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_7733_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_aca),plus_plus_int) ).

% plus_int.transfer
tff(fact_7734_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_acg),minus_minus_int) ).

% minus_int.transfer
tff(fact_7735_plus__rat__def,axiom,
    plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_adc) ).

% plus_rat_def
tff(fact_7736_inverse__rat__def,axiom,
    inverse_inverse_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_add) ).

% inverse_rat_def
tff(fact_7737_times__rat__def,axiom,
    times_times_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_adf) ).

% times_rat_def
tff(fact_7738_uminus__rat__def,axiom,
    uminus_uminus_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_ade) ).

% uminus_rat_def
tff(fact_7739_one__rat__def,axiom,
    one_one_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) ).

% one_rat_def
tff(fact_7740_zero__rat__def,axiom,
    zero_zero_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)) ).

% zero_rat_def
tff(fact_7741_plus__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,Xb: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa),Xa)
     => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xb),Xb)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa)),aa_Pro6166618844887768661nt_rat(abs_Rat,Xb)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xb)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_7742_inverse__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa),Xa)
     => ( aa_rat_rat(inverse_inverse_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa)) = aa_Pro6166618844887768661nt_rat(abs_Rat,
            $ite(aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa) = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_7743_one__rat_Orsp,axiom,
    aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) ).

% one_rat.rsp
tff(fact_7744_zero__rat_Orsp,axiom,
    aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)) ).

% zero_rat.rsp
tff(fact_7745_Fract_Orsp,axiom,
    bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_adi,aTP_Lamm_adi) ).

% Fract.rsp
tff(fact_7746_ratrel__def,axiom,
    ! [X: product_prod_int_int,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),Xa3)
    <=> ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,X) != zero_zero_int )
        & ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa3) != zero_zero_int )
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,X)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa3)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X)) ) ) ) ).

% ratrel_def
tff(fact_7747_times__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_adf,aTP_Lamm_adf) ).

% times_rat.rsp
tff(fact_7748_uminus__rat_Orsp,axiom,
    aa_fun2650603632653508759_int_o(aa_fun3734617511155415725_int_o(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_ade),aTP_Lamm_ade) ).

% uminus_rat.rsp
tff(fact_7749_positive_Orsp,axiom,
    bNF_re8699439704749558557nt_o_o(ratrel,fequal_o,aTP_Lamm_adg,aTP_Lamm_adg) ).

% positive.rsp
tff(fact_7750_inverse__rat_Orsp,axiom,
    aa_fun2650603632653508759_int_o(aa_fun3734617511155415725_int_o(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_add),aTP_Lamm_add) ).

% inverse_rat.rsp
tff(fact_7751_plus__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_adc,aTP_Lamm_adc) ).

% plus_rat.rsp
tff(fact_7752_uminus__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa),Xa)
     => ( aa_rat_rat(uminus_uminus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa))),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa))) ) ) ).

% uminus_rat.abs_eq
tff(fact_7753_times__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,Xb: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa),Xa)
     => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xb),Xb)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa)),aa_Pro6166618844887768661nt_rat(abs_Rat,Xb)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_fst_int_int,Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xb)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_7754_cr__rat__def,axiom,
    ! [X: product_prod_int_int,Xa3: rat] :
      ( aa_rat_o(cr_rat(X),Xa3)
    <=> ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
        & ( aa_Pro6166618844887768661nt_rat(abs_Rat,X) = Xa3 ) ) ) ).

% cr_rat_def
tff(fact_7755_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_abj),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_abj)) ).

% times_int.rsp
tff(fact_7756_Suc__funpow,axiom,
    ! [N: nat] : compow_fun_nat_nat(N,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N) ).

% Suc_funpow
tff(fact_7757_intrel__iff,axiom,
    ! [Xa: nat,Y3: nat,U: nat,V: nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y3)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),V) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y3) ) ) ).

% intrel_iff
tff(fact_7758_int_Oabs__eq__iff,axiom,
    ! [Xa: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( aa_Pro4240683299535347697at_int(abs_Integ,Xa) = aa_Pro4240683299535347697at_int(abs_Integ,Y3) )
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,Xa),Y3) ) ).

% int.abs_eq_iff
tff(fact_7759_zero__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(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_7760_uminus__int_Orsp,axiom,
    aa_fun3538214411680374423_nat_o(aa_fun6920661532540581549_nat_o(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_mh)),produc2626176000494625587at_nat(aTP_Lamm_mh)) ).

% uminus_int.rsp
tff(fact_7761_nat_Orsp,axiom,
    bNF_re8246922863344978751at_nat(intrel,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat.rsp
tff(fact_7762_one__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(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_7763_intrel__def,axiom,
    intrel = aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_adk) ).

% intrel_def
tff(fact_7764_less__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_abw),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_abw)) ).

% less_int.rsp
tff(fact_7765_int_Orel__eq__transfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),intrel,fequal_int) ).

% int.rel_eq_transfer
tff(fact_7766_less__eq__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_aby),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_aby)) ).

% less_eq_int.rsp
tff(fact_7767_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_acg),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_acg)) ).

% minus_int.rsp
tff(fact_7768_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_aca),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_aca)) ).

% plus_int.rsp
tff(fact_7769_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N)),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% less_eq
tff(fact_7770_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N)),transi2905341329935302413cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% pred_nat_trancl_eq_le
tff(fact_7771_upto_Opelims,axiom,
    ! [Xa: int,Xaa: int,Y3: list_int] :
      ( ( upto(Xa,Xaa) = Y3 )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Xaa))
       => ~ ( ( Y3 = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xaa),cons_int(Xa,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),one_one_int),Xaa)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_7772_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I),J))
     => ( upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_7773_list__encode_Oelims,axiom,
    ! [Xa: list_nat,Y3: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,Xa) = Y3 )
     => ( ( ( Xa = nil_nat )
         => ( Y3 != zero_zero_nat ) )
       => ~ ! [X4: nat,Xs2: list_nat] :
              ( ( Xa = cons_nat(X4,Xs2) )
             => ( Y3 != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),aa_list_nat_nat(nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_7774_list__encode_Osimps_I1_J,axiom,
    aa_list_nat_nat(nat_list_encode,nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_7775_list__encode_Osimps_I2_J,axiom,
    ! [Xa: nat,Xs: list_nat] : aa_list_nat_nat(nat_list_encode,cons_nat(Xa,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_list_nat_nat(nat_list_encode,Xs)))) ).

% list_encode.simps(2)
tff(fact_7776_inj__Suc,axiom,
    ! [N5: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N5) ).

% inj_Suc
tff(fact_7777_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,N2),N5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acr,K)),N5) ) ).

% inj_on_diff_nat
tff(fact_7778_list__encode_Opelims,axiom,
    ! [Xa: list_nat,Y3: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,Xa) = Y3 )
     => ( accp_list_nat(nat_list_encode_rel,Xa)
       => ( ( ( Xa = nil_nat )
           => ( ( Y3 = zero_zero_nat )
             => ~ accp_list_nat(nat_list_encode_rel,nil_nat) ) )
         => ~ ! [X4: nat,Xs2: list_nat] :
                ( ( Xa = cons_nat(X4,Xs2) )
               => ( ( Y3 = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),aa_list_nat_nat(nat_list_encode,Xs2)))) )
                 => ~ accp_list_nat(nat_list_encode_rel,cons_nat(X4,Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_7779_wf__int__ge__less__than,axiom,
    ! [D: int] : aa_set6170482117830564987_int_o(wf_int,int_ge_less_than(D)) ).

% wf_int_ge_less_than
tff(fact_7780_wf__int__ge__less__than2,axiom,
    ! [D: int] : aa_set6170482117830564987_int_o(wf_int,int_ge_less_than2(D)) ).

% wf_int_ge_less_than2
tff(fact_7781_wf__less,axiom,
    aa_set3702741264012733747_nat_o(wf_nat,collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat))) ).

% wf_less
tff(fact_7782_vimage__Suc__insert__0,axiom,
    ! [A3: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A3)) = vimage_nat_nat(suc,A3) ).

% vimage_Suc_insert_0
tff(fact_7783_card__length__sum__list__rec,axiom,
    ! [M: nat,N5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_adl(M),N5))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_adm(M),N5)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_adn(M),N5)))) ) ) ).

% card_length_sum_list_rec
tff(fact_7784_card__length__sum__list,axiom,
    ! [M: nat,N5: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_adl(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_7785_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_list_nat_nat(groups4561878855575611511st_nat,upt(M,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_av),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) ) ).

% sum_list_upt
tff(fact_7786_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,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_7787_take__upt,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_7788_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_7789_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_7790_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_7791_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_7792_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xa: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(Xa,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = Xa )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_7793_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_7794_map__add__upt,axiom,
    ! [N: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ado,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_7795_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_ih,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_7796_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_7797_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_7798_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_list_nat_nat(last_nat,upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_7799_Sup__int__def,axiom,
    ! [X6: set_int] : complete_Sup_Sup_int(X6) = the_int(aTP_Lamm_adp(X6)) ).

% Sup_int_def
tff(fact_7800_GreatestI__nat,axiom,
    ! [Pa: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(Pa,K)
     => ( ! [Y4: nat] :
            ( aa_nat_o(Pa,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => aa_nat_o(Pa,order_Greatest_nat(Pa)) ) ) ).

% GreatestI_nat
tff(fact_7801_Greatest__le__nat,axiom,
    ! [Pa: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(Pa,K)
     => ( ! [Y4: nat] :
            ( aa_nat_o(Pa,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(Pa)) ) ) ).

% Greatest_le_nat
tff(fact_7802_GreatestI__ex__nat,axiom,
    ! [Pa: fun_nat_o,B: nat] :
      ( ? [X_12: nat] : aa_nat_o(Pa,X_12)
     => ( ! [Y4: nat] :
            ( aa_nat_o(Pa,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => aa_nat_o(Pa,order_Greatest_nat(Pa)) ) ) ).

% GreatestI_ex_nat
tff(fact_7803_bij__betw__Suc,axiom,
    ! [M2: set_nat,N5: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M2),N5)
    <=> ( image_nat_nat(suc,M2) = N5 ) ) ).

% bij_betw_Suc
tff(fact_7804_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),T)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_7805_pair__less__iff1,axiom,
    ! [Xa: nat,Y3: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Z)),fun_pair_less)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Z) ) ).

% pair_less_iff1
tff(fact_7806_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_7807_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S),T)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_7808_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_7809_wmax__insertI,axiom,
    ! [Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,Xa: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y3),YS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(Xa,Y3),fun_pair_leq)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_max_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(Xa,XS),YS),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_7810_wmin__insertI,axiom,
    ! [Xa: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(Xa,Y3),fun_pair_leq)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y3,YS)),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_7811_smin__insertI,axiom,
    ! [Xa: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(Xa,Y3),fun_pair_less)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y3,YS)),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_7812_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_min_strict,fun_min_weak)) ).

% min_rpair_set
tff(fact_7813_wmax__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,X6)
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X6),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_7814_smin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( X6 != bot_bo2099793752762293965at_nat )
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat),fun_min_strict) ) ).

% smin_emptyI
tff(fact_7815_wmin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] : member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat),fun_min_weak) ).

% wmin_emptyI
tff(fact_7816_min__weak__def,axiom,
    fun_min_weak = sup_su5525570899277871387at_nat(min_ex6901939911449802026at_nat(fun_pair_leq),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% min_weak_def
tff(fact_7817_max__weak__def,axiom,
    fun_max_weak = sup_su5525570899277871387at_nat(max_ex8135407076693332796at_nat(fun_pair_leq),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% max_weak_def
tff(fact_7818_smax__insertI,axiom,
    ! [Y3: product_prod_nat_nat,Y6: set_Pr1261947904930325089at_nat,Xa: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y3),Y6)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(Xa,Y3),fun_pair_less)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,Y6),fun_max_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(Xa,X6),Y6),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_7819_smax__emptyI,axiom,
    ! [Y6: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Y6)
     => ( ( Y6 != bot_bo2099793752762293965at_nat )
       => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,Y6),fun_max_strict) ) ) ).

% smax_emptyI
tff(fact_7820_natLeq__on__wo__rel,axiom,
    ! [N: nat] : aa_set3702741264012733747_nat_o(bNF_We3818239936649020644el_nat,collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ada(N)))) ).

% natLeq_on_wo_rel
tff(fact_7821_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_max_strict,fun_max_weak)) ).

% max_rpair_set
tff(fact_7822_char_Osize_I2_J,axiom,
    ! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] : size_size_char(char2((X1),(X2),(X32),(X42),(X52),(X62),(X7),(X8))) = zero_zero_nat ).

% char.size(2)
tff(fact_7823_inj__on__char__of__nat,axiom,
    aa_set_nat_o(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% inj_on_char_of_nat
tff(fact_7824_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,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_7825_char_Osize__gen,axiom,
    ! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] : size_char(char2((X1),(X2),(X32),(X42),(X52),(X62),(X7),(X8))) = zero_zero_nat ).

% char.size_gen
tff(fact_7826_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,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_7827_gcd__nat_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_aas),zero_zero_nat) ).

% gcd_nat.ordering_top_axioms
tff(fact_7828_euclidean__size__int__def,axiom,
    euclid4774559944035922753ze_int = comp_int_nat_int(nat2,abs_abs_int) ).

% euclidean_size_int_def
tff(fact_7829_bot__nat__0_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_hu,aTP_Lamm_fl),zero_zero_nat) ).

% bot_nat_0.ordering_top_axioms
tff(fact_7830_division__segment__nat__def,axiom,
    ! [N: nat] : aa_nat_nat(euclid3398187327856392827nt_nat,N) = one_one_nat ).

% division_segment_nat_def
tff(fact_7831_eventually__sequentially__Suc,axiom,
    ! [Pa: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_adq(Pa),at_top_nat)
    <=> eventually_nat(Pa,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_7832_eventually__sequentially__seg,axiom,
    ! [Pa: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_adr(Pa),K),at_top_nat)
    <=> eventually_nat(Pa,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_7833_le__sequentially,axiom,
    ! [F4: filter_nat] :
      ( ord_le2510731241096832064er_nat(F4,at_top_nat)
    <=> ! [N7: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N7),F4) ) ).

% le_sequentially
tff(fact_7834_filterlim__atMost__at__top,axiom,
    filter3212408913953519116et_nat(set_ord_atMost_nat,finite3254316476582989077op_nat(top_top_set_nat),at_top_nat) ).

% filterlim_atMost_at_top
tff(fact_7835_filterlim__lessThan__at__top,axiom,
    filter3212408913953519116et_nat(set_ord_lessThan_nat,finite3254316476582989077op_nat(top_top_set_nat),at_top_nat) ).

% filterlim_lessThan_at_top
tff(fact_7836_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_7837_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_fk,at_top_nat) ).

% eventually_False_sequentially
tff(fact_7838_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_7839_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_7840_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero_nat) ).

% prod_decode_def
tff(fact_7841_list__decode_Oelims,axiom,
    ! [Xa: nat,Y3: list_nat] :
      ( ( nat_list_decode(Xa) = Y3 )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y3 != nil_nat ) )
       => ~ ! [N2: nat] :
              ( ( Xa = aa_nat_nat(suc,N2) )
             => ( Y3 != produc2761476792215241774st_nat(aTP_Lamm_ads,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) ) ) ) ) ).

% list_decode.elims
tff(fact_7842_list__decode_Osimps_I1_J,axiom,
    nat_list_decode(zero_zero_nat) = nil_nat ).

% list_decode.simps(1)
tff(fact_7843_list__decode_Osimps_I2_J,axiom,
    ! [N: nat] : nat_list_decode(aa_nat_nat(suc,N)) = produc2761476792215241774st_nat(aTP_Lamm_ads,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ).

% list_decode.simps(2)
tff(fact_7844_list__decode_Opelims,axiom,
    ! [Xa: nat,Y3: list_nat] :
      ( ( nat_list_decode(Xa) = Y3 )
     => ( aa_nat_o(accp_nat(nat_list_decode_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y3 = nil_nat )
             => ~ aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat) ) )
         => ~ ! [N2: nat] :
                ( ( Xa = aa_nat_nat(suc,N2) )
               => ( ( Y3 = produc2761476792215241774st_nat(aTP_Lamm_ads,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) )
                 => ~ aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2)) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_7845_list__decode_Opsimps_I1_J,axiom,
    ( aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat)
   => ( nat_list_decode(zero_zero_nat) = nil_nat ) ) ).

% list_decode.psimps(1)
tff(fact_7846_list__decode_Opinduct,axiom,
    ! [A0: nat,Pa: fun_nat_o] :
      ( aa_nat_o(accp_nat(nat_list_decode_rel),A0)
     => ( ( aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat)
         => aa_nat_o(Pa,zero_zero_nat) )
       => ( ! [N2: nat] :
              ( aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2))
             => ( ! [X: nat,Y5: nat] :
                    ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y5) = aa_nat7603221163087113075at_nat(nat_prod_decode,N2) )
                   => aa_nat_o(Pa,Y5) )
               => aa_nat_o(Pa,aa_nat_nat(suc,N2)) ) )
         => aa_nat_o(Pa,A0) ) ) ) ).

% list_decode.pinduct
tff(fact_7847_list__decode_Opsimps_I2_J,axiom,
    ! [N: nat] :
      ( aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N))
     => ( nat_list_decode(aa_nat_nat(suc,N)) = produc2761476792215241774st_nat(aTP_Lamm_ads,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ) ) ).

% list_decode.psimps(2)
tff(fact_7848_natLeq__on__well__order__on,axiom,
    ! [N: nat] : aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fl,N))),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ada(N)))) ).

% natLeq_on_well_order_on
tff(fact_7849_natLeq__on__Well__order,axiom,
    ! [N: nat] : aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ada(N))))),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ada(N)))) ).

% natLeq_on_Well_order
tff(fact_7850_ctwo__Cnotzero,axiom,
    ( ~ member444158400953824016od_o_o(produc763777882069021527od_o_o(bNF_Cardinal_ctwo,bNF_Cardinal_czero_o),bNF_We2654380646378065620so_o_o)
    & bNF_Ca8331644756375544342r_on_o(field_o(bNF_Cardinal_ctwo),bNF_Cardinal_ctwo) ) ).

% ctwo_Cnotzero
tff(fact_7851_ctwo__ordLess__natLeq,axiom,
    member4095101504841534314at_nat(produc8517790099723286449at_nat(bNF_Cardinal_ctwo,bNF_Ca8665028551170535155natLeq),bNF_We8182288985678559134_o_nat) ).

% ctwo_ordLess_natLeq
tff(fact_7852_natLeq__underS__less,axiom,
    ! [N: nat] : aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fl,N)) ).

% natLeq_underS_less
tff(fact_7853_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_7854_Restr__natLeq2,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N),aTP_Lamm_adt(N))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ada(N))) ).

% Restr_natLeq2
tff(fact_7855_Restr__natLeq,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fl,N)),aTP_Lamm_adu(N))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ada(N))) ).

% Restr_natLeq
tff(fact_7856_card__of__nat,axiom,
    member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(top_top_set_nat),bNF_Ca8665028551170535155natLeq),bNF_We5258908940166488438at_nat) ).

% card_of_nat
tff(fact_7857_card__of__Field__natLeq,axiom,
    member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq),bNF_We5258908940166488438at_nat) ).

% card_of_Field_natLeq
tff(fact_7858_num__of__integer_Otransfer,axiom,
    bNF_re6718328864250387230um_num(code_pcr_integer,fequal_num,comp_nat_num_int(num_of_nat,nat2),code_num_of_integer) ).

% num_of_integer.transfer
tff(fact_7859_integer_Oid__abs__transfer,axiom,
    bNF_re982302072995117890nteger(fequal_int,code_pcr_integer,aTP_Lamm_cn,code_integer_of_int) ).

% integer.id_abs_transfer
tff(fact_7860_integer_Orep__transfer,axiom,
    bNF_re3804157879324367682nt_int(code_pcr_integer,fequal_int,aTP_Lamm_cn,code_int_of_integer) ).

% integer.rep_transfer
tff(fact_7861_gcd__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),gcd_gcd_int,gcd_gcd_Code_integer) ).

% gcd_integer.transfer
tff(fact_7862_dup_Otransfer,axiom,
    aa_fun774133186966548707eger_o(aa_fun6864876441520360999eger_o(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),aTP_Lamm_acn),code_dup) ).

% dup.transfer
tff(fact_7863_sub_Otransfer,axiom,
    bNF_re4520166952630509772nteger(fequal_num,bNF_re6501075790457514782nteger(fequal_num,code_pcr_integer),aTP_Lamm_acq,code_sub) ).

% sub.transfer
tff(fact_7864_eventually__prod__sequentially,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] :
      ( eventu1038000079068216329at_nat(Pa,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N7: nat] :
        ! [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M7)
         => ! [N6: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N6)
             => aa_Pro6150637604445964947_nat_o(Pa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N6),M7)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_7865_less__than__iff,axiom,
    ! [Xa: nat,Y3: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y3)),less_than)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y3) ) ).

% less_than_iff
tff(fact_7866_sort__upt,axiom,
    ! [M: nat,N: nat] : linord738340561235409698at_nat(aTP_Lamm_av,upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_7867_sort__upto,axiom,
    ! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_cn,upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_7868_int_Oid__abs__transfer,axiom,
    bNF_re7934895593101944656at_int(basic_5328504652464829177at_nat(fequal_nat,fequal_nat),pcr_int,aTP_Lamm_adv,abs_Integ) ).

% int.id_abs_transfer
tff(fact_7869_pcr__int__def,axiom,
    pcr_int = relcom8911704206367525917at_int(basic_5328504652464829177at_nat(fequal_nat,fequal_nat),cr_int) ).

% pcr_int_def
tff(fact_7870_lcm__code__integer,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_lcm_Code_integer,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,A),B)) ).

% lcm_code_integer
tff(fact_7871_Least__eq__0,axiom,
    ! [Pa: fun_nat_o] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ord_Least_nat(Pa) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_7872_Least__Suc2,axiom,
    ! [Pa: fun_nat_o,N: nat,Q: fun_nat_o,M: nat] :
      ( aa_nat_o(Pa,N)
     => ( aa_nat_o(Q,M)
       => ( ~ aa_nat_o(Pa,zero_zero_nat)
         => ( ! [K2: nat] :
                ( aa_nat_o(Pa,aa_nat_nat(suc,K2))
              <=> aa_nat_o(Q,K2) )
           => ( ord_Least_nat(Pa) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_7873_Least__Suc,axiom,
    ! [Pa: fun_nat_o,N: nat] :
      ( aa_nat_o(Pa,N)
     => ( ~ aa_nat_o(Pa,zero_zero_nat)
       => ( ord_Least_nat(Pa) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_adq(Pa))) ) ) ) ).

% Least_Suc
tff(fact_7874_int_Opcr__cr__eq,axiom,
    pcr_int = cr_int ).

% int.pcr_cr_eq
tff(fact_7875_cr__int__def,axiom,
    ! [X: product_prod_nat_nat] : aa_Pro2912763917461432128_int_o(cr_int,X) = aa_int_fun_int_o(fequal_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) ).

% cr_int_def
tff(fact_7876_lcm__integer__def,axiom,
    gcd_lcm_Code_integer = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),gcd_lcm_int) ).

% lcm_integer_def
tff(fact_7877_Quotient__int,axiom,
    quotie1194848508323700631at_int(intrel,abs_Integ,rep_Integ,cr_int) ).

% Quotient_int
tff(fact_7878_lcm__0__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% lcm_0_iff_nat
tff(fact_7879_lcm__proj2__if__dvd__nat,axiom,
    ! [Xa: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),Y3)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,Xa),Y3) = Y3 ) ) ).

% lcm_proj2_if_dvd_nat
tff(fact_7880_lcm__proj1__if__dvd__nat,axiom,
    ! [Xa: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),Y3)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,Y3),Xa) = Y3 ) ) ).

% lcm_proj1_if_dvd_nat
tff(fact_7881_lcm__proj2__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N) = N )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N) ) ).

% lcm_proj2_iff_nat
tff(fact_7882_lcm__proj1__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N) = M )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N),M) ) ).

% lcm_proj1_iff_nat
tff(fact_7883_lcm__1__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_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) ) ) ) ).

% lcm_1_iff_nat
tff(fact_7884_lcm__0__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N) = zero_zero_int )
    <=> ( ( M = zero_zero_int )
        | ( N = zero_zero_int ) ) ) ).

% lcm_0_iff_int
tff(fact_7885_lcm__int__int__eq,axiom,
    ! [M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N)) ).

% lcm_int_int_eq
tff(fact_7886_abs__lcm__int,axiom,
    ! [I: int,J: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,I),J)) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,I),J) ).

% abs_lcm_int
tff(fact_7887_lcm__abs1__int,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_int_int(abs_abs_int,Xa)),Y3) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa),Y3) ).

% lcm_abs1_int
tff(fact_7888_lcm__abs2__int,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa),aa_int_int(abs_abs_int,Y3)) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa),Y3) ).

% lcm_abs2_int
tff(fact_7889_lcm__1__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N) = one_one_int )
    <=> ( ( ( M = one_one_int )
          | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) )
        & ( ( N = one_one_int )
          | ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% lcm_1_iff_int
tff(fact_7890_lcm__proj2__if__dvd__int,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),Y3)
     => ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa),Y3) = aa_int_int(abs_abs_int,Y3) ) ) ).

% lcm_proj2_if_dvd_int
tff(fact_7891_lcm__proj1__if__dvd__int,axiom,
    ! [Xa: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),Y3)
     => ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Y3),Xa) = aa_int_int(abs_abs_int,Y3) ) ) ).

% lcm_proj1_if_dvd_int
tff(fact_7892_lcm__proj2__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N) = aa_int_int(abs_abs_int,N) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),N) ) ).

% lcm_proj2_iff_int
tff(fact_7893_lcm__proj1__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N) = aa_int_int(abs_abs_int,M) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,N),M) ) ).

% lcm_proj1_iff_int
tff(fact_7894_lcm__nat__abs__right__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ).

% lcm_nat_abs_right_eq
tff(fact_7895_lcm__nat__abs__left__eq,axiom,
    ! [K: int,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% lcm_nat_abs_left_eq
tff(fact_7896_lcm__ge__0__int,axiom,
    ! [Xa: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa),Y3)) ).

% lcm_ge_0_int
tff(fact_7897_Inf__nat__def,axiom,
    ! [X6: set_nat] : aa_set_nat_nat(complete_Inf_Inf_nat,X6) = ord_Least_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ac,X6)) ).

% Inf_nat_def
tff(fact_7898_prod__gcd__lcm__nat,axiom,
    ! [M: nat,N: 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(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N)) ).

% prod_gcd_lcm_nat
tff(fact_7899_lcm__neg1__int,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_int_int(uminus_uminus_int,Xa)),Y3) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa),Y3) ).

% lcm_neg1_int
tff(fact_7900_lcm__neg2__int,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa),aa_int_int(uminus_uminus_int,Y3)) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa),Y3) ).

% lcm_neg2_int
tff(fact_7901_dvd__lcm__I2__int,axiom,
    ! [I: int,N: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,I),N)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,I),aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N)) ) ).

% dvd_lcm_I2_int
tff(fact_7902_dvd__lcm__I1__int,axiom,
    ! [I: int,M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,I),M)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,I),aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N)) ) ).

% dvd_lcm_I1_int
tff(fact_7903_lcm__unique__nat,axiom,
    ! [A: nat,D: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),D)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),D)
        & ! [E3: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),E3)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),E3) )
           => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),E3) ) )
    <=> ( D = aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,A),B) ) ) ).

% lcm_unique_nat
tff(fact_7904_dvd__lcm__I2__nat,axiom,
    ! [K: nat,N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N)) ) ).

% dvd_lcm_I2_nat
tff(fact_7905_dvd__lcm__I1__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N)) ) ).

% dvd_lcm_I1_nat
tff(fact_7906_lcm__int__def,axiom,
    ! [Xa: int,Y3: int] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa),Y3) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,Xa))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Y3)))) ).

% lcm_int_def
tff(fact_7907_lcm__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N)) ) ) ).

% lcm_pos_nat
tff(fact_7908_lcm__pos__int,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( N != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N)) ) ) ).

% lcm_pos_int
tff(fact_7909_lcm__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),gcd_lcm_int,gcd_lcm_int) ).

% lcm_integer.rsp
tff(fact_7910_lcm__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_lcm_Code_integer,Xa),Xaa)) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_Code_integer_int(code_int_of_integer,Xa)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% lcm_integer.rep_eq
tff(fact_7911_lcm__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_lcm_Code_integer,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa),Xb)) ).

% lcm_integer.abs_eq
tff(fact_7912_lcm__cases__int,axiom,
    ! [Xa: int,Y3: int,Pa: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
         => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa),Y3)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),zero_zero_int)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa),aa_int_int(uminus_uminus_int,Y3))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_int_int(uminus_uminus_int,Xa)),Y3)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),zero_zero_int)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_int_int(uminus_uminus_int,Xa)),aa_int_int(uminus_uminus_int,Y3))) ) )
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa),Y3)) ) ) ) ) ).

% lcm_cases_int
tff(fact_7913_lcm__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),D)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),E3)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),E3) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),E3) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,A),B) ) ) ).

% lcm_unique_int
tff(fact_7914_lcm__nat__def,axiom,
    ! [Xa: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,Xa),Y3) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Y3)) ).

% lcm_nat_def
tff(fact_7915_prod__gcd__lcm__int,axiom,
    ! [M: int,N: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,M)),aa_int_int(abs_abs_int,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N)),aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N)) ).

% prod_gcd_lcm_int
tff(fact_7916_lcm__altdef__int,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,A),B) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)) ).

% lcm_altdef_int
tff(fact_7917_lcm__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),gcd_lcm_int,gcd_lcm_Code_integer) ).

% lcm_integer.transfer
tff(fact_7918_Lcm__int__set__eq__fold,axiom,
    ! [Xs: list_int] : gcd_Lcm_int(set_int2(Xs)) = aa_int_int(fold_int_int(gcd_lcm_int,Xs),one_one_int) ).

% Lcm_int_set_eq_fold
tff(fact_7919_Lcm__eq__Max__nat,axiom,
    ! [M2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M2)
     => ( ( M2 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),M2)
         => ( ! [M4: nat,N2: nat] :
                ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,M4),M2)
               => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,N2),M2)
                 => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M4),N2)),M2) ) )
           => ( aa_set_nat_nat(gcd_Lcm_nat,M2) = aa_set_nat_nat(lattic8265883725875713057ax_nat,M2) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_7920_Lcm__eq__0__I__nat,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),A3)
     => ( aa_set_nat_nat(gcd_Lcm_nat,A3) = zero_zero_nat ) ) ).

% Lcm_eq_0_I_nat
tff(fact_7921_abs__Lcm__eq,axiom,
    ! [K5: set_int] : aa_int_int(abs_abs_int,gcd_Lcm_int(K5)) = gcd_Lcm_int(K5) ).

% abs_Lcm_eq
tff(fact_7922_Lcm__0__iff__nat,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set_nat_nat(gcd_Lcm_nat,A3) = zero_zero_nat )
      <=> aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,zero_zero_nat),A3) ) ) ).

% Lcm_0_iff_nat
tff(fact_7923_Lcm__abs__eq,axiom,
    ! [K5: set_int] : gcd_Lcm_int(image_int_int(abs_abs_int,K5)) = gcd_Lcm_int(K5) ).

% Lcm_abs_eq
tff(fact_7924_Lcm__int__eq,axiom,
    ! [N5: set_nat] : gcd_Lcm_int(image_nat_int(semiri1314217659103216013at_int,N5)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Lcm_nat,N5)) ).

% Lcm_int_eq
tff(fact_7925_Lcm__nat__abs__eq,axiom,
    ! [K5: set_int] : aa_set_nat_nat(gcd_Lcm_nat,image_int_nat(aTP_Lamm_acs,K5)) = aa_int_nat(nat2,gcd_Lcm_int(K5)) ).

% Lcm_nat_abs_eq
tff(fact_7926_Lcm__nat__insert,axiom,
    ! [N: nat,M2: set_nat] : aa_set_nat_nat(gcd_Lcm_nat,insert_nat(N,M2)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,N),aa_set_nat_nat(gcd_Lcm_nat,M2)) ).

% Lcm_nat_insert
tff(fact_7927_Lcm__least__int,axiom,
    ! [A3: set_int,A: int] :
      ( ! [B2: int] :
          ( member_int(B2,A3)
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,gcd_Lcm_int(A3)),A) ) ).

% Lcm_least_int
tff(fact_7928_dvd__Lcm__int,axiom,
    ! [M: int,M2: set_int] :
      ( member_int(M,M2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),gcd_Lcm_int(M2)) ) ).

% dvd_Lcm_int
tff(fact_7929_dvd__Lcm__nat,axiom,
    ! [M: nat,M2: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,M),M2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_set_nat_nat(gcd_Lcm_nat,M2)) ) ).

% dvd_Lcm_nat
tff(fact_7930_Lcm__dvd__nat,axiom,
    ! [M2: set_nat,N: nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X4),M2)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X4),N) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(gcd_Lcm_nat,M2)),N) ) ).

% Lcm_dvd_nat
tff(fact_7931_Lcm__int__greater__eq__0,axiom,
    ! [K5: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),gcd_Lcm_int(K5)) ).

% Lcm_int_greater_eq_0
tff(fact_7932_Lcm__nat__infinite,axiom,
    ! [M2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,M2)
     => ( aa_set_nat_nat(gcd_Lcm_nat,M2) = zero_zero_nat ) ) ).

% Lcm_nat_infinite
tff(fact_7933_Lcm__nat__empty,axiom,
    aa_set_nat_nat(gcd_Lcm_nat,bot_bot_set_nat) = one_one_nat ).

% Lcm_nat_empty
tff(fact_7934_Lcm__in__lcm__closed__set__nat,axiom,
    ! [M2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M2)
     => ( ( M2 != bot_bot_set_nat )
       => ( ! [M4: nat,N2: nat] :
              ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,M4),M2)
             => ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,N2),M2)
               => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M4),N2)),M2) ) )
         => aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_set_nat_nat(gcd_Lcm_nat,M2)),M2) ) ) ) ).

% Lcm_in_lcm_closed_set_nat
tff(fact_7935_Gcd__nat__def,axiom,
    ! [M2: set_nat] : aa_set_nat_nat(gcd_Gcd_nat,M2) = aa_set_nat_nat(gcd_Lcm_nat,collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_adw,M2))) ).

% Gcd_nat_def
tff(fact_7936_Lcm__int__def,axiom,
    ! [K5: set_int] : gcd_Lcm_int(K5) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Lcm_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),K5))) ).

% Lcm_int_def
tff(fact_7937_Lcm__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : aa_set_nat_nat(gcd_Lcm_nat,set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_lcm_nat,Xs),one_one_nat) ).

% Lcm_nat_set_eq_fold
tff(fact_7938_Lcm__nat__def,axiom,
    ! [M2: set_nat] :
      aa_set_nat_nat(gcd_Lcm_nat,M2) = $ite(aa_set_nat_o(finite_finite_nat,M2),aa_set_nat_nat(aa_nat6910189468943142497at_nat(lattic7826324295020591184_F_nat(gcd_lcm_nat),one_one_nat),M2),zero_zero_nat) ).

% Lcm_nat_def
tff(fact_7939_normalize__stable,axiom,
    ! [Q2: int,P: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2)
     => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P),Q2)
       => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q2) ) ) ) ).

% normalize_stable
tff(fact_7940_normalize__coprime,axiom,
    ! [R: product_prod_int_int,P: int,Q2: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q2) )
     => aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P),Q2) ) ).

% normalize_coprime
tff(fact_7941_coprime__crossproduct__int,axiom,
    ! [A: int,D: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,B),C)
       => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,B)),aa_int_int(abs_abs_int,D)) )
        <=> ( ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) )
            & ( aa_int_int(abs_abs_int,C) = aa_int_int(abs_abs_int,D) ) ) ) ) ) ).

% coprime_crossproduct_int
tff(fact_7942_quotient__of__coprime,axiom,
    ! [R: rat,P: int,Q2: int] :
      ( ( quotient_of(R) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q2) )
     => aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P),Q2) ) ).

% quotient_of_coprime
tff(fact_7943_quotient__of__def,axiom,
    ! [Xa: rat] : quotient_of(Xa) = the_Pr4378521158711661632nt_int(aTP_Lamm_adx(Xa)) ).

% quotient_of_def
tff(fact_7944_coprime__common__divisor__nat,axiom,
    ! [A: nat,B: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),B)
         => ( Xa = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_7945_coprime__Suc__left__nat,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,N)),N) ).

% coprime_Suc_left_nat
tff(fact_7946_coprime__Suc__right__nat,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,N),aa_nat_nat(suc,N)) ).

% coprime_Suc_right_nat
tff(fact_7947_coprime__crossproduct__nat,axiom,
    ! [A: nat,D: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,A),D)
     => ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,B),C)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D) )
        <=> ( ( A = B )
            & ( C = D ) ) ) ) ) ).

% coprime_crossproduct_nat
tff(fact_7948_coprime__Suc__0__left,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),N) ).

% coprime_Suc_0_left
tff(fact_7949_coprime__Suc__0__right,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,N),aa_nat_nat(suc,zero_zero_nat)) ).

% coprime_Suc_0_right
tff(fact_7950_quotient__of__eq,axiom,
    ! [A: int,B: int,P: int,Q2: int] :
      ( ( quotient_of(aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q2) )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,P),Q2) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).

% quotient_of_eq
tff(fact_7951_normalize__eq,axiom,
    ! [A: int,B: int,P: int,Q2: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q2) )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,P),Q2) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).

% normalize_eq
tff(fact_7952_quotient__of__Fract,axiom,
    ! [A: int,B: int] : quotient_of(aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ).

% quotient_of_Fract
tff(fact_7953_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(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_7954_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(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_7955_Fract_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] :
      aa_int_rat(aa_int_fun_int_rat(fract,Xa),Xb) = aa_Pro6166618844887768661nt_rat(abs_Rat,
        $ite(Xb = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Xb))) ).

% Fract.abs_eq
tff(fact_7956_Fract_Otransfer,axiom,
    bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_adi,fract) ).

% Fract.transfer
tff(fact_7957_Fract__def,axiom,
    fract = map_fu7945985449245211511nt_rat(id_int,map_fu3424225382358772769nt_rat(id_int,abs_Rat),aTP_Lamm_adi) ).

% Fract_def
tff(fact_7958_of__int__eq__id,axiom,
    ring_1_of_int_int = id_int ).

% of_int_eq_id
tff(fact_7959_natural__decr,axiom,
    ! [N: code_natural] :
      ( ( N != zero_z2226904508553997617atural )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Code_natural_nat(code_nat_of_natural,N)),aa_nat_nat(suc,zero_zero_nat))),aa_Code_natural_nat(code_nat_of_natural,N)) ) ).

% natural_decr
tff(fact_7960_plus__natural_Orep__eq,axiom,
    ! [Xa: code_natural,Xaa: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Xa),Xaa)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_natural_nat(code_nat_of_natural,Xa)),aa_Code_natural_nat(code_nat_of_natural,Xaa)) ).

% plus_natural.rep_eq
tff(fact_7961_zero__natural_Orep__eq,axiom,
    aa_Code_natural_nat(code_nat_of_natural,zero_z2226904508553997617atural) = zero_zero_nat ).

% zero_natural.rep_eq
tff(fact_7962_one__natural_Orep__eq,axiom,
    aa_Code_natural_nat(code_nat_of_natural,one_one_Code_natural) = one_one_nat ).

% one_natural.rep_eq
tff(fact_7963_plus__natural__def,axiom,
    plus_p4538020629002901425atural = map_fu7613509917203888380atural(code_nat_of_natural,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),plus_plus_nat) ).

% plus_natural_def
tff(fact_7964_one__natural__def,axiom,
    one_one_Code_natural = aa_nat_Code_natural(code_natural_of_nat,one_one_nat) ).

% one_natural_def
tff(fact_7965_zero__natural__def,axiom,
    zero_z2226904508553997617atural = aa_nat_Code_natural(code_natural_of_nat,zero_zero_nat) ).

% zero_natural_def
tff(fact_7966_plus__natural_Oabs__eq,axiom,
    ! [Xa: nat,Xb: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(code_natural_of_nat,Xa)),aa_nat_Code_natural(code_natural_of_nat,Xb)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Xb)) ).

% plus_natural.abs_eq
tff(fact_7967_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_7968_gcd__nat_Osemilattice__neutr__axioms,axiom,
    aa_nat_o(semila9081495762789891438tr_nat(gcd_gcd_nat),zero_zero_nat) ).

% gcd_nat.semilattice_neutr_axioms
tff(fact_7969_max__nat_Osemilattice__neutr__axioms,axiom,
    aa_nat_o(semila9081495762789891438tr_nat(ord_max_nat),zero_zero_nat) ).

% max_nat.semilattice_neutr_axioms
tff(fact_7970_Quotient3__int,axiom,
    quotie6776551016481293500at_int(intrel,abs_Integ,rep_Integ) ).

% Quotient3_int
tff(fact_7971_String_Oless__literal__def,axiom,
    ord_less_literal = map_fu6041574301308154232eral_o(explode,map_fu6698795530259886495ar_o_o(explode,id_o),lexordp_char(aTP_Lamm_ady)) ).

% String.less_literal_def
tff(fact_7972_less__literal_Orep__eq,axiom,
    ! [Xa: literal,Xaa: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Xa),Xaa)
    <=> aa_list_char_o(aa_lis7814493268959572977char_o(lexordp_char(aTP_Lamm_ady),aa_literal_list_char(explode,Xa)),aa_literal_list_char(explode,Xaa)) ) ).

% less_literal.rep_eq
tff(fact_7973_String_Oless__eq__literal__def,axiom,
    ord_less_eq_literal = map_fu6041574301308154232eral_o(explode,map_fu6698795530259886495ar_o_o(explode,id_o),lexordp_eq_char(aTP_Lamm_ady)) ).

% String.less_eq_literal_def
tff(fact_7974_less__eq__literal_Orep__eq,axiom,
    ! [Xa: literal,Xaa: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,Xa),Xaa)
    <=> aa_list_char_o(aa_lis7814493268959572977char_o(lexordp_eq_char(aTP_Lamm_ady),aa_literal_list_char(explode,Xa)),aa_literal_list_char(explode,Xaa)) ) ).

% less_eq_literal.rep_eq
tff(fact_7975_less__eq__literal_Otransfer,axiom,
    bNF_re9191414151699148995eral_o(pcr_literal,bNF_re8730477052009823892al_o_o(pcr_literal,fequal_o),lexordp_eq_char(aTP_Lamm_ady),ord_less_eq_literal) ).

% less_eq_literal.transfer
tff(fact_7976_less__literal_Otransfer,axiom,
    bNF_re9191414151699148995eral_o(pcr_literal,bNF_re8730477052009823892al_o_o(pcr_literal,fequal_o),lexordp_char(aTP_Lamm_ady),ord_less_literal) ).

% less_literal.transfer
tff(fact_7977_literal_Orep__transfer,axiom,
    bNF_re4991257583289732696t_char(pcr_literal,list_all2_char_char(fequal_char),aTP_Lamm_adz,explode) ).

% literal.rep_transfer
tff(fact_7978_String_Ocr__literal__def,axiom,
    ! [X: list_char,Xa3: literal] :
      ( aa_literal_o(aa_lis3557855670089359654eral_o(cr_literal,X),Xa3)
    <=> ( X = aa_literal_list_char(explode,Xa3) ) ) ).

% String.cr_literal_def
tff(fact_7979_cr__natural__def,axiom,
    ! [X: nat,Xa3: code_natural] :
      ( aa_Code_natural_o(code_cr_natural(X),Xa3)
    <=> ( X = aa_Code_natural_nat(code_nat_of_natural,Xa3) ) ) ).

% cr_natural_def
tff(fact_7980_cr__integer__def,axiom,
    ! [X: int,Xa3: code_integer] :
      ( aa_Code_integer_o(code_cr_integer(X),Xa3)
    <=> ( X = aa_Code_integer_int(code_int_of_integer,Xa3) ) ) ).

% cr_integer_def
tff(fact_7981_gcd__nat_Omonoid__axioms,axiom,
    aa_nat_o(monoid_nat(gcd_gcd_nat),zero_zero_nat) ).

% gcd_nat.monoid_axioms
tff(fact_7982_max__nat_Omonoid__axioms,axiom,
    aa_nat_o(monoid_nat(ord_max_nat),zero_zero_nat) ).

% max_nat.monoid_axioms
tff(fact_7983_integer__of__nat__1,axiom,
    aa_nat_Code_integer(code_integer_of_nat,one_one_nat) = one_one_Code_integer ).

% integer_of_nat_1
tff(fact_7984_integer__of__nat__0,axiom,
    aa_nat_Code_integer(code_integer_of_nat,zero_zero_nat) = zero_z3403309356797280102nteger ).

% integer_of_nat_0
tff(fact_7985_int__equivp,axiom,
    equiv_749116872481872345at_nat(intrel) ).

% int_equivp
tff(fact_7986_unit__factor__simps_I1_J,axiom,
    aa_nat_nat(unit_f2748546683901255202or_nat,zero_zero_nat) = zero_zero_nat ).

% unit_factor_simps(1)
tff(fact_7987_unit__factor__simps_I2_J,axiom,
    ! [N: nat] : aa_nat_nat(unit_f2748546683901255202or_nat,aa_nat_nat(suc,N)) = one_one_nat ).

% unit_factor_simps(2)
tff(fact_7988_unit__factor__nat__def,axiom,
    ! [N: nat] :
      aa_nat_nat(unit_f2748546683901255202or_nat,N) = $ite(N = zero_zero_nat,zero_zero_nat,one_one_nat) ).

% unit_factor_nat_def
tff(fact_7989_Literal_Otransfer,axiom,
    bNF_re7219000028618014437iteral(fequal_o,bNF_re2080241665920251617iteral(fequal_o,bNF_re2270469785393322297iteral(fequal_o,bNF_re7488407634572369889iteral(fequal_o,bNF_re6877862493807443341iteral(fequal_o,bNF_re7699176496855487201iteral(fequal_o,bNF_re4041988539332897761iteral(fequal_o,bNF_re5453467372598631581iteral(pcr_literal,pcr_literal))))))),aTP_Lamm_aea,literal2) ).

% Literal.transfer
tff(fact_7990_gcd__nat_Ocomm__monoid__axioms,axiom,
    aa_nat_o(comm_monoid_nat(gcd_gcd_nat),zero_zero_nat) ).

% gcd_nat.comm_monoid_axioms
tff(fact_7991_max__nat_Ocomm__monoid__axioms,axiom,
    aa_nat_o(comm_monoid_nat(ord_max_nat),zero_zero_nat) ).

% max_nat.comm_monoid_axioms
tff(fact_7992_String_OLiteral__def,axiom,
    literal2 = map_fu5639709166104062110iteral(id_o,map_fu3469583328629365786iteral(id_o,map_fu39305780192311794iteral(id_o,map_fu4012007721304947738iteral(id_o,map_fu7405565234982245702iteral(id_o,map_fu4664413887100446234iteral(id_o,map_fu8523206220227787418iteral(id_o,map_fu7398138089906338514iteral(explode,abs_literal))))))),aTP_Lamm_aea) ).

% String.Literal_def
tff(fact_7993_sum__encode__def,axiom,
    ! [Xa: sum_sum_nat_nat] : nat_sum_encode(Xa) = sum_ca6763686470577984908at_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aTP_Lamm_aeb,Xa) ).

% sum_encode_def
tff(fact_7994_nat__of__num__code_I3_J,axiom,
    ! [N: num] :
      aa_num_nat(nat_of_num,aa_num_num(bit1,N)) = $let(
        m: nat,
        m:= aa_num_nat(nat_of_num,N),
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m)) ) ).

% nat_of_num_code(3)
tff(fact_7995_nat__of__num__code_I1_J,axiom,
    aa_num_nat(nat_of_num,one) = one_one_nat ).

% nat_of_num_code(1)
tff(fact_7996_nat__of__num_Osimps_I2_J,axiom,
    ! [Xa: num] : aa_num_nat(nat_of_num,aa_num_num(bit0,Xa)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,Xa)),aa_num_nat(nat_of_num,Xa)) ).

% nat_of_num.simps(2)
tff(fact_7997_nat__of__num__neq__0,axiom,
    ! [Xa: num] : aa_num_nat(nat_of_num,Xa) != zero_zero_nat ).

% nat_of_num_neq_0
tff(fact_7998_nat__of__num__pos,axiom,
    ! [Xa: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(nat_of_num,Xa)) ).

% nat_of_num_pos
tff(fact_7999_nat__of__num__add,axiom,
    ! [Xa: num,Y3: num] : aa_num_nat(nat_of_num,aa_num_num(plus_plus_num(Xa),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,Xa)),aa_num_nat(nat_of_num,Y3)) ).

% nat_of_num_add
tff(fact_8000_nat__of__num__code_I2_J,axiom,
    ! [N: num] :
      aa_num_nat(nat_of_num,aa_num_num(bit0,N)) = $let(
        m: nat,
        m:= aa_num_nat(nat_of_num,N),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m) ) ).

% nat_of_num_code(2)
tff(fact_8001_nat__of__num_Osimps_I1_J,axiom,
    aa_num_nat(nat_of_num,one) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_of_num.simps(1)
tff(fact_8002_nat__of__num_Osimps_I3_J,axiom,
    ! [Xa: num] : aa_num_nat(nat_of_num,aa_num_num(bit1,Xa)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,Xa)),aa_num_nat(nat_of_num,Xa))) ).

% nat_of_num.simps(3)
tff(fact_8003_num__of__nat__inverse,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_num_nat(nat_of_num,aa_nat_num(num_of_nat,N)) = N ) ) ).

% num_of_nat_inverse
tff(fact_8004_plus__num__def,axiom,
    ! [M: num,N: num] : aa_num_num(plus_plus_num(M),N) = aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,M)),aa_num_nat(nat_of_num,N))) ).

% plus_num_def
tff(fact_8005_gcd__nat_Osemilattice__order__axioms,axiom,
    semila1248733672344298208er_nat(gcd_gcd_nat,dvd_dvd_nat,aTP_Lamm_aas) ).

% gcd_nat.semilattice_order_axioms
tff(fact_8006_int__decode__def,axiom,
    ! [N: nat] : aa_nat_int(nat_int_decode,N) = sum_ca7763040182479039464nt_nat(semiri1314217659103216013at_int,aTP_Lamm_aec,nat_sum_decode(N)) ).

% int_decode_def
tff(fact_8007_nat__to__rat__surj__def,axiom,
    ! [N: nat] : aa_nat_rat(nat_to_rat_surj,N) = aa_Pro3608043709957902237at_rat(produc6207742614233964070at_rat(aTP_Lamm_aed),aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ).

% nat_to_rat_surj_def
tff(fact_8008_less__nat__rel,axiom,
    ord_less_nat = transi2163837189807498211lp_nat(aTP_Lamm_acm) ).

% less_nat_rel
tff(fact_8009_small__lazy_H_Ocases,axiom,
    ! [Xa: product_prod_int_int] :
      ~ ! [D3: int,I2: int] : Xa != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D3),I2) ).

% small_lazy'.cases
tff(fact_8010_less__eq__int__code_I3_J,axiom,
    ! [L: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(neg,L)) ).

% less_eq_int_code(3)
tff(fact_8011_less__eq__int__code_I7_J,axiom,
    ! [K: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(neg,K)),zero_zero_int) ).

% less_eq_int_code(7)
tff(fact_8012_less__int__code_I3_J,axiom,
    ! [L: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(neg,L)) ).

% less_int_code(3)
tff(fact_8013_less__int__code_I7_J,axiom,
    ! [K: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(neg,K)),zero_zero_int) ).

% less_int_code(7)
tff(fact_8014_nat__code_I1_J,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(neg,K)) = zero_zero_nat ).

% nat_code(1)
tff(fact_8015_plus__int__code_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(neg,M)),aa_num_int(neg,N)) = aa_num_int(neg,aa_num_num(plus_plus_num(M),N)) ).

% plus_int_code(6)
tff(fact_8016_less__eq__int__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(neg,K)),aa_num_int(neg,L))
    <=> aa_num_o(ord_less_eq_num(L),K) ) ).

% less_eq_int_code(9)
tff(fact_8017_less__int__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(neg,K)),aa_num_int(neg,L))
    <=> aa_num_o(ord_less_num(L),K) ) ).

% less_int_code(9)
tff(fact_8018_Int_Osub__code_I4_J,axiom,
    ! [N: num] : aa_num_int(sub(one),aa_num_num(bit0,N)) = aa_num_int(neg,aa_num_num(bitM,N)) ).

% Int.sub_code(4)
tff(fact_8019_Int_Osub__code_I5_J,axiom,
    ! [N: num] : aa_num_int(sub(one),aa_num_num(bit1,N)) = aa_num_int(neg,aa_num_num(bit0,N)) ).

% Int.sub_code(5)
tff(fact_8020_Int_Osub__code_I1_J,axiom,
    aa_num_int(sub(one),one) = zero_zero_int ).

% Int.sub_code(1)
tff(fact_8021_Int_Osub__def,axiom,
    ! [M: num,N: num] : aa_num_int(sub(M),N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) ).

% Int.sub_def
tff(fact_8022_minus__int__code_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(neg,M)),aa_num_int(neg,N)) = aa_num_int(sub(N),M) ).

% minus_int_code(6)
tff(fact_8023_Int_Osub__code_I9_J,axiom,
    ! [M: num,N: num] : aa_num_int(sub(aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(dup,aa_num_int(sub(M),N))),one_one_int) ).

% Int.sub_code(9)
tff(fact_8024_Int_Osub__code_I8_J,axiom,
    ! [M: num,N: num] : aa_num_int(sub(aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(dup,aa_num_int(sub(M),N))),one_one_int) ).

% Int.sub_code(8)
tff(fact_8025_Int_Odup__code_I1_J,axiom,
    aa_int_int(dup,zero_zero_int) = zero_zero_int ).

% Int.dup_code(1)
tff(fact_8026_Int_Odup__def,axiom,
    ! [K: int] : aa_int_int(dup,K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),K) ).

% Int.dup_def
tff(fact_8027_Int_Odup__code_I3_J,axiom,
    ! [N: num] : aa_int_int(dup,aa_num_int(neg,N)) = aa_num_int(neg,aa_num_num(bit0,N)) ).

% Int.dup_code(3)
tff(fact_8028_Int_Osub__code_I6_J,axiom,
    ! [M: num,N: num] : aa_num_int(sub(aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_int_int(dup,aa_num_int(sub(M),N)) ).

% Int.sub_code(6)
tff(fact_8029_Int_Osub__code_I7_J,axiom,
    ! [M: num,N: num] : aa_num_int(sub(aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_int_int(dup,aa_num_int(sub(M),N)) ).

% Int.sub_code(7)
tff(fact_8030_Int_Osub__code_I2_J,axiom,
    ! [M: num] : aa_num_int(sub(aa_num_num(bit0,M)),one) = aa_num_int(pos,aa_num_num(bitM,M)) ).

% Int.sub_code(2)
tff(fact_8031_Int_Osub__code_I3_J,axiom,
    ! [M: num] : aa_num_int(sub(aa_num_num(bit1,M)),one) = aa_num_int(pos,aa_num_num(bit0,M)) ).

% Int.sub_code(3)
tff(fact_8032_Int_ONeg__def,axiom,
    ! [N: num] : aa_num_int(neg,N) = aa_int_int(uminus_uminus_int,aa_num_int(pos,N)) ).

% Int.Neg_def
tff(fact_8033_uminus__int__code_I2_J,axiom,
    ! [M: num] : aa_int_int(uminus_uminus_int,aa_num_int(pos,M)) = aa_num_int(neg,M) ).

% uminus_int_code(2)
tff(fact_8034_uminus__int__code_I3_J,axiom,
    ! [M: num] : aa_int_int(uminus_uminus_int,aa_num_int(neg,M)) = aa_num_int(pos,M) ).

% uminus_int_code(3)
tff(fact_8035_less__eq__int__code_I8_J,axiom,
    ! [K: num,L: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(neg,K)),aa_num_int(pos,L)) ).

% less_eq_int_code(8)
tff(fact_8036_less__eq__int__code_I6_J,axiom,
    ! [K: num,L: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(pos,K)),aa_num_int(neg,L)) ).

% less_eq_int_code(6)
tff(fact_8037_less__int__code_I6_J,axiom,
    ! [K: num,L: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(pos,K)),aa_num_int(neg,L)) ).

% less_int_code(6)
tff(fact_8038_less__int__code_I8_J,axiom,
    ! [K: num,L: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(neg,K)),aa_num_int(pos,L)) ).

% less_int_code(8)
tff(fact_8039_Int_OPos__def,axiom,
    pos = numeral_numeral_int ).

% Int.Pos_def
tff(fact_8040_one__int__code,axiom,
    one_one_int = aa_num_int(pos,one) ).

% one_int_code
tff(fact_8041_less__int__code_I4_J,axiom,
    ! [K: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(pos,K)),zero_zero_int) ).

% less_int_code(4)
tff(fact_8042_less__int__code_I2_J,axiom,
    ! [L: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(pos,L)) ).

% less_int_code(2)
tff(fact_8043_plus__int__code_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(pos,M)),aa_num_int(pos,N)) = aa_num_int(pos,aa_num_num(plus_plus_num(M),N)) ).

% plus_int_code(3)
tff(fact_8044_less__eq__int__code_I4_J,axiom,
    ! [K: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(pos,K)),zero_zero_int) ).

% less_eq_int_code(4)
tff(fact_8045_less__eq__int__code_I2_J,axiom,
    ! [L: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(pos,L)) ).

% less_eq_int_code(2)
tff(fact_8046_times__int__code_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(pos,M)),aa_num_int(pos,N)) = aa_num_int(pos,aa_num_num(times_times_num(M),N)) ).

% times_int_code(3)
tff(fact_8047_nat__code_I3_J,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(pos,K)) = aa_num_nat(nat_of_num,K) ).

% nat_code(3)
tff(fact_8048_less__eq__int__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(pos,K)),aa_num_int(pos,L))
    <=> aa_num_o(ord_less_eq_num(K),L) ) ).

% less_eq_int_code(5)
tff(fact_8049_less__int__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(pos,K)),aa_num_int(pos,L))
    <=> aa_num_o(ord_less_num(K),L) ) ).

% less_int_code(5)
tff(fact_8050_minus__int__code_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(pos,M)),aa_num_int(pos,N)) = aa_num_int(sub(M),N) ).

% minus_int_code(3)
tff(fact_8051_Int_Odup__code_I2_J,axiom,
    ! [N: num] : aa_int_int(dup,aa_num_int(pos,N)) = aa_num_int(pos,aa_num_num(bit0,N)) ).

% Int.dup_code(2)
tff(fact_8052_minus__int__code_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(neg,M)),aa_num_int(pos,N)) = aa_num_int(neg,aa_num_num(plus_plus_num(M),N)) ).

% minus_int_code(5)
tff(fact_8053_minus__int__code_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(pos,M)),aa_num_int(neg,N)) = aa_num_int(pos,aa_num_num(plus_plus_num(M),N)) ).

% minus_int_code(4)
tff(fact_8054_times__int__code_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(pos,M)),aa_num_int(neg,N)) = aa_num_int(neg,aa_num_num(times_times_num(M),N)) ).

% times_int_code(4)
tff(fact_8055_times__int__code_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(neg,M)),aa_num_int(pos,N)) = aa_num_int(neg,aa_num_num(times_times_num(M),N)) ).

% times_int_code(5)
tff(fact_8056_times__int__code_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(neg,M)),aa_num_int(neg,N)) = aa_num_int(pos,aa_num_num(times_times_num(M),N)) ).

% times_int_code(6)
tff(fact_8057_plus__int__code_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(neg,M)),aa_num_int(pos,N)) = aa_num_int(sub(N),M) ).

% plus_int_code(5)
tff(fact_8058_plus__int__code_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(pos,M)),aa_num_int(neg,N)) = aa_num_int(sub(M),N) ).

% plus_int_code(4)
tff(fact_8059_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( order_5726023648592871131at_nat(F)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(F,N)) ) ).

% strict_mono_imp_increasing
tff(fact_8060_Abs__int__inject,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( member2643936169264416010at_nat(Xa,collec5514110066124741708at_nat(aTP_Lamm_aee))
     => ( member2643936169264416010at_nat(Y3,collec5514110066124741708at_nat(aTP_Lamm_aee))
       => ( ( aa_set4746584241299694929at_int(abs_int,Xa) = aa_set4746584241299694929at_int(abs_int,Y3) )
        <=> ( Xa = Y3 ) ) ) ) ).

% Abs_int_inject
tff(fact_8061_Abs__int__induct,axiom,
    ! [Pa: fun_int_o,Xa: int] :
      ( ! [Y4: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(Y4,collec5514110066124741708at_nat(aTP_Lamm_aee))
         => aa_int_o(Pa,aa_set4746584241299694929at_int(abs_int,Y4)) )
     => aa_int_o(Pa,Xa) ) ).

% Abs_int_induct
tff(fact_8062_Abs__int__cases,axiom,
    ! [Xa: int] :
      ~ ! [Y4: set_Pr1261947904930325089at_nat] :
          ( ( Xa = aa_set4746584241299694929at_int(abs_int,Y4) )
         => ~ member2643936169264416010at_nat(Y4,collec5514110066124741708at_nat(aTP_Lamm_aee)) ) ).

% Abs_int_cases
tff(fact_8063_Abs__Integ__def,axiom,
    abs_Integ = quot_a8334514980957697477at_int(intrel,abs_int) ).

% Abs_Integ_def
tff(fact_8064_Abs__int__inverse,axiom,
    ! [Y3: set_Pr1261947904930325089at_nat] :
      ( member2643936169264416010at_nat(Y3,collec5514110066124741708at_nat(aTP_Lamm_aee))
     => ( aa_int8822742982810151799at_nat(rep_int,aa_set4746584241299694929at_int(abs_int,Y3)) = Y3 ) ) ).

% Abs_int_inverse
tff(fact_8065_Rep__int__inject,axiom,
    ! [Xa: int,Y3: int] :
      ( ( aa_int8822742982810151799at_nat(rep_int,Xa) = aa_int8822742982810151799at_nat(rep_int,Y3) )
    <=> ( Xa = Y3 ) ) ).

% Rep_int_inject
tff(fact_8066_Rep__int__inverse,axiom,
    ! [Xa: int] : aa_set4746584241299694929at_int(abs_int,aa_int8822742982810151799at_nat(rep_int,Xa)) = Xa ).

% Rep_int_inverse
tff(fact_8067_Rep__int__induct,axiom,
    ! [Y3: set_Pr1261947904930325089at_nat,Pa: fun_se4782206882984618170_nat_o] :
      ( member2643936169264416010at_nat(Y3,collec5514110066124741708at_nat(aTP_Lamm_aee))
     => ( ! [X4: int] : aa_set3702741264012733747_nat_o(Pa,aa_int8822742982810151799at_nat(rep_int,X4))
       => aa_set3702741264012733747_nat_o(Pa,Y3) ) ) ).

% Rep_int_induct
tff(fact_8068_Rep__int__cases,axiom,
    ! [Y3: set_Pr1261947904930325089at_nat] :
      ( member2643936169264416010at_nat(Y3,collec5514110066124741708at_nat(aTP_Lamm_aee))
     => ~ ! [X4: int] : Y3 != aa_int8822742982810151799at_nat(rep_int,X4) ) ).

% Rep_int_cases
tff(fact_8069_Rep__int,axiom,
    ! [Xa: int] : member2643936169264416010at_nat(aa_int8822742982810151799at_nat(rep_int,Xa),collec5514110066124741708at_nat(aTP_Lamm_aee)) ).

% Rep_int
tff(fact_8070_type__definition__int,axiom,
    type_d8752208705193531015at_nat(rep_int,abs_int,collec5514110066124741708at_nat(aTP_Lamm_aee)) ).

% type_definition_int
tff(fact_8071_Rep__Integ__def,axiom,
    rep_Integ = quot_r1512469584000298464at_nat(rep_int) ).

% Rep_Integ_def
tff(fact_8072_less__eq__literal_Oabs__eq,axiom,
    ! [Xa: list_char,Xb: list_char] :
      ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_aef),Xa),Xa)
     => ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_aef),Xb),Xb)
       => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_eq_literal,aa_list_char_literal(abs_literal,Xa)),aa_list_char_literal(abs_literal,Xb))
        <=> aa_list_char_o(aa_lis7814493268959572977char_o(lexordp_eq_char(aTP_Lamm_ady),Xa),Xb) ) ) ) ).

% less_eq_literal.abs_eq
tff(fact_8073_plus__literal_Orsp,axiom,
    bNF_re1514760611062797473t_char(bNF_eq_onp_list_char(aTP_Lamm_aef),bNF_re1407854171455731685t_char(bNF_eq_onp_list_char(aTP_Lamm_aef),bNF_eq_onp_list_char(aTP_Lamm_aef)),append_char,append_char) ).

% plus_literal.rsp
tff(fact_8074_less__eq__literal_Orsp,axiom,
    bNF_re6109064546783844641char_o(bNF_eq_onp_list_char(aTP_Lamm_aef),bNF_re6668474094832959905ar_o_o(bNF_eq_onp_list_char(aTP_Lamm_aef),fequal_o),lexordp_eq_char(aTP_Lamm_ady),lexordp_eq_char(aTP_Lamm_ady)) ).

% less_eq_literal.rsp
tff(fact_8075_asciis__of__literal_Orsp,axiom,
    bNF_re901197953848981245nteger(bNF_eq_onp_list_char(aTP_Lamm_aef),fequal5224421480335545304nteger,map_ch5843194146513361140nteger(comm_s4049598766089966537nteger),map_ch5843194146513361140nteger(comm_s4049598766089966537nteger)) ).

% asciis_of_literal.rsp
tff(fact_8076_size__literal_Orsp,axiom,
    bNF_re76688982720252355at_nat(bNF_eq_onp_list_char(aTP_Lamm_aef),fequal_nat,size_size_list_char,size_size_list_char) ).

% size_literal.rsp
tff(fact_8077_Literal_Orsp,axiom,
    bNF_re5337651037176608865t_char(fequal_o,bNF_re8108790427976964573t_char(fequal_o,bNF_re2075097589415080885t_char(fequal_o,bNF_re8572817166782503389t_char(fequal_o,bNF_re7288172596490189065t_char(fequal_o,bNF_re8717008942347199453t_char(fequal_o,bNF_re7974392315488110685t_char(fequal_o,bNF_re1407854171455731685t_char(bNF_eq_onp_list_char(aTP_Lamm_aef),bNF_eq_onp_list_char(aTP_Lamm_aef)))))))),aTP_Lamm_aea,aTP_Lamm_aea) ).

% Literal.rsp
tff(fact_8078_less__literal_Orsp,axiom,
    bNF_re6109064546783844641char_o(bNF_eq_onp_list_char(aTP_Lamm_aef),bNF_re6668474094832959905ar_o_o(bNF_eq_onp_list_char(aTP_Lamm_aef),fequal_o),lexordp_char(aTP_Lamm_ady),lexordp_char(aTP_Lamm_ady)) ).

% less_literal.rsp
tff(fact_8079_plus__literal_Oabs__eq,axiom,
    ! [Xa: list_char,Xb: list_char] :
      ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_aef),Xa),Xa)
     => ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_aef),Xb),Xb)
       => ( plus_plus_literal(aa_list_char_literal(abs_literal,Xa),aa_list_char_literal(abs_literal,Xb)) = aa_list_char_literal(abs_literal,aa_lis7614194356200183040t_char(aa_lis2542250943915203678t_char(append_char,Xa),Xb)) ) ) ) ).

% plus_literal.abs_eq
tff(fact_8080_literal_OAbs__literal__cases,axiom,
    ! [Xa: literal] :
      ~ ! [Y4: list_char] :
          ( ( Xa = aa_list_char_literal(abs_literal,Y4) )
         => ~ member_list_char(Y4,collect_list_char(aTP_Lamm_aef)) ) ).

% literal.Abs_literal_cases
tff(fact_8081_literal_OAbs__literal__induct,axiom,
    ! [Pa: fun_literal_o,Xa: literal] :
      ( ! [Y4: list_char] :
          ( member_list_char(Y4,collect_list_char(aTP_Lamm_aef))
         => aa_literal_o(Pa,aa_list_char_literal(abs_literal,Y4)) )
     => aa_literal_o(Pa,Xa) ) ).

% literal.Abs_literal_induct
tff(fact_8082_literal_OAbs__literal__inject,axiom,
    ! [Xa: list_char,Y3: list_char] :
      ( member_list_char(Xa,collect_list_char(aTP_Lamm_aef))
     => ( member_list_char(Y3,collect_list_char(aTP_Lamm_aef))
       => ( ( aa_list_char_literal(abs_literal,Xa) = aa_list_char_literal(abs_literal,Y3) )
        <=> ( Xa = Y3 ) ) ) ) ).

% literal.Abs_literal_inject
tff(fact_8083_size__literal_Oabs__eq,axiom,
    ! [Xa: list_char] :
      ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_aef),Xa),Xa)
     => ( size_size_literal(aa_list_char_literal(abs_literal,Xa)) = aa_list_char_nat(size_size_list_char,Xa) ) ) ).

% size_literal.abs_eq
tff(fact_8084_literal_Oexplode,axiom,
    ! [Xa: literal] : member_list_char(aa_literal_list_char(explode,Xa),collect_list_char(aTP_Lamm_aef)) ).

% literal.explode
tff(fact_8085_literal_Oexplode__cases,axiom,
    ! [Y3: list_char] :
      ( member_list_char(Y3,collect_list_char(aTP_Lamm_aef))
     => ~ ! [X4: literal] : Y3 != aa_literal_list_char(explode,X4) ) ).

% literal.explode_cases
tff(fact_8086_literal_Oexplode__induct,axiom,
    ! [Y3: list_char,Pa: fun_list_char_o] :
      ( member_list_char(Y3,collect_list_char(aTP_Lamm_aef))
     => ( ! [X4: literal] : aa_list_char_o(Pa,aa_literal_list_char(explode,X4))
       => aa_list_char_o(Pa,Y3) ) ) ).

% literal.explode_induct
tff(fact_8087_zero__literal_Orsp,axiom,
    aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_aef),nil_char),nil_char) ).

% zero_literal.rsp
tff(fact_8088_String_OQuotient__literal,axiom,
    quotie6109894551476798619iteral(bNF_eq_onp_list_char(aTP_Lamm_aef),abs_literal,explode,cr_literal) ).

% String.Quotient_literal
tff(fact_8089_Literal_Oabs__eq,axiom,
    ! [Xa: list_char,Xg: $o,Xf: $o,Xe: $o,Xd: $o,Xc: $o,Xb: $o,Xaa: $o] :
      ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_aef),Xa),Xa)
     => ( aa_literal_literal(aa_o_f1412681823949618463iteral(aa_o_f837559538564922386iteral(aa_o_f8819669424537506293iteral(aa_o_f3445383346798733800iteral(aa_o_f2755856924670595275iteral(aa_o_f293512076901925310iteral(aa_o_f6593176798326225313iteral(literal2,(Xg)),(Xf)),(Xe)),(Xd)),(Xc)),(Xb)),(Xaa)),aa_list_char_literal(abs_literal,Xa)) = aa_list_char_literal(abs_literal,aa_lis7614194356200183040t_char(cons_char(char2((Xg),(Xf),(Xe),(Xd),(Xc),(Xb),(Xaa),$false)),Xa)) ) ) ).

% Literal.abs_eq
tff(fact_8090_literal_OAbs__literal__inverse,axiom,
    ! [Y3: list_char] :
      ( member_list_char(Y3,collect_list_char(aTP_Lamm_aef))
     => ( aa_literal_list_char(explode,aa_list_char_literal(abs_literal,Y3)) = Y3 ) ) ).

% literal.Abs_literal_inverse
tff(fact_8091_literal_Otype__definition__literal,axiom,
    type_d4752411451802217481t_char(explode,abs_literal,collect_list_char(aTP_Lamm_aef)) ).

% literal.type_definition_literal
tff(fact_8092_less__literal_Oabs__eq,axiom,
    ! [Xa: list_char,Xb: list_char] :
      ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_aef),Xa),Xa)
     => ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_aef),Xb),Xb)
       => ( aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,aa_list_char_literal(abs_literal,Xa)),aa_list_char_literal(abs_literal,Xb))
        <=> aa_list_char_o(aa_lis7814493268959572977char_o(lexordp_char(aTP_Lamm_ady),Xa),Xb) ) ) ) ).

% less_literal.abs_eq
tff(fact_8093_asciis__of__literal_Oabs__eq,axiom,
    ! [Xa: list_char] :
      ( aa_list_char_o(aa_lis7814493268959572977char_o(bNF_eq_onp_list_char(aTP_Lamm_aef),Xa),Xa)
     => ( asciis_of_literal(aa_list_char_literal(abs_literal,Xa)) = aa_lis2940536026700532620nteger(map_ch5843194146513361140nteger(comm_s4049598766089966537nteger),Xa) ) ) ).

% asciis_of_literal.abs_eq
tff(fact_8094_literal__of__asciis_Orsp,axiom,
    bNF_re5828396963709814525t_char(fequal5224421480335545304nteger,bNF_eq_onp_list_char(aTP_Lamm_aef),map_Co843364623960961780r_char(comp_c4562247934908617188nteger(ascii_of,unique1708501630192214702nteger)),map_Co843364623960961780r_char(comp_c4562247934908617188nteger(ascii_of,unique1708501630192214702nteger))) ).

% literal_of_asciis.rsp
tff(fact_8095_implode_Orsp,axiom,
    aa_fun8069410965070236123char_o(aa_fun8853676540986067637char_o(bNF_re1407854171455731685t_char(fequal_list_char,bNF_eq_onp_list_char(aTP_Lamm_aef)),map_char_char(ascii_of)),map_char_char(ascii_of)) ).

% implode.rsp
tff(fact_8096_literal__of__asciis_Otransfer,axiom,
    bNF_re3035051291559180600iteral(fequal5224421480335545304nteger,pcr_literal,map_Co843364623960961780r_char(comp_c4562247934908617188nteger(ascii_of,unique1708501630192214702nteger)),literal_of_asciis) ).

% literal_of_asciis.transfer
tff(fact_8097_String_Oliteral__of__asciis__def,axiom,
    literal_of_asciis = map_fu9222038735678016191iteral(id_list_Code_integer,abs_literal,map_Co843364623960961780r_char(comp_c4562247934908617188nteger(ascii_of,unique1708501630192214702nteger))) ).

% String.literal_of_asciis_def
tff(fact_8098_literal__of__asciis_Orep__eq,axiom,
    ! [Xa: list_Code_integer] : aa_literal_list_char(explode,aa_lis5633869780665501801iteral(literal_of_asciis,Xa)) = aa_lis1268021600136961932t_char(map_Co843364623960961780r_char(comp_c4562247934908617188nteger(ascii_of,unique1708501630192214702nteger)),Xa) ).

% literal_of_asciis.rep_eq
tff(fact_8099_literal__of__asciis_Oabs__eq,axiom,
    ! [Xa: list_Code_integer] : aa_lis5633869780665501801iteral(literal_of_asciis,Xa) = aa_list_char_literal(abs_literal,aa_lis1268021600136961932t_char(map_Co843364623960961780r_char(comp_c4562247934908617188nteger(ascii_of,unique1708501630192214702nteger)),Xa)) ).

% literal_of_asciis.abs_eq
tff(fact_8100_String_Oliteral__of__asciis__Cons,axiom,
    ! [K: code_integer,Ks: list_Code_integer] : aa_lis5633869780665501801iteral(literal_of_asciis,cons_Code_integer(K,Ks)) = case_char_literal(aTP_Lamm_aeg(Ks),aa_Code_integer_char(unique1708501630192214702nteger,K)) ).

% String.literal_of_asciis_Cons
tff(fact_8101_literal_Odomain,axiom,
    ! [X: list_char] :
      ( aa_list_char_o(domain7888411570377768909iteral(pcr_literal),X)
    <=> ? [Y: list_char] :
          ( aa_list_char_o(aa_lis7814493268959572977char_o(list_all2_char_char(fequal_char),X),Y)
          & ! [Xa2: char] :
              ( member_char(Xa2,set_char2(Y))
             => ~ aa_char_o(digit7,Xa2) ) ) ) ).

% literal.domain
tff(fact_8102_literal_Odomain__eq,axiom,
    ! [X: list_char] :
      ( aa_list_char_o(domain7888411570377768909iteral(pcr_literal),X)
    <=> ! [Xa2: char] :
          ( member_char(Xa2,set_char2(X))
         => ~ aa_char_o(digit7,Xa2) ) ) ).

% literal.domain_eq
tff(fact_8103_literal_Odomain__par__left__total,axiom,
    ! [P2: fun_list_char_o] :
      ( left_t8440299596084406506t_char(list_all2_char_char(fequal_char))
     => ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(bNF_re6668474094832959905ar_o_o(list_all2_char_char(fequal_char),fequal_o),P2),aTP_Lamm_aef)
       => ( domain7888411570377768909iteral(pcr_literal) = P2 ) ) ) ).

% literal.domain_par_left_total
tff(fact_8104_rat_Odomain__par,axiom,
    ! [DR1: fun_int_o,DR2: fun_int_o,P22: fun_Pr3238374608701176162_int_o] :
      ( ( domainp_int_int(fequal_int) = DR1 )
     => ( ( domainp_int_int(fequal_int) = DR2 )
       => ( bNF_re8699439704749558557nt_o_o(basic_4387203522000727145nt_int(fequal_int,fequal_int),fequal_o,P22,aTP_Lamm_aeh)
         => ( domain9213661015745956397nt_rat(pcr_rat) = inf_in1782598092406300404_int_o(basic_1567116559311922317nt_int(DR1,DR2),P22) ) ) ) ) ).

% rat.domain_par
tff(fact_8105_digit7__def,axiom,
    ! [Char: char] :
      ( aa_char_o(digit7,Char)
    <=> aa_char_o(case_char_o(aTP_Lamm_aei),Char) ) ).

% digit7_def
tff(fact_8106_int_Odomain,axiom,
    ! [DR1: fun_nat_o,DR2: fun_nat_o] :
      ( ( domainp_nat_nat(fequal_nat) = DR1 )
     => ( ( domainp_nat_nat(fequal_nat) = DR2 )
       => ( domain7287725470393535433at_int(pcr_int) = basic_570253317919918037at_nat(DR1,DR2) ) ) ) ).

% int.domain
tff(fact_8107_int_Oright__total,axiom,
    right_1157779049177982926at_int(pcr_int) ).

% int.right_total
tff(fact_8108_rat_Odomain,axiom,
    ! [X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(domain9213661015745956397nt_rat(pcr_rat),X)
    <=> ? [Y: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(basic_4387203522000727145nt_int(fequal_int,fequal_int),X),Y)
          & aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Y),Y) ) ) ).

% rat.domain
tff(fact_8109_literal_Odomain__par,axiom,
    ! [DR: fun_char_o,P22: fun_list_char_o] :
      ( ( domainp_char_char(fequal_char) = DR )
     => ( aa_fun_list_char_o_o(aa_fun3501637381074999839ar_o_o(bNF_re6668474094832959905ar_o_o(list_all2_char_char(fequal_char),fequal_o),P22),aTP_Lamm_aef)
       => ( domain7888411570377768909iteral(pcr_literal) = inf_in94156054473216830char_o(list_all_char(DR),P22) ) ) ) ).

% literal.domain_par
tff(fact_8110_char_Odisc__eq__case,axiom,
    ! [Char: char] : aa_char_o(case_char_o(aTP_Lamm_aej),Char) ).

% char.disc_eq_case
tff(fact_8111_int_Oleft__total,axiom,
    left_t1861001511420785031at_int(pcr_int) ).

% int.left_total
tff(fact_8112_rat_Odomain__par__left__total,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o] :
      ( left_t3131394472396969446nt_int(basic_4387203522000727145nt_int(fequal_int,fequal_int))
     => ( bNF_re8699439704749558557nt_o_o(basic_4387203522000727145nt_int(fequal_int,fequal_int),fequal_o,P2,aTP_Lamm_aeh)
       => ( domain9213661015745956397nt_rat(pcr_rat) = P2 ) ) ) ).

% rat.domain_par_left_total
tff(fact_8113_Abs__rat__inject,axiom,
    ! [Xa: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
      ( member2340774599025711042nt_int(Xa,collec5210948495886036740nt_int(aTP_Lamm_aek))
     => ( member2340774599025711042nt_int(Y3,collec5210948495886036740nt_int(aTP_Lamm_aek))
       => ( ( aa_set4118085860027418549nt_rat(abs_rat,Xa) = aa_set4118085860027418549nt_rat(abs_rat,Y3) )
        <=> ( Xa = Y3 ) ) ) ) ).

% Abs_rat_inject
tff(fact_8114_Abs__rat__cases,axiom,
    ! [Xa: rat] :
      ~ ! [Y4: set_Pr958786334691620121nt_int] :
          ( ( Xa = aa_set4118085860027418549nt_rat(abs_rat,Y4) )
         => ~ member2340774599025711042nt_int(Y4,collec5210948495886036740nt_int(aTP_Lamm_aek)) ) ).

% Abs_rat_cases
tff(fact_8115_Abs__rat__induct,axiom,
    ! [Pa: fun_rat_o,Xa: rat] :
      ( ! [Y4: set_Pr958786334691620121nt_int] :
          ( member2340774599025711042nt_int(Y4,collec5210948495886036740nt_int(aTP_Lamm_aek))
         => aa_rat_o(Pa,aa_set4118085860027418549nt_rat(abs_rat,Y4)) )
     => aa_rat_o(Pa,Xa) ) ).

% Abs_rat_induct
tff(fact_8116_Abs__rat__inverse,axiom,
    ! [Y3: set_Pr958786334691620121nt_int] :
      ( member2340774599025711042nt_int(Y3,collec5210948495886036740nt_int(aTP_Lamm_aek))
     => ( aa_rat8624586542854104963nt_int(rep_rat,aa_set4118085860027418549nt_rat(abs_rat,Y3)) = Y3 ) ) ).

% Abs_rat_inverse
tff(fact_8117_type__definition__rat,axiom,
    type_d8554052265237484179nt_int(rep_rat,abs_rat,collec5210948495886036740nt_int(aTP_Lamm_aek)) ).

% type_definition_rat
tff(fact_8118_Rep__rat__induct,axiom,
    ! [Y3: set_Pr958786334691620121nt_int,Pa: fun_se7249947736802449410_int_o] :
      ( member2340774599025711042nt_int(Y3,collec5210948495886036740nt_int(aTP_Lamm_aek))
     => ( ! [X4: rat] : aa_set6170482117830564987_int_o(Pa,aa_rat8624586542854104963nt_int(rep_rat,X4))
       => aa_set6170482117830564987_int_o(Pa,Y3) ) ) ).

% Rep_rat_induct
tff(fact_8119_Rep__rat__cases,axiom,
    ! [Y3: set_Pr958786334691620121nt_int] :
      ( member2340774599025711042nt_int(Y3,collec5210948495886036740nt_int(aTP_Lamm_aek))
     => ~ ! [X4: rat] : Y3 != aa_rat8624586542854104963nt_int(rep_rat,X4) ) ).

% Rep_rat_cases
tff(fact_8120_Rep__rat,axiom,
    ! [Xa: rat] : member2340774599025711042nt_int(aa_rat8624586542854104963nt_int(rep_rat,Xa),collec5210948495886036740nt_int(aTP_Lamm_aek)) ).

% Rep_rat
tff(fact_8121_int_Obi__total,axiom,
    bi_tot896582865486249351at_int(pcr_int) ).

% int.bi_total
tff(fact_8122_gcd__nat_Oordering__axioms,axiom,
    ordering_nat(dvd_dvd_nat,aTP_Lamm_aas) ).

% gcd_nat.ordering_axioms
tff(fact_8123_typerep_Osize__neq,axiom,
    ! [Xa: typerep] : aa_typerep_nat(size_size_typerep,Xa) != zero_zero_nat ).

% typerep.size_neq
tff(fact_8124_gcd__nat_Opreordering__axioms,axiom,
    preordering_nat(dvd_dvd_nat,aTP_Lamm_aas) ).

% gcd_nat.preordering_axioms
tff(fact_8125_typerep_Osize_I2_J,axiom,
    ! [X1: literal,X2: list_typerep] : aa_typerep_nat(size_size_typerep,typerep2(X1,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_typerep(size_size_typerep,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% typerep.size(2)
tff(fact_8126_typerep_Osize__gen,axiom,
    ! [X1: literal,X2: list_typerep] : aa_typerep_nat(size_typerep,typerep2(X1,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_typerep(size_typerep,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% typerep.size_gen
tff(fact_8127_digit6__def,axiom,
    ! [Char: char] :
      ( aa_char_o(digit6,Char)
    <=> aa_char_o(case_char_o(aTP_Lamm_ael),Char) ) ).

% digit6_def
tff(fact_8128_digit5__def,axiom,
    ! [Char: char] :
      ( aa_char_o(digit5,Char)
    <=> aa_char_o(case_char_o(aTP_Lamm_aem),Char) ) ).

% digit5_def
tff(fact_8129_digit0__def,axiom,
    ! [Char: char] :
      ( aa_char_o(digit0,Char)
    <=> aa_char_o(case_char_o(aTP_Lamm_aen),Char) ) ).

% digit0_def
tff(fact_8130_digit1__def,axiom,
    ! [Char: char] :
      ( aa_char_o(digit1,Char)
    <=> aa_char_o(case_char_o(aTP_Lamm_aeo),Char) ) ).

% digit1_def
tff(fact_8131_digit4__def,axiom,
    ! [Char: char] :
      ( aa_char_o(digit4,Char)
    <=> aa_char_o(case_char_o(aTP_Lamm_aep),Char) ) ).

% digit4_def
tff(fact_8132_digit3__def,axiom,
    ! [Char: char] :
      ( aa_char_o(digit3,Char)
    <=> aa_char_o(case_char_o(aTP_Lamm_aeq),Char) ) ).

% digit3_def
tff(fact_8133_digit2__def,axiom,
    ! [Char: char] :
      ( aa_char_o(digit2,Char)
    <=> aa_char_o(case_char_o(aTP_Lamm_aer),Char) ) ).

% digit2_def
tff(fact_8134_log_Opelims,axiom,
    ! [Xa: code_natural,Xaa: code_natural,Y3: code_natural] :
      ( ( log(Xa,Xaa) = Y3 )
     => ( aa_Pro6073791466322389475ural_o(accp_P8126237942716283194atural(log_rel),produc3574140220909816553atural(Xa,Xaa))
       => ~ ( ( Y3 = $ite(
                  ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,Xa),one_one_Code_natural)
                  | aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Xaa),Xa) ),
                  one_one_Code_natural,
                  aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),log(Xa,divide5121882707175180666atural(Xaa,Xa))) ) )
           => ~ aa_Pro6073791466322389475ural_o(accp_P8126237942716283194atural(log_rel),produc3574140220909816553atural(Xa,Xaa)) ) ) ) ).

% log.pelims
tff(fact_8135_log_Ocases,axiom,
    ! [Xa: produc7822875418678951345atural] :
      ~ ! [B2: code_natural,I2: code_natural] : Xa != produc3574140220909816553atural(B2,I2) ).

% log.cases
tff(fact_8136_full__exhaustive__natural_H_Ocases,axiom,
    ! [Xa: produc9140211490168418807atural] :
      ~ ! [F5: fun_Pr285225112257423215e_term,D3: code_natural,I2: code_natural] : Xa != aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,F5),produc3574140220909816553atural(D3,I2)) ).

% full_exhaustive_natural'.cases
tff(fact_8137_exhaustive__natural_H_Ocases,axiom,
    ! [Xa: produc7476732711998919211atural] :
      ~ ! [F5: fun_Co6386927884524458043e_term,D3: code_natural,I2: code_natural] : Xa != produc3379161622255323613atural(F5,produc3574140220909816553atural(D3,I2)) ).

% exhaustive_natural'.cases
tff(fact_8138_Random__Pred_Onot__randompred__def,axiom,
    ! [Pa: fun_Pr160617923510306311atural,X: produc7822875418678951345atural] : aa_Pro6676232280479689400atural(random6974930770145893639ompred(Pa),X) = produc6665183775751917029atural(aTP_Lamm_aes,aa_Pro6676232280479689400atural(Pa,X)) ).

% Random_Pred.not_randompred_def
tff(fact_8139_ATP_Olambda__1,axiom,
    ! [Uu: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(aTP_Lamm_add,Uu) = $ite(aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu) = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu))) ).

% ATP.lambda_1
tff(fact_8140_ATP_Olambda__2,axiom,
    ! [Uu: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_bu,Uu)
    <=> ( member_Code_integer(Uu,ring_11222124179247155820nteger)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Uu) ) ) ).

% ATP.lambda_2
tff(fact_8141_ATP_Olambda__3,axiom,
    ! [Uu: rat] :
      ( aa_rat_o(aTP_Lamm_bv,Uu)
    <=> ( member_rat(Uu,ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Uu) ) ) ).

% ATP.lambda_3
tff(fact_8142_ATP_Olambda__4,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_bw,Uu)
    <=> ( member_int(Uu,ring_1_Ints_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Uu) ) ) ).

% ATP.lambda_4
tff(fact_8143_ATP_Olambda__5,axiom,
    ! [Uu: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_ade,Uu) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu))),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)) ).

% ATP.lambda_5
tff(fact_8144_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_aec,Uu) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Uu))),one_one_int) ).

% ATP.lambda_6
tff(fact_8145_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ih,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_7
tff(fact_8146_ATP_Olambda__8,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_acn,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uu) ).

% ATP.lambda_8
tff(fact_8147_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_hf,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),Uu) ).

% ATP.lambda_9
tff(fact_8148_ATP_Olambda__10,axiom,
    ! [Uu: int] : aa_int7462565939943104719nt_int(aTP_Lamm_hg,Uu) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),Uu) ).

% ATP.lambda_10
tff(fact_8149_ATP_Olambda__11,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_aeh,Uu)
    <=> aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Uu),Uu) ) ).

% ATP.lambda_11
tff(fact_8150_ATP_Olambda__12,axiom,
    ! [Uu: code_natural] : aa_Cod4272041906116450446atural(aTP_Lamm_bm,Uu) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Uu),one_one_Code_natural) ).

% ATP.lambda_12
tff(fact_8151_ATP_Olambda__13,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_bk,Uu) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),one_one_Code_integer) ).

% ATP.lambda_13
tff(fact_8152_ATP_Olambda__14,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_bl,Uu) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),one_one_rat) ).

% ATP.lambda_14
tff(fact_8153_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_bo,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_15
tff(fact_8154_ATP_Olambda__16,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_bn,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_16
tff(fact_8155_ATP_Olambda__17,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_adh,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).

% ATP.lambda_17
tff(fact_8156_ATP_Olambda__18,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_adg,Uu)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu))) ) ).

% ATP.lambda_18
tff(fact_8157_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_aeb,Uu) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)) ).

% ATP.lambda_19
tff(fact_8158_ATP_Olambda__20,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_acv,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cx,Uu)) ).

% ATP.lambda_20
tff(fact_8159_ATP_Olambda__21,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_ep,Uu) = aa_num_option_num(some_num,aa_num_num(bit1,Uu)) ).

% ATP.lambda_21
tff(fact_8160_ATP_Olambda__22,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_ec,Uu) = aa_num_option_num(some_num,aa_num_num(bit0,Uu)) ).

% ATP.lambda_22
tff(fact_8161_ATP_Olambda__23,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_abm,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_23
tff(fact_8162_ATP_Olambda__24,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_abn,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_24
tff(fact_8163_ATP_Olambda__25,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_dy,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_25
tff(fact_8164_ATP_Olambda__26,axiom,
    ! [Uu: int] : aa_int_nat(aTP_Lamm_acs,Uu) = aa_int_nat(nat2,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_26
tff(fact_8165_ATP_Olambda__27,axiom,
    ! [Uu: list_char] :
      ( aa_list_char_o(aTP_Lamm_aef,Uu)
    <=> ! [X3: char] :
          ( member_char(X3,set_char2(Uu))
         => ~ aa_char_o(digit7,X3) ) ) ).

% ATP.lambda_27
tff(fact_8166_ATP_Olambda__28,axiom,
    ! [Uu: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_aek,Uu)
    <=> ? [X3: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X3),X3)
          & ( Uu = collec213857154873943460nt_int(aa_Pro3952998155787061599_int_o(ratrel,X3)) ) ) ) ).

% ATP.lambda_28
tff(fact_8167_ATP_Olambda__29,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aTP_Lamm_aee,Uu)
    <=> ? [X3: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,X3),X3)
          & ( Uu = collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(intrel,X3)) ) ) ) ).

% ATP.lambda_29
tff(fact_8168_ATP_Olambda__30,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_es(Uu),Uua) = aa_num_option_num(case_num_option_num(aa_num_option_num(some_num,one),aTP_Lamm_eq(Uua),aTP_Lamm_er(Uua)),Uu) ).

% ATP.lambda_30
tff(fact_8169_ATP_Olambda__31,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aTP_Lamm_hk(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_z3403309356797280102nteger) ).

% ATP.lambda_31
tff(fact_8170_ATP_Olambda__32,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_hl(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_rat) ).

% ATP.lambda_32
tff(fact_8171_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_hm,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_33
tff(fact_8172_ATP_Olambda__34,axiom,
    ! [Uu: pred_Product_unit,Uua: produc7822875418678951345atural] :
      aa_Pro6676232280479689400atural(aa_pre2304647123621358054atural(aTP_Lamm_aes,Uu),Uua) = $ite(aa_Product_unit_o(eval_Product_unit(Uu),product_Unity),aa_Pro6676232280479689400atural(produc5069803637994805237atural(bot_bo2717538794563056311t_unit),Uua),aa_Pro6676232280479689400atural(produc5069803637994805237atural(single_Product_unit(product_Unity)),Uua)) ).

% ATP.lambda_34
tff(fact_8173_ATP_Olambda__35,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_acj,Uu),Uua) = $ite(Uu = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu)),Uua)),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_35
tff(fact_8174_ATP_Olambda__36,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_adi,Uu),Uua) = $ite(Uua = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),Uua)) ).

% ATP.lambda_36
tff(fact_8175_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aTP_Lamm_hh(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_z3403309356797280102nteger) ).

% ATP.lambda_37
tff(fact_8176_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_hi(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_rat) ).

% ATP.lambda_38
tff(fact_8177_ATP_Olambda__39,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_hj,Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_39
tff(fact_8178_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_et,Uu),Uua) = aa_nat_option_num(case_nat_option_num(none_num,aTP_Lamm_es(Uua)),Uu) ).

% ATP.lambda_40
tff(fact_8179_ATP_Olambda__41,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_eq(Uu),Uua) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_ec),aa_num_option_num(bit_take_bit_num(Uu),Uua)) ).

% ATP.lambda_41
tff(fact_8180_ATP_Olambda__42,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ns(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_nr(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_42
tff(fact_8181_ATP_Olambda__43,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_nq(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ir(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_43
tff(fact_8182_ATP_Olambda__44,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_is(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ir(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_44
tff(fact_8183_ATP_Olambda__45,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_adc,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua))) ).

% ATP.lambda_45
tff(fact_8184_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_kb(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_46
tff(fact_8185_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_kc(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,Uua))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_47
tff(fact_8186_ATP_Olambda__48,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_kd,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,Uua))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_48
tff(fact_8187_ATP_Olambda__49,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_ke(Uu),Uua) = divide_divide_rat(aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua),aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_49
tff(fact_8188_ATP_Olambda__50,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_fi,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Code_integer_int(code_int_of_integer,Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).

% ATP.lambda_50
tff(fact_8189_ATP_Olambda__51,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_adf,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua))) ).

% ATP.lambda_51
tff(fact_8190_ATP_Olambda__52,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_ack(Uu),Uua)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_52
tff(fact_8191_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_km(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_53
tff(fact_8192_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_kn(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_54
tff(fact_8193_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ko,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_55
tff(fact_8194_ATP_Olambda__56,axiom,
    ! [Uu: rat,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_adx(Uu),Uua)
    <=> ( ( Uu = aa_int_rat(aa_int_fun_int_rat(fract,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua)) )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua))
        & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua)) ) ) ).

% ATP.lambda_56
tff(fact_8195_ATP_Olambda__57,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ip(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ).

% ATP.lambda_57
tff(fact_8196_ATP_Olambda__58,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hn(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(Uu),Uua)),minus_minus_rat(divide_divide_rat(Uu,numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_58
tff(fact_8197_ATP_Olambda__59,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_jf(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)) ).

% ATP.lambda_59
tff(fact_8198_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aas,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_60
tff(fact_8199_ATP_Olambda__61,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jm,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_61
tff(fact_8200_ATP_Olambda__62,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_adp(Uu),Uua)
    <=> ( member_int(Uua,Uu)
        & ! [X3: int] :
            ( member_int(X3,Uu)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Uua) ) ) ) ).

% ATP.lambda_62
tff(fact_8201_ATP_Olambda__63,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_xd,Uu),Uua)
    <=> ( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(Uu),Uua)
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Uua) ) ) ).

% ATP.lambda_63
tff(fact_8202_ATP_Olambda__64,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_xb,Uu),Uua)
    <=> ( aa_set_nat_o(ord_less_set_nat(Uu),Uua)
        & aa_set_nat_o(finite_finite_nat,Uua) ) ) ).

% ATP.lambda_64
tff(fact_8203_ATP_Olambda__65,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_xc,Uu),Uua)
    <=> ( ord_less_set_int(Uu,Uua)
        & finite_finite_int(Uua) ) ) ).

% ATP.lambda_65
tff(fact_8204_ATP_Olambda__66,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_adb,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_66
tff(fact_8205_ATP_Olambda__67,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ij,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_67
tff(fact_8206_ATP_Olambda__68,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ii,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_68
tff(fact_8207_ATP_Olambda__69,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_vv,Uu),Uua)
    <=> ( member_rat(Uua,ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Uua)),Uu) ) ) ).

% ATP.lambda_69
tff(fact_8208_ATP_Olambda__70,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_mp(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_Code_integer(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_70
tff(fact_8209_ATP_Olambda__71,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mq(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_71
tff(fact_8210_ATP_Olambda__72,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_mr(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_72
tff(fact_8211_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ms(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_73
tff(fact_8212_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_jc(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_74
tff(fact_8213_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_je(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_75
tff(fact_8214_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_jd(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_76
tff(fact_8215_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_if(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(Uu,aa_nat_nat(suc,Uua))),aa_nat_Code_integer(Uu,Uua)) ).

% ATP.lambda_77
tff(fact_8216_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ig(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua)),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_78
tff(fact_8217_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ie(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_79
tff(fact_8218_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_oc(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uua)) ).

% ATP.lambda_80
tff(fact_8219_ATP_Olambda__81,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_od(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uua)) ).

% ATP.lambda_81
tff(fact_8220_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_oa(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uua)) ).

% ATP.lambda_82
tff(fact_8221_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qe(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_83
tff(fact_8222_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_iz(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(Uu,Uua)),aa_nat_Code_integer(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_84
tff(fact_8223_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ja(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,Uua),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_85
tff(fact_8224_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_iy(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_86
tff(fact_8225_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_iw(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(Uu,Uua)),aa_nat_Code_integer(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_87
tff(fact_8226_ATP_Olambda__88,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ix(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,Uua),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_88
tff(fact_8227_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_iv(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_89
tff(fact_8228_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_ic(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(Uu,Uua)),aa_nat_Code_integer(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_90
tff(fact_8229_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_id(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,Uua),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_91
tff(fact_8230_ATP_Olambda__92,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ib(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_92
tff(fact_8231_ATP_Olambda__93,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_nv(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_93
tff(fact_8232_ATP_Olambda__94,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_nx(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_94
tff(fact_8233_ATP_Olambda__95,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_nt(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_95
tff(fact_8234_ATP_Olambda__96,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ux(Uu),Uua)
    <=> ( aa_nat_Code_natural(Uu,Uua) = zero_z2226904508553997617atural ) ) ).

% ATP.lambda_96
tff(fact_8235_ATP_Olambda__97,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_uv(Uu),Uua)
    <=> ( aa_nat_Code_integer(Uu,Uua) = zero_z3403309356797280102nteger ) ) ).

% ATP.lambda_97
tff(fact_8236_ATP_Olambda__98,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_uw(Uu),Uua)
    <=> ( aa_nat_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_98
tff(fact_8237_ATP_Olambda__99,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_uy(Uu),Uua)
    <=> ( aa_nat_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_99
tff(fact_8238_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_int,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_uu(Uu),Uua)
    <=> ( aa_nat_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_100
tff(fact_8239_ATP_Olambda__101,axiom,
    ! [Uu: fun_int_Code_natural,Uua: int] :
      ( aa_int_o(aTP_Lamm_ut(Uu),Uua)
    <=> ( aa_int_Code_natural(Uu,Uua) = zero_z2226904508553997617atural ) ) ).

% ATP.lambda_101
tff(fact_8240_ATP_Olambda__102,axiom,
    ! [Uu: fun_int_Code_integer,Uua: int] :
      ( aa_int_o(aTP_Lamm_ur(Uu),Uua)
    <=> ( aa_int_Code_integer(Uu,Uua) = zero_z3403309356797280102nteger ) ) ).

% ATP.lambda_102
tff(fact_8241_ATP_Olambda__103,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_us(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_103
tff(fact_8242_ATP_Olambda__104,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_uq(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_104
tff(fact_8243_ATP_Olambda__105,axiom,
    ! [Uu: fun_int_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_uz(Uu),Uua)
    <=> ( aa_int_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_105
tff(fact_8244_ATP_Olambda__106,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_vg(Uu),Uua)
    <=> ( aa_nat_Code_natural(Uu,Uua) = one_one_Code_natural ) ) ).

% ATP.lambda_106
tff(fact_8245_ATP_Olambda__107,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ve(Uu),Uua)
    <=> ( aa_nat_Code_integer(Uu,Uua) = one_one_Code_integer ) ) ).

% ATP.lambda_107
tff(fact_8246_ATP_Olambda__108,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_vf(Uu),Uua)
    <=> ( aa_nat_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_108
tff(fact_8247_ATP_Olambda__109,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_vh(Uu),Uua)
    <=> ( aa_nat_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_109
tff(fact_8248_ATP_Olambda__110,axiom,
    ! [Uu: fun_nat_int,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_vi(Uu),Uua)
    <=> ( aa_nat_int(Uu,Uua) = one_one_int ) ) ).

% ATP.lambda_110
tff(fact_8249_ATP_Olambda__111,axiom,
    ! [Uu: fun_int_Code_natural,Uua: int] :
      ( aa_int_o(aTP_Lamm_vd(Uu),Uua)
    <=> ( aa_int_Code_natural(Uu,Uua) = one_one_Code_natural ) ) ).

% ATP.lambda_111
tff(fact_8250_ATP_Olambda__112,axiom,
    ! [Uu: fun_int_Code_integer,Uua: int] :
      ( aa_int_o(aTP_Lamm_vb(Uu),Uua)
    <=> ( aa_int_Code_integer(Uu,Uua) = one_one_Code_integer ) ) ).

% ATP.lambda_112
tff(fact_8251_ATP_Olambda__113,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_vc(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_113
tff(fact_8252_ATP_Olambda__114,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_va(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_114
tff(fact_8253_ATP_Olambda__115,axiom,
    ! [Uu: fun_int_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_vj(Uu),Uua)
    <=> ( aa_int_int(Uu,Uua) = one_one_int ) ) ).

% ATP.lambda_115
tff(fact_8254_ATP_Olambda__116,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acl,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_116
tff(fact_8255_ATP_Olambda__117,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_dx(Uu),Uua) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,Uu)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_117
tff(fact_8256_ATP_Olambda__118,axiom,
    ! [Uu: char,Uua: char] :
      ( aa_char_o(aa_char_fun_char_o(aTP_Lamm_ady,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,Uu)),aa_char_nat(comm_s629917340098488124ar_nat,Uua)) ) ).

% ATP.lambda_118
tff(fact_8257_ATP_Olambda__119,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_abs,Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Uu)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)) ).

% ATP.lambda_119
tff(fact_8258_ATP_Olambda__120,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_abt,Uu),Uua) = minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_120
tff(fact_8259_ATP_Olambda__121,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_abu,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Uu)),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_121
tff(fact_8260_ATP_Olambda__122,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aa_num_fun_num_int(aTP_Lamm_acq,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Uu)),aa_num_int(numeral_numeral_int,Uua)) ).

% ATP.lambda_122
tff(fact_8261_ATP_Olambda__123,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_aed,Uu),Uua) = aa_int_rat(aa_int_fun_int_rat(fract,aa_nat_int(nat_int_decode,Uu)),aa_nat_int(nat_int_decode,Uua)) ).

% ATP.lambda_123
tff(fact_8262_ATP_Olambda__124,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_eu,Uu),Uua) = $let(
        l2: num,
        l2:= aa_Code_integer_num(code_num_of_integer,Uu),
        $let(
          l3: num,
          l3:= aa_num_num(plus_plus_num(l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(plus_plus_num(l3),one)) ) ) ).

% ATP.lambda_124
tff(fact_8263_ATP_Olambda__125,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_fe,Uu),Uua) = $let(
        l2: nat,
        l2:= aa_Code_integer_nat(code_nat_of_integer,Uu),
        $let(
          l3: nat,
          l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).

% ATP.lambda_125
tff(fact_8264_ATP_Olambda__126,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_jz(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_126
tff(fact_8265_ATP_Olambda__127,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_eb,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),one_one_Code_integer)) ).

% ATP.lambda_127
tff(fact_8266_ATP_Olambda__128,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ea,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_128
tff(fact_8267_ATP_Olambda__129,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dz,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)) ).

% ATP.lambda_129
tff(fact_8268_ATP_Olambda__130,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_df,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_130
tff(fact_8269_ATP_Olambda__131,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_de,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_131
tff(fact_8270_ATP_Olambda__132,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dd,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_132
tff(fact_8271_ATP_Olambda__133,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kp(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua),numeral_numeral_rat(aa_num_num(bit0,one)))) ).

% ATP.lambda_133
tff(fact_8272_ATP_Olambda__134,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kk,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ).

% ATP.lambda_134
tff(fact_8273_ATP_Olambda__135,axiom,
    ! [Uu: product_unit,Uua: heap_e7401611519738050253t_unit] : aa_hea256619718444519449it_nat(aTP_Lamm_al(Uu),Uua) = produc7133225469290080770it_nat(Uu,produc584006145561248582it_nat(Uua,zero_zero_nat)) ).

% ATP.lambda_135
tff(fact_8274_ATP_Olambda__136,axiom,
    ! [Uu: a,Uua: heap_e7401611519738050253t_unit] : aa_hea5645563107516622602it_nat(aTP_Lamm_am(Uu),Uua) = produc9178034014595674355it_nat(Uu,produc584006145561248582it_nat(Uua,zero_zero_nat)) ).

% ATP.lambda_136
tff(fact_8275_ATP_Olambda__137,axiom,
    ! [Uu: product_unit,Uua: heap_e7401611519738050253t_unit] : aa_hea256619718444519449it_nat(aTP_Lamm_aj(Uu),Uua) = produc7133225469290080770it_nat(Uu,produc584006145561248582it_nat(Uua,one_one_nat)) ).

% ATP.lambda_137
tff(fact_8276_ATP_Olambda__138,axiom,
    ! [Uu: a,Uua: heap_e7401611519738050253t_unit] : aa_hea5645563107516622602it_nat(aTP_Lamm_ak(Uu),Uua) = produc9178034014595674355it_nat(Uu,produc584006145561248582it_nat(Uua,one_one_nat)) ).

% ATP.lambda_138
tff(fact_8277_ATP_Olambda__139,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aat,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Uua),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Uu),Uua))) ).

% ATP.lambda_139
tff(fact_8278_ATP_Olambda__140,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_aau,Uu),Uua) = aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Uua),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,Uu),Uua))) ).

% ATP.lambda_140
tff(fact_8279_ATP_Olambda__141,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ej,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_o_int(zero_n2684676970156552555ol_int,Uua != zero_zero_int)) ).

% ATP.lambda_141
tff(fact_8280_ATP_Olambda__142,axiom,
    ! [Uu: code_natural,Uua: nat] : aa_nat_Code_natural(aTP_Lamm_na(Uu),Uua) = minus_7197305767214868737atural(Uu,aa_nat_Code_natural(semiri3763490453095760265atural,Uua)) ).

% ATP.lambda_142
tff(fact_8281_ATP_Olambda__143,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_my(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)) ).

% ATP.lambda_143
tff(fact_8282_ATP_Olambda__144,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mz(Uu),Uua) = minus_minus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_144
tff(fact_8283_ATP_Olambda__145,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_145
tff(fact_8284_ATP_Olambda__146,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_nc,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_146
tff(fact_8285_ATP_Olambda__147,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mj(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_147
tff(fact_8286_ATP_Olambda__148,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mk,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_148
tff(fact_8287_ATP_Olambda__149,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ml,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_149
tff(fact_8288_ATP_Olambda__150,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_ads,Uu),Uua) = cons_nat(Uu,nat_list_decode(Uua)) ).

% ATP.lambda_150
tff(fact_8289_ATP_Olambda__151,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_acm,Uu),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_151
tff(fact_8290_ATP_Olambda__152,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aTP_Lamm_ho(Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Uua),Uu) ) ).

% ATP.lambda_152
tff(fact_8291_ATP_Olambda__153,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_hp(Uu),Uua)
    <=> aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(Uua),Uu) ) ).

% ATP.lambda_153
tff(fact_8292_ATP_Olambda__154,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_hq(Uu),Uua)
    <=> aa_set_nat_o(ord_less_eq_set_nat(Uua),Uu) ) ).

% ATP.lambda_154
tff(fact_8293_ATP_Olambda__155,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_vw(Uu),Uua)
    <=> ord_less_eq_set_int(Uua,Uu) ) ).

% ATP.lambda_155
tff(fact_8294_ATP_Olambda__156,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_hr,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_156
tff(fact_8295_ATP_Olambda__157,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aTP_Lamm_hs(Uu),Uua)
    <=> aa_num_o(ord_less_eq_num(Uua),Uu) ) ).

% ATP.lambda_157
tff(fact_8296_ATP_Olambda__158,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_hu,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_158
tff(fact_8297_ATP_Olambda__159,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ht,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_159
tff(fact_8298_ATP_Olambda__160,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acu,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_160
tff(fact_8299_ATP_Olambda__161,axiom,
    ! [Uu: literal,Uua: literal] :
      ( aa_literal_o(aTP_Lamm_nk(Uu),Uua)
    <=> aa_literal_o(aa_lit5986169069484365087eral_o(ord_less_literal,Uua),Uu) ) ).

% ATP.lambda_161
tff(fact_8300_ATP_Olambda__162,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aTP_Lamm_nh(Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(Uua),Uu) ) ).

% ATP.lambda_162
tff(fact_8301_ATP_Olambda__163,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_ni(Uu),Uua)
    <=> aa_set6170482117830564987_int_o(ord_le7563427860532173253nt_int(Uua),Uu) ) ).

% ATP.lambda_163
tff(fact_8302_ATP_Olambda__164,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_nj(Uu),Uua)
    <=> aa_set_nat_o(ord_less_set_nat(Uua),Uu) ) ).

% ATP.lambda_164
tff(fact_8303_ATP_Olambda__165,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_nl,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu) ) ).

% ATP.lambda_165
tff(fact_8304_ATP_Olambda__166,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aTP_Lamm_nm(Uu),Uua)
    <=> aa_num_o(ord_less_num(Uua),Uu) ) ).

% ATP.lambda_166
tff(fact_8305_ATP_Olambda__167,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_fl,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_167
tff(fact_8306_ATP_Olambda__168,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_nn,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_168
tff(fact_8307_ATP_Olambda__169,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acr,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_169
tff(fact_8308_ATP_Olambda__170,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ado,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_170
tff(fact_8309_ATP_Olambda__171,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_act,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_171
tff(fact_8310_ATP_Olambda__172,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_cy,Uu),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu) ) ).

% ATP.lambda_172
tff(fact_8311_ATP_Olambda__173,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cx,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_173
tff(fact_8312_ATP_Olambda__174,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cw,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_174
tff(fact_8313_ATP_Olambda__175,axiom,
    ! [Uu: produc7822875418678951345atural,Uua: fun_Pr285225112257423215e_term] : aa_fun7725423394461249828atural(aa_Pro4167682999780799576atural(aTP_Lamm_md,Uu),Uua) = aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,Uua),Uu) ).

% ATP.lambda_175
tff(fact_8314_ATP_Olambda__176,axiom,
    ! [Uu: produc8923325533196201883nteger,Uua: fun_Co3460408610320954864e_term] : aa_fun6004758606051371726nteger(aa_Pro6864914976855246444nteger(aTP_Lamm_mc,Uu),Uua) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Uua),Uu) ).

% ATP.lambda_176
tff(fact_8315_ATP_Olambda__177,axiom,
    ! [Uu: produc8923325533196201883nteger,Uua: fun_Pr5098669352780829690e_term] : aa_fun8425650785172306510nteger(aa_Pro5774345223241026284nteger(aTP_Lamm_me,Uu),Uua) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Uua),Uu) ).

% ATP.lambda_177
tff(fact_8316_ATP_Olambda__178,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_Pr7763680713840462443e_term] : aa_fun9040010020517048364nt_int(aa_Pro7187055436517876584nt_int(aTP_Lamm_mf,Uu),Uua) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Uua),Uu) ).

% ATP.lambda_178
tff(fact_8317_ATP_Olambda__179,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_in5730379987311621695e_term] : aa_fun3520829229142255404nt_int(aa_Pro3263785050932909160nt_int(aTP_Lamm_mg,Uu),Uua) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Uua),Uu) ).

% ATP.lambda_179
tff(fact_8318_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_mh,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_180
tff(fact_8319_ATP_Olambda__181,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_mi,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uu) ).

% ATP.lambda_181
tff(fact_8320_ATP_Olambda__182,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hy,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_182
tff(fact_8321_ATP_Olambda__183,axiom,
    ! [Uu: set_se7855581050983116737at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aTP_Lamm_a(Uu),Uua)
    <=> member2643936169264416010at_nat(Uua,Uu) ) ).

% ATP.lambda_183
tff(fact_8322_ATP_Olambda__184,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_aa(Uu),Uua)
    <=> member2340774599025711042nt_int(Uua,Uu) ) ).

% ATP.lambda_184
tff(fact_8323_ATP_Olambda__185,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ab(Uu),Uua)
    <=> member_set_nat(Uua,Uu) ) ).

% ATP.lambda_185
tff(fact_8324_ATP_Olambda__186,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ac,Uu),Uua)
    <=> aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uua),Uu) ) ).

% ATP.lambda_186
tff(fact_8325_ATP_Olambda__187,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ad(Uu),Uua)
    <=> member_int(Uua,Uu) ) ).

% ATP.lambda_187
tff(fact_8326_ATP_Olambda__188,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_adt(Uu),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu) ).

% ATP.lambda_188
tff(fact_8327_ATP_Olambda__189,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_lw(Uu),Uua) = aa_nat_Code_integer(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_189
tff(fact_8328_ATP_Olambda__190,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_adq(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_190
tff(fact_8329_ATP_Olambda__191,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hz(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_191
tff(fact_8330_ATP_Olambda__192,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_hv(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_192
tff(fact_8331_ATP_Olambda__193,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ia(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_193
tff(fact_8332_ATP_Olambda__194,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_er(Uu),Uua) = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_take_bit_num(Uu),Uua))) ).

% ATP.lambda_194
tff(fact_8333_ATP_Olambda__195,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_adk,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_adj(Uu),Uua)) ).

% ATP.lambda_195
tff(fact_8334_ATP_Olambda__196,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_acg,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_acf(Uu),Uua)) ).

% ATP.lambda_196
tff(fact_8335_ATP_Olambda__197,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_aca,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abz(Uu),Uua)) ).

% ATP.lambda_197
tff(fact_8336_ATP_Olambda__198,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_aby,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abx(Uu),Uua)) ).

% ATP.lambda_198
tff(fact_8337_ATP_Olambda__199,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_abw,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abv(Uu),Uua)) ).

% ATP.lambda_199
tff(fact_8338_ATP_Olambda__200,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_abj,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abi(Uu),Uua)) ).

% ATP.lambda_200
tff(fact_8339_ATP_Olambda__201,axiom,
    ! [Uu: fun_a_2990225006182978344t_unit,Uua: a] : aa_a_f2724909083865345251it_nat(aTP_Lamm_eh(Uu),Uua) = produc7488178964372371019it_nat(aa_a_f112692477493148739it_nat(aTP_Lamm_eg(Uu),Uua)) ).

% ATP.lambda_201
tff(fact_8340_ATP_Olambda__202,axiom,
    ! [Uu: fun_a_6093291656547882009Heap_a,Uua: a] : aa_a_f2042851359257671380it_nat(aTP_Lamm_ee(Uu),Uua) = produc5645566021302314940it_nat(aa_a_f1761598956578211508it_nat(aTP_Lamm_ed(Uu),Uua)) ).

% ATP.lambda_202
tff(fact_8341_ATP_Olambda__203,axiom,
    ! [Uu: fun_he5123621388056232515t_unit,Uua: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aTP_Lamm_ao(Uu),Uua) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(aa_hea4146273921685161788t_unit(Uu,Uua),produc584006145561248582it_nat(Uua,one_one_nat))) ).

% ATP.lambda_203
tff(fact_8342_ATP_Olambda__204,axiom,
    ! [Uu: fun_he6342020308413958580unit_a,Uua: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(aTP_Lamm_an(Uu),Uua) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,produc9178034014595674355it_nat(aa_hea5100639427662539565unit_a(Uu,Uua),produc584006145561248582it_nat(Uua,one_one_nat))) ).

% ATP.lambda_204
tff(fact_8343_ATP_Olambda__205,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cz,Uu),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ) ).

% ATP.lambda_205
tff(fact_8344_ATP_Olambda__206,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_cv(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_206
tff(fact_8345_ATP_Olambda__207,axiom,
    ! [Uu: code_natural,Uua: nat] : aa_nat5282697797315388251atural(aTP_Lamm_cq(Uu),Uua) = times_2397367101498566445atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Uu),aa_nat_Code_natural(semiri3763490453095760265atural,Uua))) ).

% ATP.lambda_207
tff(fact_8346_ATP_Olambda__208,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_cr(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))) ).

% ATP.lambda_208
tff(fact_8347_ATP_Olambda__209,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_cs(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_209
tff(fact_8348_ATP_Olambda__210,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_cu(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_210
tff(fact_8349_ATP_Olambda__211,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_ct(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_211
tff(fact_8350_ATP_Olambda__212,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aco,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_212
tff(fact_8351_ATP_Olambda__213,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acp,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_213
tff(fact_8352_ATP_Olambda__214,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yx,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_214
tff(fact_8353_ATP_Olambda__215,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yw,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_215
tff(fact_8354_ATP_Olambda__216,axiom,
    ! [Uu: nat,Uua: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aTP_Lamm_aw(Uu),Uua) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(product_Unity,produc584006145561248582it_nat(Uua,Uu))) ).

% ATP.lambda_216
tff(fact_8355_ATP_Olambda__217,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_fq(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_217
tff(fact_8356_ATP_Olambda__218,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_fp(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_218
tff(fact_8357_ATP_Olambda__219,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_fo(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_219
tff(fact_8358_ATP_Olambda__220,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_gz(Uu),Uua) = aa_int_int(uminus_uminus_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_220
tff(fact_8359_ATP_Olambda__221,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: nat] : aa_nat5282697797315388251atural(aTP_Lamm_lx(Uu),Uua) = times_2397367101498566445atural(aa_nat_Code_natural(Uu,Uua)) ).

% ATP.lambda_221
tff(fact_8360_ATP_Olambda__222,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_ly(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uu,Uua)) ).

% ATP.lambda_222
tff(fact_8361_ATP_Olambda__223,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_lz(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_223
tff(fact_8362_ATP_Olambda__224,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ma(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_224
tff(fact_8363_ATP_Olambda__225,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_mb(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_225
tff(fact_8364_ATP_Olambda__226,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_rat(aTP_Lamm_kt(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_226
tff(fact_8365_ATP_Olambda__227,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ku(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_227
tff(fact_8366_ATP_Olambda__228,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_fm(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_228
tff(fact_8367_ATP_Olambda__229,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_fn(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_229
tff(fact_8368_ATP_Olambda__230,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: nat] : aa_nat5282697797315388251atural(aTP_Lamm_in(Uu),Uua) = aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(Uu,Uua)) ).

% ATP.lambda_230
tff(fact_8369_ATP_Olambda__231,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_il(Uu),Uua) = aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(Uu,Uua)) ).

% ATP.lambda_231
tff(fact_8370_ATP_Olambda__232,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_im(Uu),Uua) = aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_232
tff(fact_8371_ATP_Olambda__233,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_io(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_233
tff(fact_8372_ATP_Olambda__234,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_ik(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_234
tff(fact_8373_ATP_Olambda__235,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_fx(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_235
tff(fact_8374_ATP_Olambda__236,axiom,
    ! [Uu: fun_Product_unit_nat,Uua: product_unit] : aa_Product_unit_nat(aTP_Lamm_za(Uu),Uua) = aa_nat_nat(suc,aa_Product_unit_nat(Uu,Uua)) ).

% ATP.lambda_236
tff(fact_8375_ATP_Olambda__237,axiom,
    ! [Uu: fun_list_nat_nat,Uua: list_nat] : aa_list_nat_nat(aTP_Lamm_zc(Uu),Uua) = aa_nat_nat(suc,aa_list_nat_nat(Uu,Uua)) ).

% ATP.lambda_237
tff(fact_8376_ATP_Olambda__238,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat] : aa_set_nat_nat(aTP_Lamm_zb(Uu),Uua) = aa_nat_nat(suc,aa_set_nat_nat(Uu,Uua)) ).

% ATP.lambda_238
tff(fact_8377_ATP_Olambda__239,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_zd(Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_239
tff(fact_8378_ATP_Olambda__240,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_adu(Uu),Uua) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fl,Uu)) ).

% ATP.lambda_240
tff(fact_8379_ATP_Olambda__241,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_adw,Uu),Uua)
    <=> ! [X3: nat] :
          ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,X3),Uu)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),X3) ) ) ).

% ATP.lambda_241
tff(fact_8380_ATP_Olambda__242,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_da(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu)),Uub),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uub),aa_num_Code_integer(numera6620942414471956472nteger,Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_242
tff(fact_8381_ATP_Olambda__243,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_db(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_243
tff(fact_8382_ATP_Olambda__244,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dc(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_244
tff(fact_8383_ATP_Olambda__245,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_natural,Uub: nat] :
      aa_nat_Code_natural(aa_fun56224542555504168atural(aTP_Lamm_ow(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_Code_natural(Uua,Uub),zero_z2226904508553997617atural) ).

% ATP.lambda_245
tff(fact_8384_ATP_Olambda__246,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_os(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_246
tff(fact_8385_ATP_Olambda__247,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ou(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_247
tff(fact_8386_ATP_Olambda__248,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_oy(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_248
tff(fact_8387_ATP_Olambda__249,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_or(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_249
tff(fact_8388_ATP_Olambda__250,axiom,
    ! [Uu: int,Uua: fun_int_Code_natural,Uub: int] :
      aa_int_Code_natural(aa_fun4771645977350521568atural(aTP_Lamm_ox(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_Code_natural(Uua,Uub),zero_z2226904508553997617atural) ).

% ATP.lambda_250
tff(fact_8389_ATP_Olambda__251,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_ot(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_251
tff(fact_8390_ATP_Olambda__252,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ov(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_252
tff(fact_8391_ATP_Olambda__253,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_oq(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_253
tff(fact_8392_ATP_Olambda__254,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] :
      aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_oz(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_254
tff(fact_8393_ATP_Olambda__255,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_natural,Uub: nat] :
      aa_nat_Code_natural(aa_fun56224542555504168atural(aTP_Lamm_pz(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_Code_natural(Uua,Uub),one_one_Code_natural) ).

% ATP.lambda_255
tff(fact_8394_ATP_Olambda__256,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_pv(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_Code_integer(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_256
tff(fact_8395_ATP_Olambda__257,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_px(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_257
tff(fact_8396_ATP_Olambda__258,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qb(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_nat(Uua,Uub),one_one_nat) ).

% ATP.lambda_258
tff(fact_8397_ATP_Olambda__259,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_qc(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_int(Uua,Uub),one_one_int) ).

% ATP.lambda_259
tff(fact_8398_ATP_Olambda__260,axiom,
    ! [Uu: int,Uua: fun_int_Code_natural,Uub: int] :
      aa_int_Code_natural(aa_fun4771645977350521568atural(aTP_Lamm_qa(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_Code_natural(Uua,Uub),one_one_Code_natural) ).

% ATP.lambda_260
tff(fact_8399_ATP_Olambda__261,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_pw(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_Code_integer(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_261
tff(fact_8400_ATP_Olambda__262,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_py(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_262
tff(fact_8401_ATP_Olambda__263,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pu(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_nat(Uua,Uub),one_one_nat) ).

% ATP.lambda_263
tff(fact_8402_ATP_Olambda__264,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] :
      aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_qd(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_int(Uua,Uub),one_one_int) ).

% ATP.lambda_264
tff(fact_8403_ATP_Olambda__265,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_natural,Uub: nat] :
      aa_nat_Code_natural(aa_fun56224542555504168atural(aTP_Lamm_pg(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_Code_natural(Uua,Uub),zero_z2226904508553997617atural) ).

% ATP.lambda_265
tff(fact_8404_ATP_Olambda__266,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_pc(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_266
tff(fact_8405_ATP_Olambda__267,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_pe(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_267
tff(fact_8406_ATP_Olambda__268,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pi(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_268
tff(fact_8407_ATP_Olambda__269,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_pb(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_269
tff(fact_8408_ATP_Olambda__270,axiom,
    ! [Uu: int,Uua: fun_int_Code_natural,Uub: int] :
      aa_int_Code_natural(aa_fun4771645977350521568atural(aTP_Lamm_ph(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_Code_natural(Uua,Uub),zero_z2226904508553997617atural) ).

% ATP.lambda_270
tff(fact_8409_ATP_Olambda__271,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_pd(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_271
tff(fact_8410_ATP_Olambda__272,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_pf(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_272
tff(fact_8411_ATP_Olambda__273,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pa(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_273
tff(fact_8412_ATP_Olambda__274,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] :
      aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_pj(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_274
tff(fact_8413_ATP_Olambda__275,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_natural,Uub: nat] :
      aa_nat_Code_natural(aa_fun56224542555504168atural(aTP_Lamm_pp(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_Code_natural(Uua,Uub),one_one_Code_natural) ).

% ATP.lambda_275
tff(fact_8414_ATP_Olambda__276,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_pl(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_Code_integer(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_276
tff(fact_8415_ATP_Olambda__277,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_pn(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_277
tff(fact_8416_ATP_Olambda__278,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pr(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),one_one_nat) ).

% ATP.lambda_278
tff(fact_8417_ATP_Olambda__279,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ps(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),one_one_int) ).

% ATP.lambda_279
tff(fact_8418_ATP_Olambda__280,axiom,
    ! [Uu: int,Uua: fun_int_Code_natural,Uub: int] :
      aa_int_Code_natural(aa_fun4771645977350521568atural(aTP_Lamm_pq(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_Code_natural(Uua,Uub),one_one_Code_natural) ).

% ATP.lambda_280
tff(fact_8419_ATP_Olambda__281,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_pm(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_Code_integer(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_281
tff(fact_8420_ATP_Olambda__282,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_po(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_282
tff(fact_8421_ATP_Olambda__283,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pk(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_nat(Uua,Uub),one_one_nat) ).

% ATP.lambda_283
tff(fact_8422_ATP_Olambda__284,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] :
      aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_pt(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_int(Uua,Uub),one_one_int) ).

% ATP.lambda_284
tff(fact_8423_ATP_Olambda__285,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_fg(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu)),Uub))) ).

% ATP.lambda_285
tff(fact_8424_ATP_Olambda__286,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_fh(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Uu)),Uub))) ).

% ATP.lambda_286
tff(fact_8425_ATP_Olambda__287,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ff(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uu),Uub))) ).

% ATP.lambda_287
tff(fact_8426_ATP_Olambda__288,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: fun_nat_o,Uub: nat] :
      aa_nat_Code_natural(aa_fun8428287193952513114atural(aTP_Lamm_um(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_Code_natural(Uu,Uub),zero_z2226904508553997617atural) ).

% ATP.lambda_288
tff(fact_8427_ATP_Olambda__289,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_o,Uub: nat] :
      aa_nat_Code_integer(aa_fun3969837373441324047nteger(aTP_Lamm_ui(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_Code_integer(Uu,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_289
tff(fact_8428_ATP_Olambda__290,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_uk(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_290
tff(fact_8429_ATP_Olambda__291,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_uo(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_291
tff(fact_8430_ATP_Olambda__292,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_o,Uub: nat] :
      aa_nat_int(aa_fun27009930614080094at_int(aTP_Lamm_uh(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_int(Uu,Uub),zero_zero_int) ).

% ATP.lambda_292
tff(fact_8431_ATP_Olambda__293,axiom,
    ! [Uu: fun_int_Code_natural,Uua: fun_int_o,Uub: int] :
      aa_int_Code_natural(aa_fun1254081949116331354atural(aTP_Lamm_un(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_Code_natural(Uu,Uub),zero_z2226904508553997617atural) ).

% ATP.lambda_293
tff(fact_8432_ATP_Olambda__294,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_o,Uub: int] :
      aa_int_Code_integer(aa_fun6019004165459918095nteger(aTP_Lamm_uj(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_Code_integer(Uu,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_294
tff(fact_8433_ATP_Olambda__295,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_ul(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_295
tff(fact_8434_ATP_Olambda__296,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      aa_int_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_ug(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_296
tff(fact_8435_ATP_Olambda__297,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_o,Uub: int] :
      aa_int_int(aa_fun8822586851003501406nt_int(aTP_Lamm_up(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_int(Uu,Uub),zero_zero_int) ).

% ATP.lambda_297
tff(fact_8436_ATP_Olambda__298,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: fun_nat_o,Uub: nat] :
      aa_nat_Code_natural(aa_fun8428287193952513114atural(aTP_Lamm_vp(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_Code_natural(Uu,Uub),one_one_Code_natural) ).

% ATP.lambda_298
tff(fact_8437_ATP_Olambda__299,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_o,Uub: nat] :
      aa_nat_Code_integer(aa_fun3969837373441324047nteger(aTP_Lamm_vl(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_Code_integer(Uu,Uub),one_one_Code_integer) ).

% ATP.lambda_299
tff(fact_8438_ATP_Olambda__300,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_vn(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_300
tff(fact_8439_ATP_Olambda__301,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_vr(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_nat(Uu,Uub),one_one_nat) ).

% ATP.lambda_301
tff(fact_8440_ATP_Olambda__302,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_o,Uub: nat] :
      aa_nat_int(aa_fun27009930614080094at_int(aTP_Lamm_vs(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_int(Uu,Uub),one_one_int) ).

% ATP.lambda_302
tff(fact_8441_ATP_Olambda__303,axiom,
    ! [Uu: fun_int_Code_natural,Uua: fun_int_o,Uub: int] :
      aa_int_Code_natural(aa_fun1254081949116331354atural(aTP_Lamm_vq(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_Code_natural(Uu,Uub),one_one_Code_natural) ).

% ATP.lambda_303
tff(fact_8442_ATP_Olambda__304,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_o,Uub: int] :
      aa_int_Code_integer(aa_fun6019004165459918095nteger(aTP_Lamm_vm(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_Code_integer(Uu,Uub),one_one_Code_integer) ).

% ATP.lambda_304
tff(fact_8443_ATP_Olambda__305,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_vo(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_305
tff(fact_8444_ATP_Olambda__306,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      aa_int_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_vk(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),one_one_nat) ).

% ATP.lambda_306
tff(fact_8445_ATP_Olambda__307,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_o,Uub: int] :
      aa_int_int(aa_fun8822586851003501406nt_int(aTP_Lamm_vt(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_int(Uu,Uub),one_one_int) ).

% ATP.lambda_307
tff(fact_8446_ATP_Olambda__308,axiom,
    ! [Uu: fun_he8413527710729349838unit_o,Uua: fun_he4944555133437423336it_nat,Uub: heap_e7401611519738050253t_unit] :
      aa_hea684056619475940329it_nat(aa_fun2986507860694160780it_nat(aTP_Lamm_ag(Uu),Uua),Uub) = $ite(aa_hea7994211679605476167unit_o(Uu,Uub),aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,aa_hea256619718444519449it_nat(Uua,Uub)),none_P9117596204409417319it_nat) ).

% ATP.lambda_308
tff(fact_8447_ATP_Olambda__309,axiom,
    ! [Uu: fun_he8413527710729349838unit_o,Uua: fun_he8889679731534893401it_nat,Uub: heap_e7401611519738050253t_unit] :
      aa_hea4724376117875161946it_nat(aa_fun3370411987892275404it_nat(aTP_Lamm_af(Uu),Uua),Uub) = $ite(aa_hea7994211679605476167unit_o(Uu,Uub),aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_hea5645563107516622602it_nat(Uua,Uub)),none_P2651198173097904984it_nat) ).

% ATP.lambda_309
tff(fact_8448_ATP_Olambda__310,axiom,
    ! [Uu: heap_Time_Heap_a,Uua: fun_a_2990225006182978344t_unit,Uub: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aa_fun372765568861188684it_nat(aTP_Lamm_ei(Uu),Uua),Uub) = case_o7727800614283616222it_nat(none_P9117596204409417319it_nat,produc5648861294512138360it_nat(aTP_Lamm_eh(Uua)),aa_hea4724376117875161946it_nat(heap_Time_execute_a(Uu),Uub)) ).

% ATP.lambda_310
tff(fact_8449_ATP_Olambda__311,axiom,
    ! [Uu: heap_Time_Heap_a,Uua: fun_a_6093291656547882009Heap_a,Uub: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(aa_fun6119220767042692876it_nat(aTP_Lamm_ef(Uu),Uua),Uub) = case_o6061097939050036047it_nat(none_P2651198173097904984it_nat,produc7850007024774191849it_nat(aTP_Lamm_ee(Uua)),aa_hea4724376117875161946it_nat(heap_Time_execute_a(Uu),Uub)) ).

% ATP.lambda_311
tff(fact_8450_ATP_Olambda__312,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ir(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_312
tff(fact_8451_ATP_Olambda__313,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_nr(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_313
tff(fact_8452_ATP_Olambda__314,axiom,
    ! [Uu: fun_se1179353343636554400at_nat,Uua: nat,Uub: set_nat] : aa_set_nat_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_rf(Uu),Uua),Uub) = aa_nat_nat(aa_set6537093552120046177at_nat(Uu,Uub),Uua) ).

% ATP.lambda_314
tff(fact_8453_ATP_Olambda__315,axiom,
    ! [Uu: fun_se7912642235559603452at_int,Uua: nat,Uub: set_nat] : aa_set_nat_int(aa_nat2732338449433945789at_int(aTP_Lamm_so(Uu),Uua),Uub) = aa_nat_int(aa_set2359242532610849469at_int(Uu,Uub),Uua) ).

% ATP.lambda_315
tff(fact_8454_ATP_Olambda__316,axiom,
    ! [Uu: fun_se876191773397849432nt_int,Uua: int,Uub: set_nat] : aa_set_nat_int(aa_int2448287375063835929at_int(aTP_Lamm_rk(Uu),Uua),Uub) = aa_int_int(aa_set3358596244511904025nt_int(Uu,Uub),Uua) ).

% ATP.lambda_316
tff(fact_8455_ATP_Olambda__317,axiom,
    ! [Uu: fun_na106483476154090400at_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_rp(Uu),Uua),Uub) = aa_set_nat_nat(aa_nat6910189468943142497at_nat(Uu,Uub),Uua) ).

% ATP.lambda_317
tff(fact_8456_ATP_Olambda__318,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_318
tff(fact_8457_ATP_Olambda__319,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ky(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_319
tff(fact_8458_ATP_Olambda__320,axiom,
    ! [Uu: fun_nat_fun_int_nat,Uua: int,Uub: nat] : aa_nat_nat(aa_int_fun_nat_nat(aTP_Lamm_qr(Uu),Uua),Uub) = aa_int_nat(aa_nat_fun_int_nat(Uu,Uub),Uua) ).

% ATP.lambda_320
tff(fact_8459_ATP_Olambda__321,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_lb(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_321
tff(fact_8460_ATP_Olambda__322,axiom,
    ! [Uu: fun_in6555721293707029592at_int,Uua: set_nat,Uub: int] : aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_ru(Uu),Uua),Uub) = aa_set_nat_int(aa_int2448287375063835929at_int(Uu,Uub),Uua) ).

% ATP.lambda_322
tff(fact_8461_ATP_Olambda__323,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_qj(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_323
tff(fact_8462_ATP_Olambda__324,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_le(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_324
tff(fact_8463_ATP_Olambda__325,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_gm(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_325
tff(fact_8464_ATP_Olambda__326,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_eo(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_326
tff(fact_8465_ATP_Olambda__327,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ny(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ky(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_327
tff(fact_8466_ATP_Olambda__328,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nw(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gj(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_328
tff(fact_8467_ATP_Olambda__329,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nu(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gj(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_329
tff(fact_8468_ATP_Olambda__330,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aci(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_ach(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_330
tff(fact_8469_ATP_Olambda__331,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ace(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_acd(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_331
tff(fact_8470_ATP_Olambda__332,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_acc(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_acb(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_332
tff(fact_8471_ATP_Olambda__333,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_abr(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_int2464330468366073224_int_o(aTP_Lamm_abq(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_333
tff(fact_8472_ATP_Olambda__334,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_abp(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_int2464330468366073224_int_o(aTP_Lamm_abo(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_334
tff(fact_8473_ATP_Olambda__335,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_abl(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_abk(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_335
tff(fact_8474_ATP_Olambda__336,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: set_int,Uub: nat] : aa_nat_int(aa_set6310687161026558745at_int(aTP_Lamm_lf(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_le(Uu),Uub),Uua) ).

% ATP.lambda_336
tff(fact_8475_ATP_Olambda__337,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: set_nat,Uub: int] : aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_lc(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_lb(Uu),Uub)),Uua) ).

% ATP.lambda_337
tff(fact_8476_ATP_Olambda__338,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: set_nat,Uub: nat] : aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_kz(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ky(Uu),Uub)),Uua) ).

% ATP.lambda_338
tff(fact_8477_ATP_Olambda__339,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_lh(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_gm(Uu),Uub),Uua) ).

% ATP.lambda_339
tff(fact_8478_ATP_Olambda__340,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_kw(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gj(Uu),Uub)),Uua) ).

% ATP.lambda_340
tff(fact_8479_ATP_Olambda__341,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_gn(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_gm(Uu),Uub),Uua) ).

% ATP.lambda_341
tff(fact_8480_ATP_Olambda__342,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_gk(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gj(Uu),Uub)),Uua) ).

% ATP.lambda_342
tff(fact_8481_ATP_Olambda__343,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_op(Uu),Uua),Uub) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_343
tff(fact_8482_ATP_Olambda__344,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_adm(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) )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_344
tff(fact_8483_ATP_Olambda__345,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_adn(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(groups4561878855575611511st_nat,Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_345
tff(fact_8484_ATP_Olambda__346,axiom,
    ! [Uu: nat,Uua: set_Product_unit,Uub: list_Product_unit] :
      ( aa_lis2302375577916549633unit_o(aa_set1739257761580076089unit_o(aTP_Lamm_abd(Uu),Uua),Uub)
    <=> ( ( size_s245203480648594047t_unit(Uub) = Uu )
        & distin4812975378522102150t_unit(Uub)
        & ord_le3507040750410214029t_unit(set_Product_unit2(Uub),Uua) ) ) ).

% ATP.lambda_346
tff(fact_8485_ATP_Olambda__347,axiom,
    ! [Uu: nat,Uua: set_char,Uub: list_char] :
      ( aa_list_char_o(aa_set3791436276075643467char_o(aTP_Lamm_abg(Uu),Uua),Uub)
    <=> ( ( aa_list_char_nat(size_size_list_char,Uub) = Uu )
        & aa_list_char_o(distinct_char,Uub)
        & ord_less_eq_set_char(set_char2(Uub),Uua) ) ) ).

% ATP.lambda_347
tff(fact_8486_ATP_Olambda__348,axiom,
    ! [Uu: nat,Uua: set_list_nat,Uub: list_list_nat] :
      ( aa_list_list_nat_o(aa_set5589568368246272301_nat_o(aTP_Lamm_abf(Uu),Uua),Uub)
    <=> ( ( size_s3023201423986296836st_nat(Uub) = Uu )
        & distinct_list_nat(Uub)
        & ord_le6045566169113846134st_nat(set_list_nat2(Uub),Uua) ) ) ).

% ATP.lambda_348
tff(fact_8487_ATP_Olambda__349,axiom,
    ! [Uu: nat,Uua: set_set_nat,Uub: list_set_nat] :
      ( aa_list_set_nat_o(aa_set474910819808044769_nat_o(aTP_Lamm_abe(Uu),Uua),Uub)
    <=> ( ( size_s3254054031482475050et_nat(Uub) = Uu )
        & distinct_set_nat(Uub)
        & ord_le6893508408891458716et_nat(set_set_nat2(Uub),Uua) ) ) ).

% ATP.lambda_349
tff(fact_8488_ATP_Olambda__350,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set3062951307625612365_nat_o(aTP_Lamm_abh(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & distinct_nat(Uub)
        & aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uua) ) ) ).

% ATP.lambda_350
tff(fact_8489_ATP_Olambda__351,axiom,
    ! [Uu: nat,Uua: set_int,Uub: list_int] :
      ( aa_list_int_o(aa_set1055383481605741461_int_o(aTP_Lamm_abc(Uu),Uua),Uub)
    <=> ( ( size_size_list_int(Uub) = Uu )
        & distinct_int(Uub)
        & ord_less_eq_set_int(set_int2(Uub),Uua) ) ) ).

% ATP.lambda_351
tff(fact_8490_ATP_Olambda__352,axiom,
    ! [Uu: set_Product_unit,Uua: nat,Uub: list_Product_unit] :
      ( aa_lis2302375577916549633unit_o(aa_nat382646585846990998unit_o(aTP_Lamm_aav(Uu),Uua),Uub)
    <=> ( ( size_s245203480648594047t_unit(Uub) = Uua )
        & distin4812975378522102150t_unit(Uub)
        & ord_le3507040750410214029t_unit(set_Product_unit2(Uub),Uu) ) ) ).

% ATP.lambda_352
tff(fact_8491_ATP_Olambda__353,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( aa_lis2793599718760257805_nat_o(aa_nat1970911183850433186_nat_o(aTP_Lamm_aaz(Uu),Uua),Uub)
    <=> ( ( size_s5460976970255530739at_nat(Uub) = Uua )
        & distin6923225563576452346at_nat(Uub)
        & aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub)),Uu) ) ) ).

% ATP.lambda_353
tff(fact_8492_ATP_Olambda__354,axiom,
    ! [Uu: set_char,Uua: nat,Uub: list_char] :
      ( aa_list_char_o(aa_nat3000828340120075458char_o(aTP_Lamm_aba(Uu),Uua),Uub)
    <=> ( ( aa_list_char_nat(size_size_list_char,Uub) = Uua )
        & aa_list_char_o(distinct_char,Uub)
        & ord_less_eq_set_char(set_char2(Uub),Uu) ) ) ).

% ATP.lambda_354
tff(fact_8493_ATP_Olambda__355,axiom,
    ! [Uu: set_list_nat,Uua: nat,Uub: list_list_nat] :
      ( aa_list_list_nat_o(aa_nat158606821896287091_nat_o(aTP_Lamm_aax(Uu),Uua),Uub)
    <=> ( ( size_s3023201423986296836st_nat(Uub) = Uua )
        & distinct_list_nat(Uub)
        & ord_le6045566169113846134st_nat(set_list_nat2(Uub),Uu) ) ) ).

% ATP.lambda_355
tff(fact_8494_ATP_Olambda__356,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: list_set_nat] :
      ( aa_list_set_nat_o(aa_nat2247512698059773005_nat_o(aTP_Lamm_aaw(Uu),Uua),Uub)
    <=> ( ( size_s3254054031482475050et_nat(Uub) = Uua )
        & distinct_set_nat(Uub)
        & ord_le6893508408891458716et_nat(set_set_nat2(Uub),Uu) ) ) ).

% ATP.lambda_356
tff(fact_8495_ATP_Olambda__357,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_abb(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uua )
        & distinct_nat(Uub)
        & aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu) ) ) ).

% ATP.lambda_357
tff(fact_8496_ATP_Olambda__358,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_aay(Uu),Uua),Uub)
    <=> ( ( size_size_list_int(Uub) = Uua )
        & distinct_int(Uub)
        & ord_less_eq_set_int(set_int2(Uub),Uu) ) ) ).

% ATP.lambda_358
tff(fact_8497_ATP_Olambda__359,axiom,
    ! [Uu: set_Product_unit,Uua: nat,Uub: list_Product_unit] :
      ( aa_lis2302375577916549633unit_o(aa_nat382646585846990998unit_o(aTP_Lamm_zo(Uu),Uua),Uub)
    <=> ( ord_le3507040750410214029t_unit(set_Product_unit2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s245203480648594047t_unit(Uub)),Uua) ) ) ).

% ATP.lambda_359
tff(fact_8498_ATP_Olambda__360,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( aa_lis2793599718760257805_nat_o(aa_nat1970911183850433186_nat_o(aTP_Lamm_zs(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s5460976970255530739at_nat(Uub)),Uua) ) ) ).

% ATP.lambda_360
tff(fact_8499_ATP_Olambda__361,axiom,
    ! [Uu: set_char,Uua: nat,Uub: list_char] :
      ( aa_list_char_o(aa_nat3000828340120075458char_o(aTP_Lamm_zt(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_char(set_char2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_char_nat(size_size_list_char,Uub)),Uua) ) ) ).

% ATP.lambda_361
tff(fact_8500_ATP_Olambda__362,axiom,
    ! [Uu: set_list_nat,Uua: nat,Uub: list_list_nat] :
      ( aa_list_list_nat_o(aa_nat158606821896287091_nat_o(aTP_Lamm_zq(Uu),Uua),Uub)
    <=> ( ord_le6045566169113846134st_nat(set_list_nat2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3023201423986296836st_nat(Uub)),Uua) ) ) ).

% ATP.lambda_362
tff(fact_8501_ATP_Olambda__363,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: list_set_nat] :
      ( aa_list_set_nat_o(aa_nat2247512698059773005_nat_o(aTP_Lamm_zp(Uu),Uua),Uub)
    <=> ( ord_le6893508408891458716et_nat(set_set_nat2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3254054031482475050et_nat(Uub)),Uua) ) ) ).

% ATP.lambda_363
tff(fact_8502_ATP_Olambda__364,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_zu(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Uub)),Uua) ) ) ).

% ATP.lambda_364
tff(fact_8503_ATP_Olambda__365,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_zr(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_int(set_int2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Uub)),Uua) ) ) ).

% ATP.lambda_365
tff(fact_8504_ATP_Olambda__366,axiom,
    ! [Uu: set_Product_unit,Uua: nat,Uub: list_Product_unit] :
      ( aa_lis2302375577916549633unit_o(aa_nat382646585846990998unit_o(aTP_Lamm_aap(Uu),Uua),Uub)
    <=> ( ord_le3507040750410214029t_unit(set_Product_unit2(Uub),Uu)
        & ( size_s245203480648594047t_unit(Uub) = Uua ) ) ) ).

% ATP.lambda_366
tff(fact_8505_ATP_Olambda__367,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( aa_lis2793599718760257805_nat_o(aa_nat1970911183850433186_nat_o(aTP_Lamm_aam(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub)),Uu)
        & ( size_s5460976970255530739at_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_367
tff(fact_8506_ATP_Olambda__368,axiom,
    ! [Uu: set_char,Uua: nat,Uub: list_char] :
      ( aa_list_char_o(aa_nat3000828340120075458char_o(aTP_Lamm_aan(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_char(set_char2(Uub),Uu)
        & ( aa_list_char_nat(size_size_list_char,Uub) = Uua ) ) ) ).

% ATP.lambda_368
tff(fact_8507_ATP_Olambda__369,axiom,
    ! [Uu: set_list_nat,Uua: nat,Uub: list_list_nat] :
      ( aa_list_list_nat_o(aa_nat158606821896287091_nat_o(aTP_Lamm_aar(Uu),Uua),Uub)
    <=> ( ord_le6045566169113846134st_nat(set_list_nat2(Uub),Uu)
        & ( size_s3023201423986296836st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_369
tff(fact_8508_ATP_Olambda__370,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: list_set_nat] :
      ( aa_list_set_nat_o(aa_nat2247512698059773005_nat_o(aTP_Lamm_aaq(Uu),Uua),Uub)
    <=> ( ord_le6893508408891458716et_nat(set_set_nat2(Uub),Uu)
        & ( size_s3254054031482475050et_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_370
tff(fact_8509_ATP_Olambda__371,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aao(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & ( aa_list_nat_nat(size_size_list_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_371
tff(fact_8510_ATP_Olambda__372,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_aal(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_int(set_int2(Uub),Uu)
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_372
tff(fact_8511_ATP_Olambda__373,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_adl(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_373
tff(fact_8512_ATP_Olambda__374,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_yy(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_374
tff(fact_8513_ATP_Olambda__375,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_acz(Uu),Uua),Uub)
    <=> ( 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_375
tff(fact_8514_ATP_Olambda__376,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_on(Uu),Uua),Uub) = divide_divide_rat(minus_minus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,Uub)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_376
tff(fact_8515_ATP_Olambda__377,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ada(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uub) ) ) ).

% ATP.lambda_377
tff(fact_8516_ATP_Olambda__378,axiom,
    ! [Uu: set_Product_unit,Uua: nat,Uub: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_nat5385523842031975612unit_o(aTP_Lamm_xr(Uu),Uua),Uub)
    <=> ( ord_le3507040750410214029t_unit(Uub,Uu)
        & ( finite410649719033368117t_unit(Uub) = Uua ) ) ) ).

% ATP.lambda_378
tff(fact_8517_ATP_Olambda__379,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_nat1727191234135941576_nat_o(aTP_Lamm_xv(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Uub),Uu)
        & ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_379
tff(fact_8518_ATP_Olambda__380,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: nat,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_nat4194932087953772816_int_o(aTP_Lamm_xq(Uu),Uua),Uub)
    <=> ( aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(Uub),Uu)
        & ( aa_set4753215920113914285nt_nat(finite6756421564338198497nt_int,Uub) = Uua ) ) ) ).

% ATP.lambda_380
tff(fact_8519_ATP_Olambda__381,axiom,
    ! [Uu: set_list_nat,Uua: nat,Uub: set_list_nat] :
      ( aa_set_list_nat_o(aa_nat7721107121239158477_nat_o(aTP_Lamm_xt(Uu),Uua),Uub)
    <=> ( ord_le6045566169113846134st_nat(Uub,Uu)
        & ( finite_card_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_381
tff(fact_8520_ATP_Olambda__382,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: set_set_nat] :
      ( aa_set_set_nat_o(aa_nat5160323483289836711_nat_o(aTP_Lamm_xs(Uu),Uua),Uub)
    <=> ( ord_le6893508408891458716et_nat(Uub,Uu)
        & ( finite_card_set_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_382
tff(fact_8521_ATP_Olambda__383,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_xw(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Uub),Uu)
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_383
tff(fact_8522_ATP_Olambda__384,axiom,
    ! [Uu: set_int,Uua: nat,Uub: set_int] :
      ( aa_set_int_o(aa_nat_fun_set_int_o(aTP_Lamm_xu(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_int(Uub,Uu)
        & ( finite_card_int(Uub) = Uua ) ) ) ).

% ATP.lambda_384
tff(fact_8523_ATP_Olambda__385,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_yz(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_385
tff(fact_8524_ATP_Olambda__386,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_it(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_386
tff(fact_8525_ATP_Olambda__387,axiom,
    ! [Uu: product_unit,Uua: set_Product_unit,Uub: product_unit] :
      ( aa_Product_unit_o(aa_set2953008695252327103unit_o(aTP_Lamm_zv(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member_Product_unit(Uub,Uua) ) ) ).

% ATP.lambda_387
tff(fact_8526_ATP_Olambda__388,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_set6353399069419443519_nat_o(aTP_Lamm_zx(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member8757157785044589968at_nat(Uub,Uua) ) ) ).

% ATP.lambda_388
tff(fact_8527_ATP_Olambda__389,axiom,
    ! [Uu: product_prod_nat_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_zw(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_389
tff(fact_8528_ATP_Olambda__390,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_se7855581050983116737at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7533644503045286079_nat_o(aTP_Lamm_zy(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member2643936169264416010at_nat(Uub,Uua) ) ) ).

% ATP.lambda_390
tff(fact_8529_ATP_Olambda__391,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_se6260736226359567993nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4062574822058255039_int_o(aTP_Lamm_zz(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member2340774599025711042nt_int(Uub,Uua) ) ) ).

% ATP.lambda_391
tff(fact_8530_ATP_Olambda__392,axiom,
    ! [Uu: set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_aaa(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member_set_nat(Uub,Uua) ) ) ).

% ATP.lambda_392
tff(fact_8531_ATP_Olambda__393,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_aab(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua) ) ) ).

% ATP.lambda_393
tff(fact_8532_ATP_Olambda__394,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_aac(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member_int(Uub,Uua) ) ) ).

% ATP.lambda_394
tff(fact_8533_ATP_Olambda__395,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ek(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uua)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_395
tff(fact_8534_ATP_Olambda__396,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_we(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_396
tff(fact_8535_ATP_Olambda__397,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_el(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_397
tff(fact_8536_ATP_Olambda__398,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_wg(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_398
tff(fact_8537_ATP_Olambda__399,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_wh(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_399
tff(fact_8538_ATP_Olambda__400,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_wf(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_400
tff(fact_8539_ATP_Olambda__401,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_acx(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uua)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uu) ) ) ).

% ATP.lambda_401
tff(fact_8540_ATP_Olambda__402,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_acy(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uua)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uu) ) ) ).

% ATP.lambda_402
tff(fact_8541_ATP_Olambda__403,axiom,
    ! [Uu: set_se7855581050983116737at_nat,Uua: fun_se4782206882984618170_nat_o,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_fun3315923329236548914_nat_o(aTP_Lamm_cc(Uu),Uua),Uub)
    <=> ( member2643936169264416010at_nat(Uub,Uu)
        & aa_set3702741264012733747_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_403
tff(fact_8542_ATP_Olambda__404,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_cd(Uu),Uua),Uub)
    <=> ( member2340774599025711042nt_int(Uub,Uu)
        & aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_404
tff(fact_8543_ATP_Olambda__405,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ce(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_405
tff(fact_8544_ATP_Olambda__406,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cg(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_406
tff(fact_8545_ATP_Olambda__407,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_cf(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_407
tff(fact_8546_ATP_Olambda__408,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_natural,Uub: nat] :
      ( aa_nat_o(aa_fun440078198722963374_nat_o(aTP_Lamm_ti(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_Code_natural(Uua,Uub) != zero_z2226904508553997617atural ) ) ) ).

% ATP.lambda_408
tff(fact_8547_ATP_Olambda__409,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aTP_Lamm_ta(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_Code_integer(Uua,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_409
tff(fact_8548_ATP_Olambda__410,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_te(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_410
tff(fact_8549_ATP_Olambda__411,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_ss(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_411
tff(fact_8550_ATP_Olambda__412,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      ( aa_nat_o(aa_fun2180201864171524266_nat_o(aTP_Lamm_sw(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_int(Uua,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_412
tff(fact_8551_ATP_Olambda__413,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_natural,Uub: int] :
      ( aa_int_o(aa_fun296549085072024822_int_o(aTP_Lamm_tk(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_Code_natural(Uua,Uub) != zero_z2226904508553997617atural ) ) ) ).

% ATP.lambda_413
tff(fact_8552_ATP_Olambda__414,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aTP_Lamm_tc(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_Code_integer(Uua,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_414
tff(fact_8553_ATP_Olambda__415,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_tg(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_415
tff(fact_8554_ATP_Olambda__416,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_su(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_416
tff(fact_8555_ATP_Olambda__417,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: int] :
      ( aa_int_o(aa_fun8155382727227203058_int_o(aTP_Lamm_sy(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_int(Uua,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_417
tff(fact_8556_ATP_Olambda__418,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_natural,Uub: nat] :
      ( aa_nat_o(aa_fun440078198722963374_nat_o(aTP_Lamm_tm(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_Code_natural(Uua,Uub) != one_one_Code_natural ) ) ) ).

% ATP.lambda_418
tff(fact_8557_ATP_Olambda__419,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aTP_Lamm_ty(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_Code_integer(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_419
tff(fact_8558_ATP_Olambda__420,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_uc(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_420
tff(fact_8559_ATP_Olambda__421,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_tq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_nat(Uua,Uub) != one_one_nat ) ) ) ).

% ATP.lambda_421
tff(fact_8560_ATP_Olambda__422,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      ( aa_nat_o(aa_fun2180201864171524266_nat_o(aTP_Lamm_tu(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ( aa_nat_int(Uua,Uub) != one_one_int ) ) ) ).

% ATP.lambda_422
tff(fact_8561_ATP_Olambda__423,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_natural,Uub: int] :
      ( aa_int_o(aa_fun296549085072024822_int_o(aTP_Lamm_to(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_Code_natural(Uua,Uub) != one_one_Code_natural ) ) ) ).

% ATP.lambda_423
tff(fact_8562_ATP_Olambda__424,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aTP_Lamm_ua(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_Code_integer(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_424
tff(fact_8563_ATP_Olambda__425,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_ue(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_425
tff(fact_8564_ATP_Olambda__426,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_ts(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_nat(Uua,Uub) != one_one_nat ) ) ) ).

% ATP.lambda_426
tff(fact_8565_ATP_Olambda__427,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: int] :
      ( aa_int_o(aa_fun8155382727227203058_int_o(aTP_Lamm_tw(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_int(Uua,Uub) != one_one_int ) ) ) ).

% ATP.lambda_427
tff(fact_8566_ATP_Olambda__428,axiom,
    ! [Uu: set_Product_unit,Uua: fun_Pr1820225556744648189nteger,Uub: product_unit] :
      ( aa_Product_unit_o(aa_fun7797430150634904527unit_o(aTP_Lamm_wv(Uu),Uua),Uub)
    <=> ( member_Product_unit(Uub,Uu)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Pro1705297278780387758nteger(Uua,Uub)) ) ) ).

% ATP.lambda_428
tff(fact_8567_ATP_Olambda__429,axiom,
    ! [Uu: set_Product_unit,Uua: fun_Product_unit_nat,Uub: product_unit] :
      ( aa_Product_unit_o(aa_fun9021389585855579330unit_o(aTP_Lamm_wt(Uu),Uua),Uub)
    <=> ( member_Product_unit(Uub,Uu)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_Product_unit_nat(Uua,Uub)) ) ) ).

% ATP.lambda_429
tff(fact_8568_ATP_Olambda__430,axiom,
    ! [Uu: set_Product_unit,Uua: fun_Product_unit_int,Uub: product_unit] :
      ( aa_Product_unit_o(aa_fun693351410520992030unit_o(aTP_Lamm_wr(Uu),Uua),Uub)
    <=> ( member_Product_unit(Uub,Uu)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Product_unit_int(Uua,Uub)) ) ) ).

% ATP.lambda_430
tff(fact_8569_ATP_Olambda__431,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_int,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun1141455960422277694_nat_o(aTP_Lamm_xa(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(Uua,Uub)) ) ) ).

% ATP.lambda_431
tff(fact_8570_ATP_Olambda__432,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aTP_Lamm_ww(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(Uua,Uub)) ) ) ).

% ATP.lambda_432
tff(fact_8571_ATP_Olambda__433,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_wy(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(Uua,Uub)) ) ) ).

% ATP.lambda_433
tff(fact_8572_ATP_Olambda__434,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      ( aa_nat_o(aa_fun2180201864171524266_nat_o(aTP_Lamm_ws(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(Uua,Uub)) ) ) ).

% ATP.lambda_434
tff(fact_8573_ATP_Olambda__435,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aTP_Lamm_wx(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(Uua,Uub)) ) ) ).

% ATP.lambda_435
tff(fact_8574_ATP_Olambda__436,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_wu(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(Uua,Uub)) ) ) ).

% ATP.lambda_436
tff(fact_8575_ATP_Olambda__437,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: int] :
      ( aa_int_o(aa_fun8155382727227203058_int_o(aTP_Lamm_wz(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(Uua,Uub)) ) ) ).

% ATP.lambda_437
tff(fact_8576_ATP_Olambda__438,axiom,
    ! [Uu: set_se7855581050983116737at_nat,Uua: set_se7855581050983116737at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7533644503045286079_nat_o(aTP_Lamm_wi(Uu),Uua),Uub)
    <=> ( member2643936169264416010at_nat(Uub,Uu)
        & ~ member2643936169264416010at_nat(Uub,Uua) ) ) ).

% ATP.lambda_438
tff(fact_8577_ATP_Olambda__439,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: set_se6260736226359567993nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4062574822058255039_int_o(aTP_Lamm_wj(Uu),Uua),Uub)
    <=> ( member2340774599025711042nt_int(Uub,Uu)
        & ~ member2340774599025711042nt_int(Uub,Uua) ) ) ).

% ATP.lambda_439
tff(fact_8578_ATP_Olambda__440,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_wk(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu)
        & ~ member_set_nat(Uub,Uua) ) ) ).

% ATP.lambda_440
tff(fact_8579_ATP_Olambda__441,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_wm(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uu)
        & ~ aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uub),Uua) ) ) ).

% ATP.lambda_441
tff(fact_8580_ATP_Olambda__442,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_wl(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ~ member_int(Uub,Uua) ) ) ).

% ATP.lambda_442
tff(fact_8581_ATP_Olambda__443,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_acw(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uu) ) ).

% ATP.lambda_443
tff(fact_8582_ATP_Olambda__444,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_iq(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_444
tff(fact_8583_ATP_Olambda__445,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_nd(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_445
tff(fact_8584_ATP_Olambda__446,axiom,
    ! [Uu: set_Pr1200682188532975830nteger,Uua: fun_Co3460408610320954864e_term,Uub: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(aTP_Lamm_bp(Uu),Uua),Uub)
    <=> member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Uua),Uub),Uu) ) ).

% ATP.lambda_446
tff(fact_8585_ATP_Olambda__447,axiom,
    ! [Uu: set_Pr620054291978045485atural,Uua: fun_Pr285225112257423215e_term,Uub: produc7822875418678951345atural] :
      ( aa_Pro6073791466322389475ural_o(aa_fun9089857049899810327ural_o(aTP_Lamm_bq(Uu),Uua),Uub)
    <=> member6441364043385305230atural(aa_Pro3608106868210220652atural(aa_fun3413103502972484686atural(produc3399137497101226409atural,Uua),Uub),Uu) ) ).

% ATP.lambda_447
tff(fact_8586_ATP_Olambda__448,axiom,
    ! [Uu: set_Pr3486312109254157132nteger,Uua: fun_Pr5098669352780829690e_term,Uub: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(aTP_Lamm_br(Uu),Uua),Uub)
    <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Uua),Uub),Uu) ) ).

% ATP.lambda_448
tff(fact_8587_ATP_Olambda__449,axiom,
    ! [Uu: set_Pr6700895561065273913nt_int,Uua: fun_Pr7763680713840462443e_term,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(aTP_Lamm_bs(Uu),Uua),Uub)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Uua),Uub),Uu) ) ).

% ATP.lambda_449
tff(fact_8588_ATP_Olambda__450,axiom,
    ! [Uu: set_Pr4166680992137330533nt_int,Uua: fun_in5730379987311621695e_term,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(aTP_Lamm_bt(Uu),Uua),Uub)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Uua),Uub),Uu) ) ).

% ATP.lambda_450
tff(fact_8589_ATP_Olambda__451,axiom,
    ! [Uu: rat,Uua: rat,Uub: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_vu(Uu),Uua),Uub)
    <=> ( member_rat(Uub,ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uu),Uub)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uub),Uua) ) ) ).

% ATP.lambda_451
tff(fact_8590_ATP_Olambda__452,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_qf(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_452
tff(fact_8591_ATP_Olambda__453,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_li(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_453
tff(fact_8592_ATP_Olambda__454,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_lj(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_454
tff(fact_8593_ATP_Olambda__455,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_lk(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_455
tff(fact_8594_ATP_Olambda__456,axiom,
    ! [Uu: fun_se2775971902884351953nteger,Uua: fun_se2775971902884351953nteger,Uub: set_Pr1261947904930325089at_nat] : aa_set5601480140167866242nteger(aa_fun5683560775029648636nteger(aTP_Lamm_gh(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set5601480140167866242nteger(Uua,Uub)),aa_set5601480140167866242nteger(Uu,Uub)) ).

% ATP.lambda_456
tff(fact_8595_ATP_Olambda__457,axiom,
    ! [Uu: fun_se7586849273513198688nteger,Uua: fun_se7586849273513198688nteger,Uub: set_nat] : aa_set871585252387937313nteger(aa_fun4587287018503165694nteger(aTP_Lamm_ge(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(Uua,Uub)),aa_set871585252387937313nteger(Uu,Uub)) ).

% ATP.lambda_457
tff(fact_8596_ATP_Olambda__458,axiom,
    ! [Uu: fun_set_nat_rat,Uua: fun_set_nat_rat,Uub: set_nat] : aa_set_nat_rat(aa_fun8478617152544252148at_rat(aTP_Lamm_gf(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(Uua,Uub)),aa_set_nat_rat(Uu,Uub)) ).

% ATP.lambda_458
tff(fact_8597_ATP_Olambda__459,axiom,
    ! [Uu: fun_set_nat_int,Uua: fun_set_nat_int,Uub: set_nat] : aa_set_nat_int(aa_fun3600621996379952540at_int(aTP_Lamm_gg(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_459
tff(fact_8598_ATP_Olambda__460,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_fy(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_460
tff(fact_8599_ATP_Olambda__461,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ga(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_461
tff(fact_8600_ATP_Olambda__462,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_gc(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_462
tff(fact_8601_ATP_Olambda__463,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_fz(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_463
tff(fact_8602_ATP_Olambda__464,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_gb(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uua,Uub)),aa_int_rat(Uu,Uub)) ).

% ATP.lambda_464
tff(fact_8603_ATP_Olambda__465,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gd(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_465
tff(fact_8604_ATP_Olambda__466,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gy(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_466
tff(fact_8605_ATP_Olambda__467,axiom,
    ! [Uu: fun_se4097146774670233028at_nat,Uua: fun_se4097146774670233028at_nat,Uub: set_Pr1261947904930325089at_nat] : aa_set4749074711808745205at_nat(aa_fun4978281634197224252at_nat(aTP_Lamm_ha(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4749074711808745205at_nat(Uua,Uub)),aa_set4749074711808745205at_nat(Uu,Uub)) ).

% ATP.lambda_467
tff(fact_8606_ATP_Olambda__468,axiom,
    ! [Uu: fun_se4101287982975402108nt_nat,Uua: fun_se4101287982975402108nt_nat,Uub: set_Pr958786334691620121nt_int] : aa_set4753215920113914285nt_nat(aa_fun209864160807355708nt_nat(aTP_Lamm_hb(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4753215920113914285nt_nat(Uua,Uub)),aa_set4753215920113914285nt_nat(Uu,Uub)) ).

% ATP.lambda_468
tff(fact_8607_ATP_Olambda__469,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_hc(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_469
tff(fact_8608_ATP_Olambda__470,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_he(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_470
tff(fact_8609_ATP_Olambda__471,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_hd(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_471
tff(fact_8610_ATP_Olambda__472,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gw(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_472
tff(fact_8611_ATP_Olambda__473,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gx(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_473
tff(fact_8612_ATP_Olambda__474,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_wd(Uu),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        | aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_474
tff(fact_8613_ATP_Olambda__475,axiom,
    ! [Uu: fun_se4782206882984618170_nat_o,Uua: fun_se4782206882984618170_nat_o,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_fun3315923329236548914_nat_o(aTP_Lamm_vy(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(Uu,Uub)
        | aa_set3702741264012733747_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_475
tff(fact_8614_ATP_Olambda__476,axiom,
    ! [Uu: fun_se7249947736802449410_int_o,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_vz(Uu),Uua),Uub)
    <=> ( aa_set6170482117830564987_int_o(Uu,Uub)
        | aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_476
tff(fact_8615_ATP_Olambda__477,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_wa(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        | aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_477
tff(fact_8616_ATP_Olambda__478,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_wb(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_478
tff(fact_8617_ATP_Olambda__479,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_wc(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_479
tff(fact_8618_ATP_Olambda__480,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_vx(Uu),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_480
tff(fact_8619_ATP_Olambda__481,axiom,
    ! [Uu: fun_se4782206882984618170_nat_o,Uua: fun_se4782206882984618170_nat_o,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_fun3315923329236548914_nat_o(aTP_Lamm_bx(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(Uu,Uub)
        & aa_set3702741264012733747_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_481
tff(fact_8620_ATP_Olambda__482,axiom,
    ! [Uu: fun_se7249947736802449410_int_o,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_by(Uu),Uua),Uub)
    <=> ( aa_set6170482117830564987_int_o(Uu,Uub)
        & aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_482
tff(fact_8621_ATP_Olambda__483,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_bz(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_483
tff(fact_8622_ATP_Olambda__484,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cb(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_484
tff(fact_8623_ATP_Olambda__485,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ca(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_485
tff(fact_8624_ATP_Olambda__486,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_kv(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_486
tff(fact_8625_ATP_Olambda__487,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: set_nat,Uub: nat] : aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_kx(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_487
tff(fact_8626_ATP_Olambda__488,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: set_int,Uub: nat] : aa_nat_int(aa_set6310687161026558745at_int(aTP_Lamm_la(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_488
tff(fact_8627_ATP_Olambda__489,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: set_nat,Uub: int] : aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_ld(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_489
tff(fact_8628_ATP_Olambda__490,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_lg(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_490
tff(fact_8629_ATP_Olambda__491,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_gi(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_491
tff(fact_8630_ATP_Olambda__492,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_gl(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_492
tff(fact_8631_ATP_Olambda__493,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_qg(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_493
tff(fact_8632_ATP_Olambda__494,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fr(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_494
tff(fact_8633_ATP_Olambda__495,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_lo(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_495
tff(fact_8634_ATP_Olambda__496,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_fs(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_496
tff(fact_8635_ATP_Olambda__497,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gq(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_497
tff(fact_8636_ATP_Olambda__498,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_gr(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_498
tff(fact_8637_ATP_Olambda__499,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ll(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_499
tff(fact_8638_ATP_Olambda__500,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_lm(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_500
tff(fact_8639_ATP_Olambda__501,axiom,
    ! [Uu: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_ln(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_501
tff(fact_8640_ATP_Olambda__502,axiom,
    ! [Uu: product_unit,Uua: fun_Product_unit_o,Uub: product_unit] :
      ( aa_Product_unit_o(aa_fun644178081534259290unit_o(aTP_Lamm_aad(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Product_unit_o(Uua,Uub) ) ) ).

% ATP.lambda_502
tff(fact_8641_ATP_Olambda__503,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_aaf(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_503
tff(fact_8642_ATP_Olambda__504,axiom,
    ! [Uu: product_prod_nat_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_aae(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_504
tff(fact_8643_ATP_Olambda__505,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_se4782206882984618170_nat_o,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_fun3315923329236548914_nat_o(aTP_Lamm_aag(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_set3702741264012733747_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_505
tff(fact_8644_ATP_Olambda__506,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_aah(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_506
tff(fact_8645_ATP_Olambda__507,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_aai(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_507
tff(fact_8646_ATP_Olambda__508,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_aaj(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_508
tff(fact_8647_ATP_Olambda__509,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_aak(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_509
tff(fact_8648_ATP_Olambda__510,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_oo(Uu),Uua),Uub) = divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub))) ).

% ATP.lambda_510
tff(fact_8649_ATP_Olambda__511,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nf(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(Uu),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ).

% ATP.lambda_511
tff(fact_8650_ATP_Olambda__512,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ng(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(Uu),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ).

% ATP.lambda_512
tff(fact_8651_ATP_Olambda__513,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ne(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(Uu),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ).

% ATP.lambda_513
tff(fact_8652_ATP_Olambda__514,axiom,
    ! [Uu: code_natural,Uua: code_natural,Uub: nat] : aa_nat_Code_natural(aa_Cod5320533616963873627atural(aTP_Lamm_kf(Uu),Uua),Uub) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Uu),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,Uub)),Uua)) ).

% ATP.lambda_514
tff(fact_8653_ATP_Olambda__515,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_kg(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Uub)),Uua)) ).

% ATP.lambda_515
tff(fact_8654_ATP_Olambda__516,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_kh(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_516
tff(fact_8655_ATP_Olambda__517,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).

% ATP.lambda_517
tff(fact_8656_ATP_Olambda__518,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ki(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).

% ATP.lambda_518
tff(fact_8657_ATP_Olambda__519,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_ol(Uu),Uua),Uub) = aa_nat_Code_integer(power_8256067586552552935nteger(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_519
tff(fact_8658_ATP_Olambda__520,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_om(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_520
tff(fact_8659_ATP_Olambda__521,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ok(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_521
tff(fact_8660_ATP_Olambda__522,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_fv(Uu),Uua),Uub) = aa_nat_Code_integer(power_8256067586552552935nteger(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_522
tff(fact_8661_ATP_Olambda__523,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_fw(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_523
tff(fact_8662_ATP_Olambda__524,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_fu(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_524
tff(fact_8663_ATP_Olambda__525,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kl(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_525
tff(fact_8664_ATP_Olambda__526,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_go(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_526
tff(fact_8665_ATP_Olambda__527,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gp(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu),aa_int_int(Uua,Uub)) ).

% ATP.lambda_527
tff(fact_8666_ATP_Olambda__528,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ls(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_528
tff(fact_8667_ATP_Olambda__529,axiom,
    ! [Uu: int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_lt(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_529
tff(fact_8668_ATP_Olambda__530,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_lu(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_530
tff(fact_8669_ATP_Olambda__531,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_mm(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_531
tff(fact_8670_ATP_Olambda__532,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mn(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_532
tff(fact_8671_ATP_Olambda__533,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_mo(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_533
tff(fact_8672_ATP_Olambda__534,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_no(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_534
tff(fact_8673_ATP_Olambda__535,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_np(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_535
tff(fact_8674_ATP_Olambda__536,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_adr(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_536
tff(fact_8675_ATP_Olambda__537,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hw(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_537
tff(fact_8676_ATP_Olambda__538,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_lr(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_538
tff(fact_8677_ATP_Olambda__539,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ds(Uu),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_539
tff(fact_8678_ATP_Olambda__540,axiom,
    ! [Uu: fun_Pr2028404973800837934_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_dt(Uu),Uua),Uub) = aa_Pro989595756709598047_nat_o(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_540
tff(fact_8679_ATP_Olambda__541,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_du(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_541
tff(fact_8680_ATP_Olambda__542,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dv(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_542
tff(fact_8681_ATP_Olambda__543,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_dw(Uu),Uua),Uub) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_543
tff(fact_8682_ATP_Olambda__544,axiom,
    ! [Uu: fun_op8989173174992673105on_num,Uua: fun_num_option_num,Uub: num] : aa_num_option_num(aa_fun3636977833914307392on_num(aTP_Lamm_bb(Uu),Uua),Uub) = aa_opt5369159329878751634on_num(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_544
tff(fact_8683_ATP_Olambda__545,axiom,
    ! [Uu: fun_op8989173174992673105on_num,Uua: fun_nat_option_num,Uub: nat] : aa_nat_option_num(aa_fun7716400287237121964on_num(aTP_Lamm_ev(Uu),Uua),Uub) = aa_opt5369159329878751634on_num(Uu,aa_nat_option_num(Uua,Uub)) ).

% ATP.lambda_545
tff(fact_8684_ATP_Olambda__546,axiom,
    ! [Uu: fun_option_num_o,Uua: fun_nat_option_num,Uub: nat] :
      ( aa_nat_o(aa_fun1675964276245824108_nat_o(aTP_Lamm_ew(Uu),Uua),Uub)
    <=> aa_option_num_o(Uu,aa_nat_option_num(Uua,Uub)) ) ).

% ATP.lambda_546
tff(fact_8685_ATP_Olambda__547,axiom,
    ! [Uu: fun_option_num_num,Uua: fun_num_option_num,Uub: num] : aa_num_num(aa_fun7142274290070961008um_num(aTP_Lamm_bc(Uu),Uua),Uub) = aa_option_num_num(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_547
tff(fact_8686_ATP_Olambda__548,axiom,
    ! [Uu: fun_option_num_nat,Uua: fun_nat_option_num,Uub: nat] : aa_nat_nat(aa_fun7387201323300553874at_nat(aTP_Lamm_ex(Uu),Uua),Uub) = aa_option_num_nat(Uu,aa_nat_option_num(Uua,Uub)) ).

% ATP.lambda_548
tff(fact_8687_ATP_Olambda__549,axiom,
    ! [Uu: fun_option_num_int,Uua: fun_num_option_num,Uub: num] : aa_num_int(aa_fun2256179333612926594um_int(aTP_Lamm_ba(Uu),Uua),Uub) = aa_option_num_int(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_549
tff(fact_8688_ATP_Olambda__550,axiom,
    ! [Uu: fun_o_option_num,Uua: fun_nat_o,Uub: nat] : aa_nat_option_num(aa_fun6990899178391151900on_num(aTP_Lamm_ey(Uu),Uua),Uub) = aa_o_option_num(Uu,aa_nat_o(Uua,Uub)) ).

% ATP.lambda_550
tff(fact_8689_ATP_Olambda__551,axiom,
    ! [Uu: fun_o_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ez(Uu),Uua),Uub)
    <=> aa_o_o(Uu,aa_nat_o(Uua,Uub)) ) ).

% ATP.lambda_551
tff(fact_8690_ATP_Olambda__552,axiom,
    ! [Uu: fun_o_nat,Uua: fun_nat_o,Uub: nat] : aa_nat_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_fa(Uu),Uua),Uub) = aa_o_nat(Uu,aa_nat_o(Uua,Uub)) ).

% ATP.lambda_552
tff(fact_8691_ATP_Olambda__553,axiom,
    ! [Uu: fun_num_option_num,Uua: fun_num_num,Uub: num] : aa_num_option_num(aa_fun612943910911948912on_num(aTP_Lamm_be(Uu),Uua),Uub) = aa_num_option_num(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_553
tff(fact_8692_ATP_Olambda__554,axiom,
    ! [Uu: fun_num_num,Uua: fun_num_num,Uub: num] : aa_num_num(aa_fun8048995283964520608um_num(aTP_Lamm_bf(Uu),Uua),Uub) = aa_num_num(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_554
tff(fact_8693_ATP_Olambda__555,axiom,
    ! [Uu: fun_num_int,Uua: fun_num_num,Uub: num] : aa_num_int(aa_fun3162900327506486194um_int(aTP_Lamm_bd(Uu),Uua),Uub) = aa_num_int(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_555
tff(fact_8694_ATP_Olambda__556,axiom,
    ! [Uu: fun_nat_option_num,Uua: fun_nat_nat,Uub: nat] : aa_nat_option_num(aa_fun6517009837249254290on_num(aTP_Lamm_fb(Uu),Uua),Uub) = aa_nat_option_num(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_556
tff(fact_8695_ATP_Olambda__557,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_fc(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_557
tff(fact_8696_ATP_Olambda__558,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fd(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_558
tff(fact_8697_ATP_Olambda__559,axiom,
    ! [Uu: fun_int_option_num,Uua: fun_num_int,Uub: num] : aa_num_option_num(aa_fun3858197967482829954on_num(aTP_Lamm_ay(Uu),Uua),Uub) = aa_int_option_num(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_559
tff(fact_8698_ATP_Olambda__560,axiom,
    ! [Uu: fun_int_num,Uua: fun_num_int,Uub: num] : aa_num_num(aa_fun5215869348392759218um_num(aTP_Lamm_az(Uu),Uua),Uub) = aa_int_num(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_560
tff(fact_8699_ATP_Olambda__561,axiom,
    ! [Uu: fun_int_int,Uua: fun_num_int,Uub: num] : aa_num_int(aa_fun329774391934724804um_int(aTP_Lamm_ax(Uu),Uua),Uub) = aa_int_int(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_561
tff(fact_8700_ATP_Olambda__562,axiom,
    ! [Uu: set_set_nat,Uua: fun_Pr1646868699942875873_nat_o,Uub: product_unit] : aa_Product_unit_nat(aa_fun6562703090490685585it_nat(aTP_Lamm_ys(Uu),Uua),Uub) = finite_card_set_nat(collect_set_nat(aa_Pro5573132868698907866_nat_o(aa_fun4072893971397127680_nat_o(aTP_Lamm_yr(Uu),Uua),Uub))) ).

% ATP.lambda_562
tff(fact_8701_ATP_Olambda__563,axiom,
    ! [Uu: set_Product_unit,Uua: fun_na5971062356810558811unit_o,Uub: nat] : aa_nat_nat(aa_fun8994848686300240182at_nat(aTP_Lamm_yn(Uu),Uua),Uub) = finite410649719033368117t_unit(collect_Product_unit(aa_nat5955485856412068252unit_o(aa_fun8483372159859046644unit_o(aTP_Lamm_ym(Uu),Uua),Uub))) ).

% ATP.lambda_563
tff(fact_8702_ATP_Olambda__564,axiom,
    ! [Uu: set_nat,Uua: fun_Pr4402817529748954391_nat_o,Uub: product_unit] : aa_Product_unit_nat(aa_fun1580523165164119515it_nat(aTP_Lamm_yj(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_Pro847370707247602192_nat_o(aa_fun1884058921492388972_nat_o(aTP_Lamm_yi(Uu),Uua),Uub))) ).

% ATP.lambda_564
tff(fact_8703_ATP_Olambda__565,axiom,
    ! [Uu: set_Product_unit,Uua: fun_in8984369272479023031unit_o,Uub: int] : aa_int_nat(aa_fun5797050646745590254nt_nat(aTP_Lamm_yf(Uu),Uua),Uub) = finite410649719033368117t_unit(collect_Product_unit(aa_int8968792772080532472unit_o(aa_fun5573515973562188204unit_o(aTP_Lamm_ye(Uu),Uua),Uub))) ).

% ATP.lambda_565
tff(fact_8704_ATP_Olambda__566,axiom,
    ! [Uu: set_int,Uua: fun_Pr2157672562078508475_int_o,Uub: product_unit] : aa_Product_unit_nat(aa_fun4006079477318474295it_nat(aTP_Lamm_yc(Uu),Uua),Uub) = finite_card_int(collect_int(aa_Pro8249700398669690036_int_o(aa_fun2646465928216627636_int_o(aTP_Lamm_yb(Uu),Uua),Uub))) ).

% ATP.lambda_566
tff(fact_8705_ATP_Olambda__567,axiom,
    ! [Uu: set_Product_unit,Uua: fun_Pr1520749766685617582unit_o,Uub: product_unit] : aa_Product_unit_nat(aa_fun2167693721382921342it_nat(aTP_Lamm_xz(Uu),Uua),Uub) = finite410649719033368117t_unit(collect_Product_unit(aa_Pro4895447120693452255unit_o(aa_fun7194487344118582332unit_o(aTP_Lamm_xy(Uu),Uua),Uub))) ).

% ATP.lambda_567
tff(fact_8706_ATP_Olambda__568,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_o,Uub: int] : aa_int_nat(aa_fun4460091444687325999nt_nat(aTP_Lamm_yg(Uu),Uua),Uub) = finite_card_int(collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_qz(Uu),Uua),Uub))) ).

% ATP.lambda_568
tff(fact_8707_ATP_Olambda__569,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_o,Uub: nat] : aa_nat_nat(aa_fun8857679445107423627at_nat(aTP_Lamm_yp(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_qt(Uu),Uua),Uub))) ).

% ATP.lambda_569
tff(fact_8708_ATP_Olambda__570,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_o,Uub: nat] : aa_nat_nat(aa_fun8666973529875159599at_nat(aTP_Lamm_yo(Uu),Uua),Uub) = finite_card_int(collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_qm(Uu),Uua),Uub))) ).

% ATP.lambda_570
tff(fact_8709_ATP_Olambda__571,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_o,Uub: int] : aa_int_nat(aa_fun4650797359919590027nt_nat(aTP_Lamm_yk(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_qh(Uu),Uua),Uub))) ).

% ATP.lambda_571
tff(fact_8710_ATP_Olambda__572,axiom,
    ! [Uu: fun_na1668100129766719005at_nat,Uua: product_prod_nat_nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_Pro6895555690143131534at_nat(aTP_Lamm_dg(Uu),Uua),Uub),Uuc) = aa_Pro262675715343573244at_nat(aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Uu,Uub),Uuc),Uua) ).

% ATP.lambda_572
tff(fact_8711_ATP_Olambda__573,axiom,
    ! [Uu: fun_na5914473681949881652_nat_o,Uua: product_prod_nat_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_Pro1957341097241355813_nat_o(aTP_Lamm_dh(Uu),Uua),Uub),Uuc)
    <=> aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uu,Uub),Uuc),Uua) ) ).

% ATP.lambda_573
tff(fact_8712_ATP_Olambda__574,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_natural,Uub: fun_nat_Code_natural,Uuc: nat] :
      aa_nat_Code_natural(aa_fun56224542555504168atural(aa_fun1651958757531339674atural(aTP_Lamm_jt(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_Code_natural(Uua,Uuc),
        $ite(Uuc = Uu,zero_z2226904508553997617atural,aa_nat_Code_natural(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_574
tff(fact_8713_ATP_Olambda__575,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_jp(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_Code_integer(Uua,Uuc),
        $ite(Uuc = Uu,zero_z3403309356797280102nteger,aa_nat_Code_integer(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_575
tff(fact_8714_ATP_Olambda__576,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_jr(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_rat(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_rat,aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_576
tff(fact_8715_ATP_Olambda__577,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_jv(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_577
tff(fact_8716_ATP_Olambda__578,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_jn(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_578
tff(fact_8717_ATP_Olambda__579,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_natural,Uub: fun_nat_Code_natural,Uuc: nat] :
      aa_nat_Code_natural(aa_fun56224542555504168atural(aa_fun1651958757531339674atural(aTP_Lamm_mv(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_Code_natural(Uua,Uuc),
        $ite(Uuc = Uu,one_one_Code_natural,aa_nat_Code_natural(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_579
tff(fact_8718_ATP_Olambda__580,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_mt(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_Code_integer(Uua,Uuc),
        $ite(Uuc = Uu,one_one_Code_integer,aa_nat_Code_integer(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_580
tff(fact_8719_ATP_Olambda__581,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_mu(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_rat(Uua,Uuc),
        $ite(Uuc = Uu,one_one_rat,aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_581
tff(fact_8720_ATP_Olambda__582,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_mw(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu,one_one_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_582
tff(fact_8721_ATP_Olambda__583,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_mx(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu,one_one_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_583
tff(fact_8722_ATP_Olambda__584,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_natural,Uub: fun_nat_Code_natural,Uuc: nat] :
      aa_nat_Code_natural(aa_fun56224542555504168atural(aa_fun1651958757531339674atural(aTP_Lamm_ju(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_Code_natural(Uua,Uuc),aa_nat_Code_natural(Uub,Uuc)) ).

% ATP.lambda_584
tff(fact_8723_ATP_Olambda__585,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_jq(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_Code_integer(Uua,Uuc),aa_nat_Code_integer(Uub,Uuc)) ).

% ATP.lambda_585
tff(fact_8724_ATP_Olambda__586,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_js(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_586
tff(fact_8725_ATP_Olambda__587,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_jw(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_587
tff(fact_8726_ATP_Olambda__588,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_jo(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_588
tff(fact_8727_ATP_Olambda__589,axiom,
    ! [Uu: product_unit,Uua: fun_Pr1820225556744648189nteger,Uub: code_integer,Uuc: product_unit] :
      aa_Pro1705297278780387758nteger(aa_Cod3669976074529508130nteger(aa_fun3412420425980457528nteger(aTP_Lamm_zh(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Pro1705297278780387758nteger(Uua,Uuc),Uub) ).

% ATP.lambda_589
tff(fact_8728_ATP_Olambda__590,axiom,
    ! [Uu: product_unit,Uua: fun_Product_unit_rat,Uub: rat,Uuc: product_unit] :
      aa_Product_unit_rat(aa_rat6507435564262174562it_rat(aa_fun5217320890768598205it_rat(aTP_Lamm_zk(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Product_unit_rat(Uua,Uuc),Uub) ).

% ATP.lambda_590
tff(fact_8729_ATP_Olambda__591,axiom,
    ! [Uu: product_unit,Uua: fun_Product_unit_nat,Uub: nat,Uuc: product_unit] :
      aa_Product_unit_nat(aa_nat7626694396248770914it_nat(aa_fun277559562889681605it_nat(aTP_Lamm_ze(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Product_unit_nat(Uua,Uuc),Uub) ).

% ATP.lambda_591
tff(fact_8730_ATP_Olambda__592,axiom,
    ! [Uu: product_unit,Uua: fun_Product_unit_int,Uub: int,Uuc: product_unit] :
      aa_Product_unit_int(aa_int1856962249220300386it_int(aa_fun4982840383720920169it_int(aTP_Lamm_zg(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Product_unit_int(Uua,Uuc),Uub) ).

% ATP.lambda_592
tff(fact_8731_ATP_Olambda__593,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: code_integer,Uuc: nat] :
      aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_fun8469111845059130348nteger(aTP_Lamm_zi(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_nat_Code_integer(Uua,Uuc),Uub) ).

% ATP.lambda_593
tff(fact_8732_ATP_Olambda__594,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: rat,Uuc: nat] :
      aa_nat_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_zl(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_nat_rat(Uua,Uuc),Uub) ).

% ATP.lambda_594
tff(fact_8733_ATP_Olambda__595,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_zn(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_nat_nat(Uua,Uuc),Uub) ).

% ATP.lambda_595
tff(fact_8734_ATP_Olambda__596,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: code_integer,Uuc: int] :
      aa_int_Code_integer(aa_Cod1949805079358391713nteger(aa_fun2884842898402423460nteger(aTP_Lamm_zj(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_Code_integer(Uua,Uuc),Uub) ).

% ATP.lambda_596
tff(fact_8735_ATP_Olambda__597,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: rat,Uuc: int] :
      aa_int_rat(aa_rat_fun_int_rat(aa_fun9155288305517341845nt_rat(aTP_Lamm_zm(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_rat(Uua,Uuc),Uub) ).

% ATP.lambda_597
tff(fact_8736_ATP_Olambda__598,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: nat,Uuc: int] :
      aa_int_nat(aa_nat_fun_int_nat(aa_fun1022895345097303677nt_nat(aTP_Lamm_zf(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_nat(Uua,Uuc),Uub) ).

% ATP.lambda_598
tff(fact_8737_ATP_Olambda__599,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : aa_int_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_gv(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_gu(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_599
tff(fact_8738_ATP_Olambda__600,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: set_nat,Uuc: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_gt(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_gs(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_600
tff(fact_8739_ATP_Olambda__601,axiom,
    ! [Uu: set_set_nat,Uua: fun_se7912642235559603452at_int,Uub: fun_se382663106265794036_nat_o,Uuc: nat] : aa_nat_int(aa_fun6199171195344186545at_int(aa_fun2727794920753907526at_int(aTP_Lamm_sp(Uu),Uua),Uub),Uuc) = groups4246057289670975065at_int(aa_nat2732338449433945789at_int(aTP_Lamm_so(Uua),Uuc),collect_set_nat(aa_nat_fun_set_nat_o(aa_fun9105140013458882876_nat_o(aTP_Lamm_rg(Uu),Uub),Uuc))) ).

% ATP.lambda_601
tff(fact_8740_ATP_Olambda__602,axiom,
    ! [Uu: set_set_nat,Uua: fun_se876191773397849432nt_int,Uub: fun_se7360890175450123928_int_o,Uuc: int] : aa_int_int(aa_fun7007818992012977073nt_int(aa_fun7130641361752856810nt_int(aTP_Lamm_sr(Uu),Uua),Uub),Uuc) = groups4246057289670975065at_int(aa_int2448287375063835929at_int(aTP_Lamm_rk(Uua),Uuc),collect_set_nat(aa_int_fun_set_nat_o(aa_fun9086678093774146108_nat_o(aTP_Lamm_rl(Uu),Uub),Uuc))) ).

% ATP.lambda_602
tff(fact_8741_ATP_Olambda__603,axiom,
    ! [Uu: set_set_nat,Uua: fun_se1179353343636554400at_nat,Uub: fun_se382663106265794036_nat_o,Uuc: nat] : aa_nat_nat(aa_fun1153650177998607445at_nat(aa_fun5868473082678171206at_nat(aTP_Lamm_sm(Uu),Uua),Uub),Uuc) = groups4248547760180025341at_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_rf(Uua),Uuc),collect_set_nat(aa_nat_fun_set_nat_o(aa_fun9105140013458882876_nat_o(aTP_Lamm_rg(Uu),Uub),Uuc))) ).

% ATP.lambda_603
tff(fact_8742_ATP_Olambda__604,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_nat,Uub: fun_nat_fun_int_o,Uuc: int] : aa_int_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_sa(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_int_fun_nat_nat(aTP_Lamm_qr(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_qo(Uu),Uub),Uuc))) ).

% ATP.lambda_604
tff(fact_8743_ATP_Olambda__605,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_ry(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aTP_Lamm_qj(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_qk(Uu),Uub),Uuc))) ).

% ATP.lambda_605
tff(fact_8744_ATP_Olambda__606,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_int,Uub: fun_int_fun_nat_o,Uuc: nat] : aa_nat_int(aa_fun8696964665364114571at_int(aa_fun3875905286069234426at_int(aTP_Lamm_si(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_le(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_qk(Uu),Uub),Uuc))) ).

% ATP.lambda_606
tff(fact_8745_ATP_Olambda__607,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_o,Uuc: int] : aa_int_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_sg(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_lb(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_qo(Uu),Uub),Uuc))) ).

% ATP.lambda_607
tff(fact_8746_ATP_Olambda__608,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_int,Uub: fun_nat_fun_nat_o,Uuc: nat] : aa_nat_int(aa_fun4679828425598226919at_int(aa_fun5734951013428723122at_int(aTP_Lamm_se(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ky(Uua),Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_qv(Uu),Uub),Uuc))) ).

% ATP.lambda_608
tff(fact_8747_ATP_Olambda__609,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_int,Uub: fun_int_fun_int_o,Uuc: int] : aa_int_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_sk(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_gm(Uua),Uuc),collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_rb(Uu),Uub),Uuc))) ).

% ATP.lambda_609
tff(fact_8748_ATP_Olambda__610,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: fun_nat_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_sc(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gj(Uua),Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_qv(Uu),Uub),Uuc))) ).

% ATP.lambda_610
tff(fact_8749_ATP_Olambda__611,axiom,
    ! [Uu: set_int,Uua: fun_in6555721293707029592at_int,Uub: fun_in7136481858371106712_nat_o,Uuc: set_nat] : aa_set_nat_int(aa_fun5805668902077597195at_int(aa_fun5576367408028688196at_int(aTP_Lamm_rw(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_ru(Uua),Uuc),collect_int(aa_set_nat_fun_int_o(aa_fun8216954948888934460_int_o(aTP_Lamm_rv(Uu),Uub),Uuc))) ).

% ATP.lambda_611
tff(fact_8750_ATP_Olambda__612,axiom,
    ! [Uu: set_nat,Uua: fun_na106483476154090400at_nat,Uub: fun_na8563630686737515764_nat_o,Uuc: set_nat] : aa_set_nat_nat(aa_fun255155394271802123at_nat(aa_fun6284103964554420476at_nat(aTP_Lamm_rr(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_rp(Uua),Uuc)),collect_nat(aa_set_nat_fun_nat_o(aa_fun2835368149466334012_nat_o(aTP_Lamm_rq(Uu),Uub),Uuc))) ).

% ATP.lambda_612
tff(fact_8751_ATP_Olambda__613,axiom,
    ! [Uu: set_set_nat,Uua: fun_se876191773397849432nt_int,Uub: fun_se7360890175450123928_int_o,Uuc: int] : aa_int_int(aa_fun7007818992012977073nt_int(aa_fun7130641361752856810nt_int(aTP_Lamm_rm(Uu),Uua),Uub),Uuc) = groups8292507037921071086at_int(aa_int2448287375063835929at_int(aTP_Lamm_rk(Uua),Uuc),collect_set_nat(aa_int_fun_set_nat_o(aa_fun9086678093774146108_nat_o(aTP_Lamm_rl(Uu),Uub),Uuc))) ).

% ATP.lambda_613
tff(fact_8752_ATP_Olambda__614,axiom,
    ! [Uu: set_set_nat,Uua: fun_se1179353343636554400at_nat,Uub: fun_se382663106265794036_nat_o,Uuc: nat] : aa_nat_nat(aa_fun1153650177998607445at_nat(aa_fun5868473082678171206at_nat(aTP_Lamm_rh(Uu),Uua),Uub),Uuc) = groups8294997508430121362at_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_rf(Uua),Uuc),collect_set_nat(aa_nat_fun_set_nat_o(aa_fun9105140013458882876_nat_o(aTP_Lamm_rg(Uu),Uub),Uuc))) ).

% ATP.lambda_614
tff(fact_8753_ATP_Olambda__615,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_nat,Uub: fun_nat_fun_int_o,Uuc: int] : aa_int_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_qs(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(aTP_Lamm_qr(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_qo(Uu),Uub),Uuc))) ).

% ATP.lambda_615
tff(fact_8754_ATP_Olambda__616,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_ql(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_qj(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_qk(Uu),Uub),Uuc))) ).

% ATP.lambda_616
tff(fact_8755_ATP_Olambda__617,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_int,Uub: fun_int_fun_nat_o,Uuc: nat] : aa_nat_int(aa_fun8696964665364114571at_int(aa_fun3875905286069234426at_int(aTP_Lamm_qy(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_nat_fun_int_int(aTP_Lamm_le(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_qk(Uu),Uub),Uuc))) ).

% ATP.lambda_617
tff(fact_8756_ATP_Olambda__618,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_o,Uuc: int] : aa_int_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_qp(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_lb(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_qo(Uu),Uub),Uuc))) ).

% ATP.lambda_618
tff(fact_8757_ATP_Olambda__619,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_int,Uub: fun_int_fun_int_o,Uuc: int] : aa_int_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_rc(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_gm(Uua),Uuc),collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_rb(Uu),Uub),Uuc))) ).

% ATP.lambda_619
tff(fact_8758_ATP_Olambda__620,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: fun_nat_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_qw(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gj(Uua),Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_qv(Uu),Uub),Uuc))) ).

% ATP.lambda_620
tff(fact_8759_ATP_Olambda__621,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abi(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_621
tff(fact_8760_ATP_Olambda__622,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_jj(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Uu),Uuc))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_622
tff(fact_8761_ATP_Olambda__623,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_jk(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uu),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_623
tff(fact_8762_ATP_Olambda__624,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_jl(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Uu),Uuc))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_624
tff(fact_8763_ATP_Olambda__625,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_jg(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uu),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_625
tff(fact_8764_ATP_Olambda__626,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_jh(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc))),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_626
tff(fact_8765_ATP_Olambda__627,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jb(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_627
tff(fact_8766_ATP_Olambda__628,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ji(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_628
tff(fact_8767_ATP_Olambda__629,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_of(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_Code_integer(power_8256067586552552935nteger(Uu),Uuc)) ).

% ATP.lambda_629
tff(fact_8768_ATP_Olambda__630,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_og(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu),Uuc)) ).

% ATP.lambda_630
tff(fact_8769_ATP_Olambda__631,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_oe(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uuc)) ).

% ATP.lambda_631
tff(fact_8770_ATP_Olambda__632,axiom,
    ! [Uu: set_Product_unit,Uua: fun_Pr1520749766685617582unit_o,Uub: product_unit,Uuc: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(aa_fun7194487344118582332unit_o(aTP_Lamm_xy(Uu),Uua),Uub),Uuc)
    <=> ( member_Product_unit(Uuc,Uu)
        & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_632
tff(fact_8771_ATP_Olambda__633,axiom,
    ! [Uu: set_Product_unit,Uua: fun_na5971062356810558811unit_o,Uub: nat,Uuc: product_unit] :
      ( aa_Product_unit_o(aa_nat5955485856412068252unit_o(aa_fun8483372159859046644unit_o(aTP_Lamm_ym(Uu),Uua),Uub),Uuc)
    <=> ( member_Product_unit(Uuc,Uu)
        & aa_Product_unit_o(aa_nat5955485856412068252unit_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_633
tff(fact_8772_ATP_Olambda__634,axiom,
    ! [Uu: set_Product_unit,Uua: fun_in8984369272479023031unit_o,Uub: int,Uuc: product_unit] :
      ( aa_Product_unit_o(aa_int8968792772080532472unit_o(aa_fun5573515973562188204unit_o(aTP_Lamm_ye(Uu),Uua),Uub),Uuc)
    <=> ( member_Product_unit(Uuc,Uu)
        & aa_Product_unit_o(aa_int8968792772080532472unit_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_634
tff(fact_8773_ATP_Olambda__635,axiom,
    ! [Uu: set_set_nat,Uua: fun_Pr1646868699942875873_nat_o,Uub: product_unit,Uuc: set_nat] :
      ( aa_set_nat_o(aa_Pro5573132868698907866_nat_o(aa_fun4072893971397127680_nat_o(aTP_Lamm_yr(Uu),Uua),Uub),Uuc)
    <=> ( member_set_nat(Uuc,Uu)
        & aa_set_nat_o(aa_Pro5573132868698907866_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_635
tff(fact_8774_ATP_Olambda__636,axiom,
    ! [Uu: set_set_nat,Uua: fun_na8563630686737515764_nat_o,Uub: nat,Uuc: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aa_fun1792963693083279932_nat_o(aTP_Lamm_rn(Uu),Uua),Uub),Uuc)
    <=> ( member_set_nat(Uuc,Uu)
        & aa_set_nat_o(aa_nat_fun_set_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_636
tff(fact_8775_ATP_Olambda__637,axiom,
    ! [Uu: set_set_nat,Uua: fun_in7136481858371106712_nat_o,Uub: int,Uuc: set_nat] :
      ( aa_set_nat_o(aa_int_fun_set_nat_o(aa_fun7992546631809917244_nat_o(aTP_Lamm_rs(Uu),Uua),Uub),Uuc)
    <=> ( member_set_nat(Uuc,Uu)
        & aa_set_nat_o(aa_int_fun_set_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_637
tff(fact_8776_ATP_Olambda__638,axiom,
    ! [Uu: set_nat,Uua: fun_Pr4402817529748954391_nat_o,Uub: product_unit,Uuc: nat] :
      ( aa_nat_o(aa_Pro847370707247602192_nat_o(aa_fun1884058921492388972_nat_o(aTP_Lamm_yi(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_nat_o(aa_Pro847370707247602192_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_638
tff(fact_8777_ATP_Olambda__639,axiom,
    ! [Uu: set_nat,Uua: fun_se382663106265794036_nat_o,Uub: set_nat,Uuc: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aa_fun924172432987161148_nat_o(aTP_Lamm_rd(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_nat_o(aa_set_nat_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_639
tff(fact_8778_ATP_Olambda__640,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_qt(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_640
tff(fact_8779_ATP_Olambda__641,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_qh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_641
tff(fact_8780_ATP_Olambda__642,axiom,
    ! [Uu: set_int,Uua: fun_Pr2157672562078508475_int_o,Uub: product_unit,Uuc: int] :
      ( aa_int_o(aa_Pro8249700398669690036_int_o(aa_fun2646465928216627636_int_o(aTP_Lamm_yb(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_int_o(aa_Pro8249700398669690036_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_642
tff(fact_8781_ATP_Olambda__643,axiom,
    ! [Uu: set_int,Uua: fun_se7360890175450123928_int_o,Uub: set_nat,Uuc: int] :
      ( aa_int_o(aa_set_nat_fun_int_o(aa_fun87714373998387516_int_o(aTP_Lamm_ri(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_int_o(aa_set_nat_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_643
tff(fact_8782_ATP_Olambda__644,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_o,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_qm(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_644
tff(fact_8783_ATP_Olambda__645,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_qz(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_645
tff(fact_8784_ATP_Olambda__646,axiom,
    ! [Uu: set_Product_unit,Uua: fun_Pr1520749766685617582unit_o,Uub: product_unit,Uuc: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(aa_fun7194487344118582332unit_o(aTP_Lamm_xx(Uu),Uua),Uub),Uuc)
    <=> ( member_Product_unit(Uuc,Uu)
        & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_646
tff(fact_8785_ATP_Olambda__647,axiom,
    ! [Uu: set_Product_unit,Uua: fun_Pr1646868699942875873_nat_o,Uub: set_nat,Uuc: product_unit] :
      ( aa_Product_unit_o(aa_set3064922119574097746unit_o(aa_fun6872337739698859568unit_o(aTP_Lamm_yq(Uu),Uua),Uub),Uuc)
    <=> ( member_Product_unit(Uuc,Uu)
        & aa_set_nat_o(aa_Pro5573132868698907866_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_647
tff(fact_8786_ATP_Olambda__648,axiom,
    ! [Uu: set_Product_unit,Uua: fun_Pr4402817529748954391_nat_o,Uub: nat,Uuc: product_unit] :
      ( aa_Product_unit_o(aa_nat5955485856412068252unit_o(aa_fun3452303748553993392unit_o(aTP_Lamm_yh(Uu),Uua),Uub),Uuc)
    <=> ( member_Product_unit(Uuc,Uu)
        & aa_nat_o(aa_Pro847370707247602192_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_648
tff(fact_8787_ATP_Olambda__649,axiom,
    ! [Uu: set_Product_unit,Uua: fun_Pr2157672562078508475_int_o,Uub: int,Uuc: product_unit] :
      ( aa_Product_unit_o(aa_int8968792772080532472unit_o(aa_fun249790601762366384unit_o(aTP_Lamm_ya(Uu),Uua),Uub),Uuc)
    <=> ( member_Product_unit(Uuc,Uu)
        & aa_int_o(aa_Pro8249700398669690036_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_649
tff(fact_8788_ATP_Olambda__650,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr6984107991879650211_nat_o,Uub: nat,Uuc: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_fun2489310642977492936_nat_o(aTP_Lamm_wp(Uu),Uua),Uub),Uuc)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uuc),Uu)
        & aa_nat_o(aa_Pro4733806262894120092_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_650
tff(fact_8789_ATP_Olambda__651,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr4738963024209204295_int_o,Uub: int,Uuc: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_int6101145382336329604_nat_o(aa_fun1144309370076579528_nat_o(aTP_Lamm_wq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uuc),Uu)
        & aa_int_o(aa_Pro2912763917461432128_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_651
tff(fact_8790_ATP_Olambda__652,axiom,
    ! [Uu: set_set_nat,Uua: fun_se382663106265794036_nat_o,Uub: nat,Uuc: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aa_fun9105140013458882876_nat_o(aTP_Lamm_rg(Uu),Uua),Uub),Uuc)
    <=> ( member_set_nat(Uuc,Uu)
        & aa_nat_o(aa_set_nat_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_652
tff(fact_8791_ATP_Olambda__653,axiom,
    ! [Uu: set_set_nat,Uua: fun_se7360890175450123928_int_o,Uub: int,Uuc: set_nat] :
      ( aa_set_nat_o(aa_int_fun_set_nat_o(aa_fun9086678093774146108_nat_o(aTP_Lamm_rl(Uu),Uua),Uub),Uuc)
    <=> ( member_set_nat(Uuc,Uu)
        & aa_int_o(aa_set_nat_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_653
tff(fact_8792_ATP_Olambda__654,axiom,
    ! [Uu: set_nat,Uua: fun_na5971062356810558811unit_o,Uub: product_unit,Uuc: nat] :
      ( aa_nat_o(aa_Pro847370707247602192_nat_o(aa_fun6915127332797442224_nat_o(aTP_Lamm_yl(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_Product_unit_o(aa_nat5955485856412068252unit_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_654
tff(fact_8793_ATP_Olambda__655,axiom,
    ! [Uu: set_nat,Uua: fun_na3970208099681795815_nat_o,Uub: product_prod_nat_nat,Uuc: nat] :
      ( aa_nat_o(aa_Pro4733806262894120092_nat_o(aa_fun5389978208584994248_nat_o(aTP_Lamm_wn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_655
tff(fact_8794_ATP_Olambda__656,axiom,
    ! [Uu: set_nat,Uua: fun_na8563630686737515764_nat_o,Uub: set_nat,Uuc: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aa_fun2835368149466334012_nat_o(aTP_Lamm_rq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_set_nat_o(aa_nat_fun_set_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_656
tff(fact_8795_ATP_Olambda__657,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_qv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_657
tff(fact_8796_ATP_Olambda__658,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_qo(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_658
tff(fact_8797_ATP_Olambda__659,axiom,
    ! [Uu: set_int,Uua: fun_in8984369272479023031unit_o,Uub: product_unit,Uuc: int] :
      ( aa_int_o(aa_Pro8249700398669690036_int_o(aa_fun7970191300016449456_int_o(aTP_Lamm_yd(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_Product_unit_o(aa_int8968792772080532472unit_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_659
tff(fact_8798_ATP_Olambda__660,axiom,
    ! [Uu: set_int,Uua: fun_in2375804543815556931_nat_o,Uub: product_prod_nat_nat,Uuc: int] :
      ( aa_int_o(aa_Pro2912763917461432128_int_o(aa_fun8623471109766342856_int_o(aTP_Lamm_wo(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_Pro6150637604445964947_nat_o(aa_int6101145382336329604_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_660
tff(fact_8799_ATP_Olambda__661,axiom,
    ! [Uu: set_int,Uua: fun_in7136481858371106712_nat_o,Uub: set_nat,Uuc: int] :
      ( aa_int_o(aa_set_nat_fun_int_o(aa_fun8216954948888934460_int_o(aTP_Lamm_rv(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_set_nat_o(aa_int_fun_set_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_661
tff(fact_8800_ATP_Olambda__662,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_o,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_qk(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_662
tff(fact_8801_ATP_Olambda__663,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_rb(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_663
tff(fact_8802_ATP_Olambda__664,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_oi(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Uu),Uuc)),aa_nat_Code_integer(power_8256067586552552935nteger(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_664
tff(fact_8803_ATP_Olambda__665,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_oj(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_665
tff(fact_8804_ATP_Olambda__666,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_oh(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uuc)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_666
tff(fact_8805_ATP_Olambda__667,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_iu(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_667
tff(fact_8806_ATP_Olambda__668,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_abo(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_668
tff(fact_8807_ATP_Olambda__669,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abx(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_669
tff(fact_8808_ATP_Olambda__670,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_abq(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_670
tff(fact_8809_ATP_Olambda__671,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abv(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_671
tff(fact_8810_ATP_Olambda__672,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abz(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_672
tff(fact_8811_ATP_Olambda__673,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_acf(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_673
tff(fact_8812_ATP_Olambda__674,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_adj(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_674
tff(fact_8813_ATP_Olambda__675,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_natural,Uub: fun_nat_Code_natural,Uuc: nat] :
      ( aa_nat_o(aa_fun440078198722963374_nat_o(aa_fun7611442044815192786_nat_o(aTP_Lamm_tn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(Uua,Uuc)),aa_nat_Code_natural(Uub,Uuc)) != one_one_Code_natural ) ) ) ).

% ATP.lambda_675
tff(fact_8814_ATP_Olambda__676,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aa_fun2745573882733218024_nat_o(aTP_Lamm_tz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uuc)),aa_nat_Code_integer(Uub,Uuc)) != one_one_Code_integer ) ) ) ).

% ATP.lambda_676
tff(fact_8815_ATP_Olambda__677,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_ud(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_677
tff(fact_8816_ATP_Olambda__678,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_tr(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(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_678
tff(fact_8817_ATP_Olambda__679,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      ( aa_nat_o(aa_fun2180201864171524266_nat_o(aa_fun8566290670789036234_nat_o(aTP_Lamm_tv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(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_679
tff(fact_8818_ATP_Olambda__680,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_natural,Uub: fun_int_Code_natural,Uuc: int] :
      ( aa_int_o(aa_fun296549085072024822_int_o(aa_fun4460768838848905406_int_o(aTP_Lamm_tp(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_int_Code_natural(Uua,Uuc)),aa_int_Code_natural(Uub,Uuc)) != one_one_Code_natural ) ) ) ).

% ATP.lambda_680
tff(fact_8819_ATP_Olambda__681,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: fun_int_Code_integer,Uuc: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aa_fun8818272713621706452_int_o(aTP_Lamm_ub(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uuc)),aa_int_Code_integer(Uub,Uuc)) != one_one_Code_integer ) ) ) ).

% ATP.lambda_681
tff(fact_8820_ATP_Olambda__682,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_uf(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_682
tff(fact_8821_ATP_Olambda__683,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aa_fun8703205213358249326_int_o(aTP_Lamm_tt(Uu),Uua),Uub),Uuc)
    <=> ( 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_683
tff(fact_8822_ATP_Olambda__684,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: fun_int_int,Uuc: int] :
      ( aa_int_o(aa_fun8155382727227203058_int_o(aa_fun1697523107123627702_int_o(aTP_Lamm_tx(Uu),Uua),Uub),Uuc)
    <=> ( 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_684
tff(fact_8823_ATP_Olambda__685,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_natural,Uub: fun_nat_Code_natural,Uuc: nat] :
      ( aa_nat_o(aa_fun440078198722963374_nat_o(aa_fun7611442044815192786_nat_o(aTP_Lamm_tj(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(Uua,Uuc)),aa_nat_Code_natural(Uub,Uuc)) != zero_z2226904508553997617atural ) ) ) ).

% ATP.lambda_685
tff(fact_8824_ATP_Olambda__686,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aa_fun2745573882733218024_nat_o(aTP_Lamm_tb(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(Uua,Uuc)),aa_nat_Code_integer(Uub,Uuc)) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_686
tff(fact_8825_ATP_Olambda__687,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_tf(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_687
tff(fact_8826_ATP_Olambda__688,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_st(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(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_688
tff(fact_8827_ATP_Olambda__689,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      ( aa_nat_o(aa_fun2180201864171524266_nat_o(aa_fun8566290670789036234_nat_o(aTP_Lamm_sx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(aa_nat_fun_set_nat_o(member_nat,Uuc),Uu)
        & ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uua,Uuc)),aa_nat_int(Uub,Uuc)) != zero_zero_int ) ) ) ).

% ATP.lambda_689
tff(fact_8828_ATP_Olambda__690,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_natural,Uub: fun_int_Code_natural,Uuc: int] :
      ( aa_int_o(aa_fun296549085072024822_int_o(aa_fun4460768838848905406_int_o(aTP_Lamm_tl(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_int_Code_natural(Uua,Uuc)),aa_int_Code_natural(Uub,Uuc)) != zero_z2226904508553997617atural ) ) ) ).

% ATP.lambda_690
tff(fact_8829_ATP_Olambda__691,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: fun_int_Code_integer,Uuc: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aa_fun8818272713621706452_int_o(aTP_Lamm_td(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(Uua,Uuc)),aa_int_Code_integer(Uub,Uuc)) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_691
tff(fact_8830_ATP_Olambda__692,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_th(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_692
tff(fact_8831_ATP_Olambda__693,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aa_fun8703205213358249326_int_o(aTP_Lamm_sv(Uu),Uua),Uub),Uuc)
    <=> ( 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_693
tff(fact_8832_ATP_Olambda__694,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: fun_int_int,Uuc: int] :
      ( aa_int_o(aa_fun8155382727227203058_int_o(aa_fun1697523107123627702_int_o(aTP_Lamm_sz(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uua,Uuc)),aa_int_int(Uub,Uuc)) != zero_zero_int ) ) ) ).

% ATP.lambda_694
tff(fact_8833_ATP_Olambda__695,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_gs(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uuc)) ).

% ATP.lambda_695
tff(fact_8834_ATP_Olambda__696,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_gu(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_696
tff(fact_8835_ATP_Olambda__697,axiom,
    ! [Uu: set_int,Uua: fun_se876191773397849432nt_int,Uub: fun_se7360890175450123928_int_o,Uuc: set_nat] : aa_set_nat_int(aa_fun5327250345464651019at_int(aa_fun3157859007690329412at_int(aTP_Lamm_sq(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(Uua,Uuc),collect_int(aa_set_nat_fun_int_o(aa_fun87714373998387516_int_o(aTP_Lamm_ri(Uu),Uub),Uuc))) ).

% ATP.lambda_697
tff(fact_8836_ATP_Olambda__698,axiom,
    ! [Uu: set_nat,Uua: fun_se1179353343636554400at_nat,Uub: fun_se382663106265794036_nat_o,Uuc: set_nat] : aa_set_nat_nat(aa_fun6238879321244136459at_nat(aa_fun1036415571424436988at_nat(aTP_Lamm_sl(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_set6537093552120046177at_nat(Uua,Uuc)),collect_nat(aa_set_nat_fun_nat_o(aa_fun924172432987161148_nat_o(aTP_Lamm_rd(Uu),Uub),Uuc))) ).

% ATP.lambda_698
tff(fact_8837_ATP_Olambda__699,axiom,
    ! [Uu: set_nat,Uua: fun_se7912642235559603452at_int,Uub: fun_se382663106265794036_nat_o,Uuc: set_nat] : aa_set_nat_int(aa_fun2061028301734939751at_int(aa_fun8609273137425597436at_int(aTP_Lamm_sn(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_set2359242532610849469at_int(Uua,Uuc)),collect_nat(aa_set_nat_fun_nat_o(aa_fun924172432987161148_nat_o(aTP_Lamm_rd(Uu),Uub),Uuc))) ).

% ATP.lambda_699
tff(fact_8838_ATP_Olambda__700,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_int,Uub: fun_int_fun_int_o,Uuc: int] : aa_int_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_sj(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_int_fun_int_int(Uua,Uuc),collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_qz(Uu),Uub),Uuc))) ).

% ATP.lambda_700
tff(fact_8839_ATP_Olambda__701,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: fun_nat_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_sb(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uua,Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_qt(Uu),Uub),Uuc))) ).

% ATP.lambda_701
tff(fact_8840_ATP_Olambda__702,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_int,Uub: fun_nat_fun_nat_o,Uuc: nat] : aa_nat_int(aa_fun4679828425598226919at_int(aa_fun5734951013428723122at_int(aTP_Lamm_sd(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uua,Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_qt(Uu),Uub),Uuc))) ).

% ATP.lambda_702
tff(fact_8841_ATP_Olambda__703,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_nat,Uub: fun_nat_fun_int_o,Uuc: nat] : aa_nat_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_rz(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(Uua,Uuc),collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_qm(Uu),Uub),Uuc))) ).

% ATP.lambda_703
tff(fact_8842_ATP_Olambda__704,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_o,Uuc: nat] : aa_nat_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_sf(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_nat_fun_int_int(Uua,Uuc),collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_qm(Uu),Uub),Uuc))) ).

% ATP.lambda_704
tff(fact_8843_ATP_Olambda__705,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: int] : aa_int_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_rx(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_qh(Uu),Uub),Uuc))) ).

% ATP.lambda_705
tff(fact_8844_ATP_Olambda__706,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_int,Uub: fun_int_fun_nat_o,Uuc: int] : aa_int_int(aa_fun472946340410393319nt_int(aa_fun6062826860762256214nt_int(aTP_Lamm_sh(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_qh(Uu),Uub),Uuc))) ).

% ATP.lambda_706
tff(fact_8845_ATP_Olambda__707,axiom,
    ! [Uu: set_set_nat,Uua: fun_in6555721293707029592at_int,Uub: fun_in7136481858371106712_nat_o,Uuc: int] : aa_int_int(aa_fun1461796484577889457nt_int(aa_fun1283830802835827434nt_int(aTP_Lamm_rt(Uu),Uua),Uub),Uuc) = groups8292507037921071086at_int(aa_int2448287375063835929at_int(Uua,Uuc),collect_set_nat(aa_int_fun_set_nat_o(aa_fun7992546631809917244_nat_o(aTP_Lamm_rs(Uu),Uub),Uuc))) ).

% ATP.lambda_707
tff(fact_8846_ATP_Olambda__708,axiom,
    ! [Uu: set_set_nat,Uua: fun_na106483476154090400at_nat,Uub: fun_na8563630686737515764_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7187098123162309461at_nat(aa_fun585184524491904070at_nat(aTP_Lamm_ro(Uu),Uua),Uub),Uuc) = groups8294997508430121362at_nat(aa_nat6910189468943142497at_nat(Uua,Uuc),collect_set_nat(aa_nat_fun_set_nat_o(aa_fun1792963693083279932_nat_o(aTP_Lamm_rn(Uu),Uub),Uuc))) ).

% ATP.lambda_708
tff(fact_8847_ATP_Olambda__709,axiom,
    ! [Uu: set_int,Uua: fun_se876191773397849432nt_int,Uub: fun_se7360890175450123928_int_o,Uuc: set_nat] : aa_set_nat_int(aa_fun5327250345464651019at_int(aa_fun3157859007690329412at_int(aTP_Lamm_rj(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_set3358596244511904025nt_int(Uua,Uuc),collect_int(aa_set_nat_fun_int_o(aa_fun87714373998387516_int_o(aTP_Lamm_ri(Uu),Uub),Uuc))) ).

% ATP.lambda_709
tff(fact_8848_ATP_Olambda__710,axiom,
    ! [Uu: set_nat,Uua: fun_se1179353343636554400at_nat,Uub: fun_se382663106265794036_nat_o,Uuc: set_nat] : aa_set_nat_nat(aa_fun6238879321244136459at_nat(aa_fun1036415571424436988at_nat(aTP_Lamm_re(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(Uua,Uuc)),collect_nat(aa_set_nat_fun_nat_o(aa_fun924172432987161148_nat_o(aTP_Lamm_rd(Uu),Uub),Uuc))) ).

% ATP.lambda_710
tff(fact_8849_ATP_Olambda__711,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_int,Uub: fun_int_fun_int_o,Uuc: int] : aa_int_int(aa_fun282240425178129291nt_int(aa_fun1486234411616799902nt_int(aTP_Lamm_ra(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(Uua,Uuc),collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_qz(Uu),Uub),Uuc))) ).

% ATP.lambda_711
tff(fact_8850_ATP_Olambda__712,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: fun_nat_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_qu(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uua,Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_qt(Uu),Uub),Uuc))) ).

% ATP.lambda_712
tff(fact_8851_ATP_Olambda__713,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_nat,Uub: fun_nat_fun_int_o,Uuc: nat] : aa_nat_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_qq(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(Uua,Uuc),collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_qm(Uu),Uub),Uuc))) ).

% ATP.lambda_713
tff(fact_8852_ATP_Olambda__714,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_o,Uuc: nat] : aa_nat_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_qn(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_nat_fun_int_int(Uua,Uuc),collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_qm(Uu),Uub),Uuc))) ).

% ATP.lambda_714
tff(fact_8853_ATP_Olambda__715,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: int] : aa_int_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_qi(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_qh(Uu),Uub),Uuc))) ).

% ATP.lambda_715
tff(fact_8854_ATP_Olambda__716,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_int,Uub: fun_int_fun_nat_o,Uuc: int] : aa_int_int(aa_fun472946340410393319nt_int(aa_fun6062826860762256214nt_int(aTP_Lamm_qx(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_qh(Uu),Uub),Uuc))) ).

% ATP.lambda_716
tff(fact_8855_ATP_Olambda__717,axiom,
    ! [Uu: $o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_em((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_717
tff(fact_8856_ATP_Olambda__718,axiom,
    ! [Uu: fun_a_2990225006182978344t_unit,Uua: a,Uub: heap_e7401611519738050253t_unit,Uuc: nat] : aa_nat3804989708481012326it_nat(aa_hea5083147195637387702it_nat(aa_a_f112692477493148739it_nat(aTP_Lamm_eg(Uu),Uua),Uub),Uuc) = heap_T3616969660504097270t_unit(Uuc,aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_a_H4862568901524997081t_unit(Uu,Uua)),Uub)) ).

% ATP.lambda_718
tff(fact_8857_ATP_Olambda__719,axiom,
    ! [Uu: fun_a_6093291656547882009Heap_a,Uua: a,Uub: heap_e7401611519738050253t_unit,Uuc: nat] : aa_nat3190776082673941207it_nat(aa_hea414212936809421479it_nat(aa_a_f1761598956578211508it_nat(aTP_Lamm_ed(Uu),Uua),Uub),Uuc) = heap_T7616092557645711335rame_a(Uuc,aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(Uu,Uua)),Uub)) ).

% ATP.lambda_719
tff(fact_8858_ATP_Olambda__720,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_dq(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_720
tff(fact_8859_ATP_Olambda__721,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_dm(Uu),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_721
tff(fact_8860_ATP_Olambda__722,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7461304428735181397nt_int(aTP_Lamm_dn(Uu),Uua),Uub),Uuc) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_722
tff(fact_8861_ATP_Olambda__723,axiom,
    ! [Uu: fun_o_8988054265482332260nt_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_do(Uu),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_723
tff(fact_8862_ATP_Olambda__724,axiom,
    ! [Uu: fun_o_o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_di(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_724
tff(fact_8863_ATP_Olambda__725,axiom,
    ! [Uu: fun_o_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7323606162024495668nt_int(aTP_Lamm_dj(Uu),Uua),Uub),Uuc) = aa_o_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_725
tff(fact_8864_ATP_Olambda__726,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun1034830265760468567nt_int(aTP_Lamm_dp(Uu),Uua),Uub),Uuc) = aa_int7462565939943104719nt_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_726
tff(fact_8865_ATP_Olambda__727,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun82482788810740534_int_o(aTP_Lamm_dk(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_727
tff(fact_8866_ATP_Olambda__728,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun6119123525526402106nt_int(aTP_Lamm_dl(Uu),Uua),Uub),Uuc) = aa_int_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_728
tff(fact_8867_ATP_Olambda__729,axiom,
    ! [Uu: fun_fu3275390434952296369_nat_o,Uua: fun_na5914473681949881652_nat_o,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_dr(Uu),Uua),Uub),Uuc) = aa_fun359341089595923442_nat_o(Uu,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_729
tff(fact_8868_ATP_Olambda__730,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_nz(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),aa_nat_nat(suc,Uuc))) ).

% ATP.lambda_730
tff(fact_8869_ATP_Olambda__731,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ob(Uu),Uua),Uub),Uuc) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),aa_nat_nat(suc,Uuc))) ).

% ATP.lambda_731
tff(fact_8870_ATP_Olambda__732,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_hx(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_732
tff(fact_8871_ATP_Olambda__733,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_lv(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_733
tff(fact_8872_ATP_Olambda__734,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_abk(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_734
tff(fact_8873_ATP_Olambda__735,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ach(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_735
tff(fact_8874_ATP_Olambda__736,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_acb(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_736
tff(fact_8875_ATP_Olambda__737,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_acd(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).

% ATP.lambda_737
tff(fact_8876_ATP_Olambda__738,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ka(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Uud)),Uua),one_one_rat)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_738
tff(fact_8877_ATP_Olambda__739,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_jx(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Uu)),Uua)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_739
tff(fact_8878_ATP_Olambda__740,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_jy(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,Uua)),Uud)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Uub)),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_740
tff(fact_8879_ATP_Olambda__741,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o] : aa_o_f2266643866902735003t_char(aa_o_f5688925798504779150t_char(aa_o_f2285770652090220913t_char(aa_o_f2419597222644285284t_char(aa_o_f1772367729240163399t_char(aa_o_f5510610115356171066t_char(aa_o_f4853929186051764509t_char(aTP_Lamm_aea,(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)) = cons_char(char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf),$false)) ).

% ATP.lambda_741
tff(fact_8880_ATP_Olambda__742,axiom,
    ! [Uu: list_Code_integer,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: $o,Uuh: $o] : aa_o_literal(aa_o_fun_o_literal(aa_o_f2313209444760553806iteral(aa_o_f1022258164736565553iteral(aa_o_f139249553318509348iteral(aa_o_f1662653615792776199iteral(aa_o_f557580081657435898iteral(aa_o_f7455844346370101469iteral(aTP_Lamm_aeg(Uu),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),(Uug)),(Uuh)) = aa_literal_literal(aa_o_f1412681823949618463iteral(aa_o_f837559538564922386iteral(aa_o_f8819669424537506293iteral(aa_o_f3445383346798733800iteral(aa_o_f2755856924670595275iteral(aa_o_f293512076901925310iteral(aa_o_f6593176798326225313iteral(literal2,(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),(Uug)),aa_lis5633869780665501801iteral(literal_of_asciis,Uu)) ).

% ATP.lambda_742
tff(fact_8881_ATP_Olambda__743,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_o_fun_o_fun_o_o(aa_o_f2738860948074597109un_o_o(aa_o_f1377459968498492136un_o_o(aa_o_f3273824232735246795un_o_o(aa_o_f213386970520222398un_o_o(aa_o_f3238628652476363425un_o_o(aTP_Lamm_aen,(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),(Uug))
    <=> (Uu) ) ).

% ATP.lambda_743
tff(fact_8882_ATP_Olambda__744,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_o_fun_o_fun_o_o(aa_o_f2738860948074597109un_o_o(aa_o_f1377459968498492136un_o_o(aa_o_f3273824232735246795un_o_o(aa_o_f213386970520222398un_o_o(aa_o_f3238628652476363425un_o_o(aTP_Lamm_aeo,(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),(Uug))
    <=> (Uua) ) ).

% ATP.lambda_744
tff(fact_8883_ATP_Olambda__745,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_o_fun_o_fun_o_o(aa_o_f2738860948074597109un_o_o(aa_o_f1377459968498492136un_o_o(aa_o_f3273824232735246795un_o_o(aa_o_f213386970520222398un_o_o(aa_o_f3238628652476363425un_o_o(aTP_Lamm_aer,(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),(Uug))
    <=> (Uub) ) ).

% ATP.lambda_745
tff(fact_8884_ATP_Olambda__746,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_o_fun_o_fun_o_o(aa_o_f2738860948074597109un_o_o(aa_o_f1377459968498492136un_o_o(aa_o_f3273824232735246795un_o_o(aa_o_f213386970520222398un_o_o(aa_o_f3238628652476363425un_o_o(aTP_Lamm_aeq,(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),(Uug))
    <=> (Uuc) ) ).

% ATP.lambda_746
tff(fact_8885_ATP_Olambda__747,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_o_fun_o_fun_o_o(aa_o_f2738860948074597109un_o_o(aa_o_f1377459968498492136un_o_o(aa_o_f3273824232735246795un_o_o(aa_o_f213386970520222398un_o_o(aa_o_f3238628652476363425un_o_o(aTP_Lamm_aep,(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),(Uug))
    <=> (Uud) ) ).

% ATP.lambda_747
tff(fact_8886_ATP_Olambda__748,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_o_fun_o_fun_o_o(aa_o_f2738860948074597109un_o_o(aa_o_f1377459968498492136un_o_o(aa_o_f3273824232735246795un_o_o(aa_o_f213386970520222398un_o_o(aa_o_f3238628652476363425un_o_o(aTP_Lamm_aem,(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),(Uug))
    <=> (Uue) ) ).

% ATP.lambda_748
tff(fact_8887_ATP_Olambda__749,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_o_fun_o_fun_o_o(aa_o_f2738860948074597109un_o_o(aa_o_f1377459968498492136un_o_o(aa_o_f3273824232735246795un_o_o(aa_o_f213386970520222398un_o_o(aa_o_f3238628652476363425un_o_o(aTP_Lamm_ael,(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),(Uug))
    <=> (Uuf) ) ).

% ATP.lambda_749
tff(fact_8888_ATP_Olambda__750,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_o_fun_o_fun_o_o(aa_o_f2738860948074597109un_o_o(aa_o_f1377459968498492136un_o_o(aa_o_f3273824232735246795un_o_o(aa_o_f213386970520222398un_o_o(aa_o_f3238628652476363425un_o_o(aTP_Lamm_aei,(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),(Uug))
    <=> (Uug) ) ).

% ATP.lambda_750
tff(fact_8889_ATP_Olambda__751,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_o_fun_o_fun_o_o(aa_o_f2738860948074597109un_o_o(aa_o_f1377459968498492136un_o_o(aa_o_f3273824232735246795un_o_o(aa_o_f213386970520222398un_o_o(aa_o_f3238628652476363425un_o_o(aTP_Lamm_aej,(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),(Uug))
    <=> $true ) ).

% ATP.lambda_751
tff(fact_8890_ATP_Olambda__752,axiom,
    ! [Uu: heap_Time_Heap_a,Uua: product_unit] : aa_Pro7886978635576167513Heap_a(aTP_Lamm_ae(Uu),Uua) = Uu ).

% ATP.lambda_752
tff(fact_8891_ATP_Olambda__753,axiom,
    ! [Uu: code_integer,Uua: product_unit] : aa_Pro1705297278780387758nteger(aTP_Lamm_xm(Uu),Uua) = Uu ).

% ATP.lambda_753
tff(fact_8892_ATP_Olambda__754,axiom,
    ! [Uu: code_integer,Uua: set_nat] : aa_set871585252387937313nteger(aTP_Lamm_xp(Uu),Uua) = Uu ).

% ATP.lambda_754
tff(fact_8893_ATP_Olambda__755,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_xl(Uu),Uua) = Uu ).

% ATP.lambda_755
tff(fact_8894_ATP_Olambda__756,axiom,
    ! [Uu: char,Uua: char] : aa_char_char(aTP_Lamm_ap(Uu),Uua) = Uu ).

% ATP.lambda_756
tff(fact_8895_ATP_Olambda__757,axiom,
    ! [Uu: rat,Uua: product_unit] : aa_Product_unit_rat(aTP_Lamm_xo(Uu),Uua) = Uu ).

% ATP.lambda_757
tff(fact_8896_ATP_Olambda__758,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_xn(Uu),Uua) = Uu ).

% ATP.lambda_758
tff(fact_8897_ATP_Olambda__759,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_num(aTP_Lamm_aq(Uu),Uua) = Uu ).

% ATP.lambda_759
tff(fact_8898_ATP_Olambda__760,axiom,
    ! [Uu: nat,Uua: product_unit] : aa_Product_unit_nat(aTP_Lamm_xe(Uu),Uua) = Uu ).

% ATP.lambda_760
tff(fact_8899_ATP_Olambda__761,axiom,
    ! [Uu: nat,Uua: list_nat] : aa_list_nat_nat(aTP_Lamm_xg(Uu),Uua) = Uu ).

% ATP.lambda_761
tff(fact_8900_ATP_Olambda__762,axiom,
    ! [Uu: nat,Uua: set_nat] : aa_set_nat_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_xf,Uu),Uua) = Uu ).

% ATP.lambda_762
tff(fact_8901_ATP_Olambda__763,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ar,Uu),Uua) = Uu ).

% ATP.lambda_763
tff(fact_8902_ATP_Olambda__764,axiom,
    ! [Uu: nat,Uua: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_as,Uu),Uua) = Uu ).

% ATP.lambda_764
tff(fact_8903_ATP_Olambda__765,axiom,
    ! [Uu: int,Uua: product_unit] : aa_Product_unit_int(aTP_Lamm_xh(Uu),Uua) = Uu ).

% ATP.lambda_765
tff(fact_8904_ATP_Olambda__766,axiom,
    ! [Uu: int,Uua: list_nat] : aa_list_nat_int(aTP_Lamm_xj(Uu),Uua) = Uu ).

% ATP.lambda_766
tff(fact_8905_ATP_Olambda__767,axiom,
    ! [Uu: int,Uua: set_nat] : aa_set_nat_int(aa_int2448287375063835929at_int(aTP_Lamm_xi,Uu),Uua) = Uu ).

% ATP.lambda_767
tff(fact_8906_ATP_Olambda__768,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_xk,Uu),Uua) = Uu ).

% ATP.lambda_768
tff(fact_8907_ATP_Olambda__769,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ft,Uu),Uua) = Uu ).

% ATP.lambda_769
tff(fact_8908_ATP_Olambda__770,axiom,
    ! [Uu: fun_Pr5876161306168176259nteger,Uua: fun_Co8499238673658658103nteger] : aa_fun2509385170507305080nteger(aTP_Lamm_at(Uu),Uua) = Uu ).

% ATP.lambda_770
tff(fact_8909_ATP_Olambda__771,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lp,Uu),Uua) = Uua ).

% ATP.lambda_771
tff(fact_8910_ATP_Olambda__772,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_lq,Uu),Uua) = Uua ).

% ATP.lambda_772
tff(fact_8911_ATP_Olambda__773,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_en,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_773
tff(fact_8912_ATP_Olambda__774,axiom,
    ! [Uu: code_natural] : aa_Cod4272041906116450446atural(aTP_Lamm_cm,Uu) = Uu ).

% ATP.lambda_774
tff(fact_8913_ATP_Olambda__775,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_co,Uu) = Uu ).

% ATP.lambda_775
tff(fact_8914_ATP_Olambda__776,axiom,
    ! [Uu: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aTP_Lamm_adv,Uu) = Uu ).

% ATP.lambda_776
tff(fact_8915_ATP_Olambda__777,axiom,
    ! [Uu: char] : aa_char_char(aTP_Lamm_au,Uu) = Uu ).

% ATP.lambda_777
tff(fact_8916_ATP_Olambda__778,axiom,
    ! [Uu: list_char] : aa_lis7614194356200183040t_char(aTP_Lamm_adz,Uu) = Uu ).

% ATP.lambda_778
tff(fact_8917_ATP_Olambda__779,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_cp,Uu) = Uu ).

% ATP.lambda_779
tff(fact_8918_ATP_Olambda__780,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_av,Uu) = Uu ).

% ATP.lambda_780
tff(fact_8919_ATP_Olambda__781,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_cn,Uu) = Uu ).

% ATP.lambda_781
tff(fact_8920_ATP_Olambda__782,axiom,
    ! [Uu: code_natural] : aa_Cod4272041906116450446atural(aTP_Lamm_ch,Uu) = zero_z2226904508553997617atural ).

% ATP.lambda_782
tff(fact_8921_ATP_Olambda__783,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_ck,Uu) = zero_z3403309356797280102nteger ).

% ATP.lambda_783
tff(fact_8922_ATP_Olambda__784,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_cl,Uu) = zero_zero_rat ).

% ATP.lambda_784
tff(fact_8923_ATP_Olambda__785,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ci,Uu) = zero_zero_nat ).

% ATP.lambda_785
tff(fact_8924_ATP_Olambda__786,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_cj,Uu) = zero_zero_int ).

% ATP.lambda_786
tff(fact_8925_ATP_Olambda__787,axiom,
    ! [Uu: product_unit] : aa_Product_unit_nat(aTP_Lamm_yt,Uu) = one_one_nat ).

% ATP.lambda_787
tff(fact_8926_ATP_Olambda__788,axiom,
    ! [Uu: list_nat] : aa_list_nat_nat(aTP_Lamm_yv,Uu) = one_one_nat ).

% ATP.lambda_788
tff(fact_8927_ATP_Olambda__789,axiom,
    ! [Uu: set_nat] : aa_set_nat_nat(aTP_Lamm_yu,Uu) = one_one_nat ).

% ATP.lambda_789
tff(fact_8928_ATP_Olambda__790,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_kq,Uu) = one_one_nat ).

% ATP.lambda_790
tff(fact_8929_ATP_Olambda__791,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_kr,Uu) = one_one_int ).

% ATP.lambda_791
tff(fact_8930_ATP_Olambda__792,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ks,Uu) = one_one_int ).

% ATP.lambda_792
tff(fact_8931_ATP_Olambda__793,axiom,
    ! [Uu: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aTP_Lamm_ai,Uu) = none_P9117596204409417319it_nat ).

% ATP.lambda_793
tff(fact_8932_ATP_Olambda__794,axiom,
    ! [Uu: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(aTP_Lamm_ah,Uu) = none_P2651198173097904984it_nat ).

% ATP.lambda_794
tff(fact_8933_ATP_Olambda__795,axiom,
    ! [Uu: produc3260487557148687353it_nat] :
      ( aa_Pro6426884464393220769_nat_o(aTP_Lamm_bh,Uu)
    <=> $false ) ).

% ATP.lambda_795
tff(fact_8934_ATP_Olambda__796,axiom,
    ! [Uu: num] :
      ( aa_num_o(aTP_Lamm_bg,Uu)
    <=> $false ) ).

% ATP.lambda_796
tff(fact_8935_ATP_Olambda__797,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_fk,Uu)
    <=> $false ) ).

% ATP.lambda_797
tff(fact_8936_ATP_Olambda__798,axiom,
    ! [Uu: produc3260487557148687353it_nat] :
      ( aa_Pro6426884464393220769_nat_o(aTP_Lamm_bj,Uu)
    <=> $true ) ).

% ATP.lambda_798
tff(fact_8937_ATP_Olambda__799,axiom,
    ! [Uu: num] :
      ( aa_num_o(aTP_Lamm_bi,Uu)
    <=> $true ) ).

% ATP.lambda_799
tff(fact_8938_ATP_Olambda__800,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_fj,Uu)
    <=> $true ) ).

% ATP.lambda_800

% Helper facts (14)
tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X9: $o,Y3: $o] :
      ( ( (X9) != (Y3) )
      | aa_o_o(aa_o_fun_o_o(fequal_o,(X9)),(Y3)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X9: $o,Y3: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X9)),(Y3))
      | ( (X9) = (Y3) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X9: int,Y3: int] :
      ( ( X9 != Y3 )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X9),Y3) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X9: int,Y3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X9),Y3)
      | ( X9 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X9: nat,Y3: nat] :
      ( ( X9 != Y3 )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X9),Y3) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X9: nat,Y3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X9),Y3)
      | ( X9 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__Num__Onum_T,axiom,
    ! [X9: num,Y3: num] :
      ( ( X9 != Y3 )
      | aa_num_o(aa_num_fun_num_o(fequal_num,X9),Y3) ) ).

tff(help_fequal_1_1_fequal_001t__Num__Onum_T,axiom,
    ! [X9: num,Y3: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(fequal_num,X9),Y3)
      | ( X9 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__String__Ochar_T,axiom,
    ! [X9: char,Y3: char] :
      ( ( X9 != Y3 )
      | aa_char_o(aa_char_fun_char_o(fequal_char,X9),Y3) ) ).

tff(help_fequal_1_1_fequal_001t__String__Ochar_T,axiom,
    ! [X9: char,Y3: char] :
      ( ~ aa_char_o(aa_char_fun_char_o(fequal_char,X9),Y3)
      | ( X9 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__String__Ochar_J_T,axiom,
    ! [X9: list_char,Y3: list_char] :
      ( ( X9 != Y3 )
      | aa_list_char_o(aa_lis7814493268959572977char_o(fequal_list_char,X9),Y3) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__String__Ochar_J_T,axiom,
    ! [X9: list_char,Y3: list_char] :
      ( ~ aa_list_char_o(aa_lis7814493268959572977char_o(fequal_list_char,X9),Y3)
      | ( X9 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Code____Numeral__Ointeger_J_T,axiom,
    ! [X9: list_Code_integer,Y3: list_Code_integer] :
      ( ( X9 != Y3 )
      | aa_lis2813719629158071341eger_o(aa_lis7224287298985949145eger_o(fequal5224421480335545304nteger,X9),Y3) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Code____Numeral__Ointeger_J_T,axiom,
    ! [X9: list_Code_integer,Y3: list_Code_integer] :
      ( ~ aa_lis2813719629158071341eger_o(aa_lis7224287298985949145eger_o(fequal5224421480335545304nteger,X9),Y3)
      | ( X9 = Y3 ) ) ).

% Conjectures (2)
tff(conj_0,hypothesis,
    aa_a_o(p,x) ).

tff(conj_1,conjecture,
    aa_hea7994211679605476167unit_o(heap_Time_success_a(aa_a_H8829438773920479562Heap_a(heap_Time_assert_a(p),x)),h) ).

%------------------------------------------------------------------------------